Gaussian elimination
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Solve Ax=b using Gaussian elimination then backwards substitution.
A being an n by n matrix.
Also, x and b are n by 1 vectors.
To improve accuracy, please use partial pivoting and scaling.
- See also
-
- the Wikipedia entry: Gaussian elimination
11l
F swap_row(&a, &b, r1, r2)
I r1 != r2
swap(&a[r1], &a[r2])
swap(&b[r1], &b[r2])
F gauss_eliminate(&a, &b)
L(dia) 0 .< a.len
V (max_row, max) = (dia, a[dia][dia])
L(row) dia+1 .< a.len
V tmp = abs(a[row][dia])
I tmp > max
(max_row, max) = (row, tmp)
swap_row(&a, &b, dia, max_row)
L(row) dia+1 .< a.len
V tmp = a[row][dia] / a[dia][dia]
L(col) dia+1 .< a.len
a[row][col] -= tmp * a[dia][col]
a[row][dia] = 0
b[row] -= tmp * b[dia]
V r = [0.0] * a.len
L(row) (a.len-1 .. 0).step(-1)
V tmp = b[row]
L(j) (a.len-1 .< row).step(-1)
tmp -= r[j] * a[row][j]
r[row] = tmp / a[row][row]
R r
V a = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
V b = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
print(gauss_eliminate(&a, &b))- Output:
[-0.01, 1.60279, -1.6132, 1.24549, -0.49099, 0.0657607]
360 Assembly
* Gaussian elimination 09/02/2019
GAUSSEL CSECT
USING GAUSSEL,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n
LA R9,1(R7) j+1
LR R6,R9 i=j+1
DO WHILE=(C,R6,LE,N) do i=j+1 to n
LR R1,R7 j
MH R1,=AL2(NN) *n
AR R1,R7 +j
BCTR R1,0 j*n+j-1
SLA R1,2 ~
LE F0,A-(NN*4)(R1) a(j,j)
LR R1,R6 i
MH R1,=AL2(NN) *n
AR R1,R7 j
BCTR R1,0 i*n+j-1
SLA R1,2 ~
LE F2,A-(NN*4)(R1) a(i,j)
DER F0,F2 a(j,j)/a(i,j)
STE F0,W w=a(j,j)/a(i,j)
LR R8,R9 k=j+1
DO WHILE=(C,R8,LE,N) do k=j+1 to n
LR R1,R7 j
MH R1,=AL2(NN) *n
AR R1,R8 +k
BCTR R1,0 j*n+k-1
SLA R1,2 ~
LE F0,A-(NN*4)(R1) a(j,k)
LR R1,R6 i
MH R1,=AL2(NN) *n
AR R1,R8 +k
BCTR R1,0 i*n+k-1
SLA R1,2 ~
LE F2,A-(NN*4)(R1) a(i,k)
LE F6,W w
MER F6,F2 *a(i,k)
SER F0,F6 a(j,k)-w*a(i,k)
STE F0,A-(NN*4)(R1) a(i,k)=a(j,k)-w*a(i,k)
LA R8,1(R8) k=k+1
ENDDO , end do k
LR R1,R7 j
SLA R1,2 ~
LE F0,B-4(R1) b(j)
LR R1,R6 i
SLA R1,2 ~
LE F2,B-4(R1) b(i)
LE F6,W w
MER F6,F2 *b(i)
SER F0,F6 b(j)-w*b(i)
STE F0,B-4(R1) b(i)=b(j)-w*b(i)
LA R6,1(R6) i=i+1
ENDDO , end do i
LA R7,1(R7) j=j+1
ENDDO , end do j
L R2,N n
SLA R2,2 ~
LE F0,B-4(R1) b(n)
L R1,N n
MH R1,=AL2(NN) *n
A R1,N n
BCTR R1,0 n*n+n-1
SLA R1,2 ~
LE F2,A-(NN*4)(R1) a(n,n)
DER F0,F2 b(n)/a(n,n)
STE F0,X-4(R2) x(n)=b(n)/a(n,n)
L R7,N n
BCTR R7,0 j=n-1
DO WHILE=(C,R7,GE,=F'1') do j=n-1 to 1 by -1
LE F0,=E'0' 0
STE F0,W w=0
LA R9,1(R7) j+1
LR R6,R9 i=j+1
DO WHILE=(C,R6,LE,N) do i=j+1 to n
LR R1,R7 j
MH R1,=AL2(NN) *n
AR R1,R6 i
BCTR R1,0 j*n+i-1
SLA R1,2 ~
LE F0,A-(NN*4)(R1) a(j,i)
LR R1,R6 i
SLA R1,2 ~
LE F2,X-4(R1) x(i)
MER F0,F2 a(j,i)*x(i)
LE F6,W w
AER F6,F0 +a(j,i)*x(i)
STE F6,W w=w+a(j,i)*x(i)
LA R6,1(R6) i=i+1
ENDDO , end do i
LR R2,R7 j
SLA R2,2 ~
LE F0,B-4(R2) b(j)
SE F0,W -w
LR R1,R7 j
MH R1,=AL2(NN) *n
AR R1,R7 j
BCTR R1,0 j*n+j-1
SLA R1,2 ~
LE F2,A-(NN*4)(R1) a(j,j)
DER F0,F2 (b(j)-w)/a(j,j)
STE F0,X-4(R2) x(j)=(b(j)-w)/a(j,j)
BCTR R7,0 j=j-1
ENDDO , end do j
XPRNT =CL8'SOLUTION',8 print
MVC PG,=CL91' ' clear buffer
LA R6,1 i=1
DO WHILE=(C,R6,LE,N) do i=1 to n
LR R1,R6 i
SLA R1,2 ~
LE F0,X-4(R1) x(i)
LA R0,5 number of decimals
BAL R14,FORMATF edit
MVC PG(13),0(R1) output
XPRNT PG,L'PG print
LA R6,1(R6) i=i+1
ENDDO , end do i
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling sav
COPY plig\$_FORMATF.MLC format F13.n
NN EQU (X-B)/4 n
N DC A(NN) n
A DC E'1',E'0',E'0',E'0',E'0',E'0'
DC E'1',E'0.63',E'0.39',E'0.25',E'0.16',E'0.10'
DC E'1',E'1.26',E'1.58',E'1.98',E'2.49',E'3.13'
DC E'1',E'1.88',E'3.55',E'6.70',E'12.62',E'23.80'
DC E'1',E'2.51',E'6.32',E'15.88',E'39.90',E'100.28'
DC E'1',E'3.14',E'9.87',E'31.01',E'97.41',E'306.02'
B DC E'-0.01',E'0.61',E'0.91',E'0.99',E'0.60',E'0.02'
X DS (NN)E x(n)
W DS E w
PG DC CL91' ' buffer
REGEQU
END GAUSSEL- Output:
SOLUTION
-0.00999
1.60279
-1.61322
1.24552
-0.49100
0.06576
Ada
with Ada.Text_IO;
with Ada.Numerics.Generic_Real_Arrays;
procedure Gaussian_Eliminations is
type Real is new Float;
package Real_Arrays is
new Ada.Numerics.Generic_Real_Arrays (Real);
use Real_Arrays;
function Gaussian_Elimination (A : in Real_Matrix;
B : in Real_Vector) return Real_Vector
is
procedure Swap_Row (A : in out Real_Matrix;
B : in out Real_Vector;
R_1, R_2 : in Integer)
is
Temp : Real;
begin
if R_1 = R_2 then return; end if;
-- Swal matrix row
for Col in A'Range (1) loop
Temp := A (R_1, Col);
A (R_1, Col) := A (R_2, Col);
A (R_2, Col) := Temp;
end loop;
-- Swap vector row
Temp := B (R_1);
B (R_1) := B (R_2);
B (R_2) := Temp;
end Swap_Row;
AC : Real_Matrix := A;
BC : Real_Vector := B;
X : Real_Vector (A'Range (1)) := BC;
Max, Tmp : Real;
Max_Row : Integer;
begin
if
A'Length (1) /= A'Length (2) or
A'Length (1) /= B'Length
then
raise Constraint_Error with "Dimensions do not match";
end if;
if
A'First (1) /= A'First (2) or
A'First (1) /= B'First
then
raise Constraint_Error with "First index must be same";
end if;
for Dia in Ac'Range (1) loop
Max_Row := Dia;
Max := Ac (Dia, Dia);
for Row in Dia + 1 .. Ac'Last (1) loop
Tmp := abs (Ac (Row, Dia));
if Tmp > Max then
Max_Row := Row;
Max := Tmp;
end if;
end loop;
Swap_Row (Ac, Bc, Dia, Max_Row);
for Row in Dia + 1 .. Ac'Last (1) loop
Tmp := Ac (Row, Dia) / Ac (Dia, Dia);
for Col in Dia + 1 .. Ac'Last (1) loop
Ac (Row, Col) := Ac (Row, Col) - Tmp * Ac (Dia, Col);
end loop;
Ac (Row, Dia) := 0.0;
Bc (Row) := Bc (Row) - Tmp * Bc (Dia);
end loop;
end loop;
for Row in reverse Ac'Range (1) loop
Tmp := Bc (Row);
for J in reverse Row + 1 .. Ac'Last (1) loop
Tmp := Tmp - X (J) * Ac (Row, J);
end loop;
X (Row) := Tmp / Ac (Row, Row);
end loop;
return X;
end Gaussian_Elimination;
procedure Put (V : in Real_Vector) is
use Ada.Text_IO;
package Real_IO is
new Ada.Text_IO.Float_IO (Real);
begin
Put ("[ ");
for E of V loop
Real_IO.Put (E, Exp => 0, Aft => 6);
Put (" ");
end loop;
Put (" ]");
New_Line;
end Put;
A : constant Real_Matrix :=
((1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
(1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
(1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
(1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
(1.00, 3.14, 9.87, 31.01, 97.41, 306.02));
B : constant Real_Vector :=
( -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 );
X : constant Real_Vector := Gaussian_Elimination (A, B);
begin
Put (X);
end Gaussian_Eliminations;
- Output:
[ -0.010000 1.602774 -1.613148 1.245437 -0.490967 0.065758 ]
ALGOL 68
File: prelude_exception.a68
# -*- coding: utf-8 -*- #
COMMENT PROVIDES
MODE FIXED; INT fixed exception, unfixed exception;
PROC (STRING message) FIXED raise, raise value error
END COMMENT
# Note: ℵ indicates attribute is "private", and
should not be used outside of this prelude #
MODE FIXED = BOOL; # if an exception is detected, can it be fixed "on-site"? #
FIXED fixed exception = TRUE, unfixed exception = FALSE;
MODE #ℵ#SIMPLEOUTV = [0]UNION(CHAR, STRING, INT, REAL, BOOL, BITS);
MODE #ℵ#SIMPLEOUTM = [0]#ℵ#SIMPLEOUTV;
MODE #ℵ#SIMPLEOUTT = [0]#ℵ#SIMPLEOUTM;
MODE SIMPLEOUT = [0]#ℵ#SIMPLEOUTT;
PROC raise = (#ℵ#SIMPLEOUT message)FIXED: (
putf(stand error, ($"Exception:"$, $xg$, message, $l$));
stop
);
PROC raise value error = (#ℵ#SIMPLEOUT message)FIXED:
IF raise(message) NE fixed exception THEN exception value error; FALSE FI;
SKIPFile: prelude_mat_lib.a68
# -*- coding: utf-8 -*- #
COMMENT PRELUDE REQUIRES
MODE SCAL = REAL;
FORMAT scal repr = real repr
# and various SCAL OPerators #
END COMMENT
COMMENT PRELUDE PROIVIDES
MODE VEC, MAT;
OP :=:, -:=, +:=, *:=, /:=;
FORMAT sub, sep, bus;
FORMAT vec repr, mat repr
END COMMENT
# Note: ℵ indicates attribute is "private", and
should not be used outside of this prelude #
INT #ℵ#lwb vec := 1, #ℵ#upb vec := 0;
INT #ℵ#lwb mat := 1, #ℵ#upb mat := 0;
MODE VEC = [lwb vec:upb vec]SCAL,
MAT = [lwb mat:upb mat,lwb vec:upb vec]SCAL;
FORMAT sub := $"( "$, sep := $", "$, bus := $")"$, nl:=$lxx$;
FORMAT vec repr := $f(sub)n(upb vec - lwb vec)(f(scal repr)f(sep))f(scal repr)f(bus)$;
FORMAT mat repr := $f(sub)n(upb mat - lwb mat)(f( vec repr)f(nl))f( vec repr)f(bus)$;
# OPerators to swap the contents of two VECtors #
PRIO =:= = 1;
OP =:= = (REF VEC u, v)VOID:
FOR i TO UPB u DO SCAL scal=u[i]; u[i]:=v[i]; v[i]:=scal OD;
OP +:= = (REF VEC lhs, VEC rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] +:= rhs[i] OD;
lhs
);
OP -:= = (REF VEC lhs, VEC rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] -:= rhs[i] OD;
lhs
);
OP *:= = (REF VEC lhs, SCAL rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] *:= rhs OD;
lhs
);
OP /:= = (REF VEC lhs, SCAL rhs)REF VEC: (
SCAL inv = 1 / rhs; # multiplication is faster #
FOR i TO UPB lhs DO lhs[i] *:= inv OD;
lhs
);
SKIPFile: prelude_gaussian_elimination.a68
# -*- coding: utf-8 -*- #
COMMENT PRELUDE REQUIRES
MODE SCAL = REAL,
REAL near min scal = min real ** 0.99,
MODE VEC = []REAL,
MODE MAT = [,]REAL,
FORMAT scal repr = real repr,
and various OPerators of MAT and VEC
END COMMENT
COMMENT PRELUDE PROVIDES
PROC(MAT a, b)MAT gaussian elimination;
PROC(REF MAT a, b)REF MAT in situ gaussian elimination
END COMMENT
####################################################
# using Gaussian elimination, find x where A*x = b #
####################################################
PROC in situ gaussian elimination = (REF MAT a, b)REF MAT: (
# Note: a and b are modified "in situ", and b is returned as x #
FOR diag TO UPB a-1 DO
INT pivot row := diag; SCAL pivot factor := ABS a[diag,diag];
FOR row FROM diag + 1 TO UPB a DO # Full pivoting #
SCAL abs a diag = ABS a[row,diag];
IF abs a diag>=pivot factor THEN
pivot row := row; pivot factor := abs a diag FI
OD;
# now we have the "best" diag to full pivot, do the actual pivot #
IF diag NE pivot row THEN
# a[pivot row,] =:= a[diag,]; XXX: unoptimised # #DB#
a[pivot row,diag:] =:= a[diag,diag:]; # XXX: optimised #
b[pivot row,] =:= b[diag,] # swap/pivot the diags of a & b #
FI;
IF ABS a[diag,diag] <= near min scal THEN
raise value error("singular matrix") FI;
SCAL a diag reciprocal := 1 / a[diag, diag];
FOR row FROM diag+1 TO UPB a DO
SCAL factor = a[row,diag] * a diag reciprocal;
# a[row,] -:= factor * a[diag,] XXX: "unoptimised" # #DB#
a[row,diag+1:] -:= factor * a[diag,diag+1:];# XXX: "optimised" #
b[row,] -:= factor * b[diag,]
OD
OD;
# We have a triangular matrix, at this point we can traverse backwards
up the diagonal calculating b\A Converting it initial to a diagonal
matrix, then to the identity. #
FOR diag FROM UPB a BY -1 TO 1+LWB a DO
IF ABS a[diag,diag] <= near min scal THEN
raise value error("Zero pivot encountered?") FI;
SCAL a diag reciprocal = 1 / a[diag,diag];
FOR row TO diag-1 DO
SCAL factor = a[row,diag] * a diag reciprocal;
# a[row,diag] -:= factor * a[diag,diag]; XXX: "unoptimised" so remove # #DB#
b[row,] -:= factor * b[diag,]
OD;
# Now we have only diagonal elements we can simply divide b
by the values along the diagonal of A. #
b[diag,] *:= a diag reciprocal
OD;
b # EXIT #
);
PROC gaussian elimination = (MAT in a, in b)MAT: (
# Note: a and b are cloned and not modified "in situ" #
[UPB in a, 2 UPB in a]SCAL a := in a;
[UPB in b, 2 UPB in b]SCAL b := in b;
in situ gaussian elimination(a,b)
);
SKIPFile: postlude_exception.a68
# -*- coding: utf-8 -*- #
COMMENT POSTLUDE PROIVIDES
PROC VOID exception too many iterations, exception value error;
END COMMENT
SKIP EXIT
exception too many iterations:
exception value error:
stopFile: test_Gaussian_elimination.a68
#!/usr/bin/algol68g-full --script #
# -*- coding: utf-8 -*- #
PR READ "prelude_exception.a68" PR;
# define the attributes of the scalar field being used #
MODE SCAL = REAL;
FORMAT scal repr = $g(-0,real width)$;
# create "near min scal" as is scales better then small real #
SCAL near min scal = min real ** 0.99;
PR READ "prelude_mat_lib.a68" PR;
PR READ "prelude_gaussian_elimination.a68" PR;
MAT a =(( 1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
( 1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
( 1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
( 1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
( 1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
( 1.00, 3.14, 9.87, 31.01, 97.41, 306.02));
VEC b = (-0.01, 0.61, 0.91, 0.99, 0.60, 0.02);
[UPB b,1]SCAL col b; col b[,1]:= b;
upb vec := 2 UPB a;
printf((vec repr, gaussian elimination(a,col b)));
PR READ "postlude_exception.a68" PROutput:
( -.010000000000002, 1.602790394502130, -1.613203059905640, 1.245494121371510, -.490989719584686, .065760696175236)
ATS
This program was written by modifying Gauss-Jordan_matrix_inversion#ATS. There is a commented out portion of the code, whose removal makes this "Gaussian" elimination (with back substitution) rather than "Gauss-Jordan" elimination (without the need for back substitution).
(* There is a "little matrix library" in the code below. Not all of it
will be used, but it travels from task to task. Furthermore, the
"unused" parts are useful during the debugging phase. Also, reading
them may make it easier to understand other parts of the code. (For
instance, seeing how "block" and "transpose" work may help one
understand how "apply_index_map" works.) *)
(* Set to 1 for debugging: to fill in ones and zeros and other values
that are not actually used but are part of the theory of the
Gaussian elimination algorithm. *)
#define DO_THINGS_THAT_DO_NOT_NEED_TO_BE_DONE 0
(* Setting this to 1 may cause rounding to change, and the change in
rounding is not unlikely to cause detection of singularity of a
matrix to change. (To invert a matrix that might be nearly
singular, the SVD seems a popular method.) The
-fexpensive-optimizations option to GCC also may cause the same
rounding changes (due to fused-multiply-and-add instructions being
generated). *)
#define USE_MULTIPLY_AND_ADD 0
%{^
#include <math.h>
#include <float.h>
%}
#include "share/atspre_staload.hats"
macdef NAN = g0f2f ($extval (float, "NAN"))
macdef Zero = g0i2f 0
macdef One = g0i2f 1
macdef Two = g0i2f 2
#if USE_MULTIPLY_AND_ADD #then
(* "fma" from the C math library, although your system may have it as
a built-in. *)
extern fn {tk : tkind} g0float_fma : (g0float tk, g0float tk, g0float tk) -<> g0float tk
implement g0float_fma<fltknd> (x, y, z) = $extfcall (float, "fmaf", x, y, z)
implement g0float_fma<dblknd> (x, y, z) = $extfcall (double, "fma", x, y, z)
implement g0float_fma<ldblknd> (x, y, z) = $extfcall (ldouble, "fmal", x, y, z)
overload fma with g0float_fma
macdef multiply_and_add = fma
#else
macdef multiply_and_add (x, y, z) = (,(x) * ,(y)) + ,(z)
#endif
(*------------------------------------------------------------------*)
(* A "little matrix library" *)
typedef Matrix_Index_Map (m1 : int, n1 : int, m0 : int, n0 : int) =
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(int i1, int j1) -<cloref0>
[i0, j0 : pos | i0 <= m0; j0 <= n0]
@(int i0, int j0)
datatype Real_Matrix (tk : tkind,
m1 : int, n1 : int,
m0 : int, n0 : int) =
| Real_Matrix of (matrixref (g0float tk, m0, n0),
int m1, int n1, int m0, int n0,
Matrix_Index_Map (m1, n1, m0, n0))
typedef Real_Matrix (tk : tkind, m1 : int, n1 : int) =
[m0, n0 : pos] Real_Matrix (tk, m1, n1, m0, n0)
typedef Real_Vector (tk : tkind, m1 : int, n1 : int) =
[m1 == 1 || n1 == 1] Real_Matrix (tk, m1, n1)
typedef Real_Row (tk : tkind, n1 : int) = Real_Vector (tk, 1, n1)
typedef Real_Column (tk : tkind, m1 : int) = Real_Vector (tk, m1, 1)
extern fn {tk : tkind}
Real_Matrix_make_elt :
{m0, n0 : pos}
(int m0, int n0, g0float tk) -< !wrt >
Real_Matrix (tk, m0, n0, m0, n0)
extern fn {tk : tkind}
Real_Matrix_copy :
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -< !refwrt > Real_Matrix (tk, m1, n1)
extern fn {tk : tkind}
Real_Matrix_copy_to :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1), (* destination *)
Real_Matrix (tk, m1, n1)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_fill_with_elt :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1), g0float tk) -< !refwrt > void
extern fn {}
Real_Matrix_dimension :
{tk : tkind}
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -<> @(int m1, int n1)
extern fn {tk : tkind}
Real_Matrix_get_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1) -< !ref > g0float tk
extern fn {tk : tkind}
Real_Matrix_set_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1, g0float tk) -< !refwrt >
void
extern fn {}
Real_Matrix_apply_index_map :
{tk : tkind}
{m1, n1 : pos}
{m0, n0 : pos}
(Real_Matrix (tk, m0, n0), int m1, int n1,
Matrix_Index_Map (m1, n1, m0, n0)) -<>
Real_Matrix (tk, m1, n1)
extern fn {}
Real_Matrix_transpose :
(* This is transposed INDEXING. It does NOT copy the data. *)
{tk : tkind}
{m1, n1 : pos}
{m0, n0 : pos}
Real_Matrix (tk, m1, n1, m0, n0) -<>
Real_Matrix (tk, n1, m1, m0, n0)
extern fn {}
Real_Matrix_block :
(* This is block (submatrix) INDEXING. It does NOT copy the data. *)
{tk : tkind}
{p0, p1 : pos | p0 <= p1}
{q0, q1 : pos | q0 <= q1}
{m1, n1 : pos | p1 <= m1; q1 <= n1}
{m0, n0 : pos}
(Real_Matrix (tk, m1, n1, m0, n0),
int p0, int p1, int q0, int q1) -<>
Real_Matrix (tk, p1 - p0 + 1, q1 - q0 + 1, m0, n0)
extern fn {tk : tkind}
Real_Matrix_unit_matrix :
{m : pos}
int m -< !refwrt > Real_Matrix (tk, m, m)
extern fn {tk : tkind}
Real_Matrix_unit_matrix_to :
{m : pos}
Real_Matrix (tk, m, m) -< !refwrt > void
extern fn {tk : tkind}
Real_Matrix_matrix_sum :
{m, n : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_matrix_sum_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, m, n)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_matrix_difference :
{m, n : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_matrix_difference_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, m, n)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_matrix_product :
{m, n, p : pos}
(Real_Matrix (tk, m, n), Real_Matrix (tk, n, p)) -< !refwrt >
Real_Matrix (tk, m, p)
extern fn {tk : tkind}
Real_Matrix_matrix_product_to :
(* For the matrix product, the destination should not be the same as
either of the other matrices. *)
{m, n, p : pos}
(Real_Matrix (tk, m, p), (* destination*)
Real_Matrix (tk, m, n),
Real_Matrix (tk, n, p)) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_scalar_product :
{m, n : pos}
(Real_Matrix (tk, m, n), g0float tk) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product_2 :
{m, n : pos}
(g0float tk, Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product :
{m, n : pos}
(Real_Matrix (tk, m, n), g0float tk) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product_2 :
{m, n : pos}
(g0float tk, Real_Matrix (tk, m, n)) -< !refwrt >
Real_Matrix (tk, m, n)
extern fn {tk : tkind}
Real_Matrix_scalar_product_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
g0float tk) -< !refwrt >
void
extern fn {tk : tkind}
Real_Matrix_scalar_multiply_and_add_to :
{m, n : pos}
(Real_Matrix (tk, m, n), (* destination*)
Real_Matrix (tk, m, n),
g0float tk,
Real_Matrix (tk, m, n)) -< !refwrt >
void
extern fn {tk : tkind} (* Useful for debugging. *)
Real_Matrix_fprint :
{m, n : pos}
(FILEref, Real_Matrix (tk, m, n)) -<1> void
overload copy with Real_Matrix_copy
overload copy_to with Real_Matrix_copy_to
overload fill_with_elt with Real_Matrix_fill_with_elt
overload dimension with Real_Matrix_dimension
overload [] with Real_Matrix_get_at
overload [] with Real_Matrix_set_at
overload apply_index_map with Real_Matrix_apply_index_map
overload transpose with Real_Matrix_transpose
overload block with Real_Matrix_block
overload unit_matrix with Real_Matrix_unit_matrix
overload unit_matrix_to with Real_Matrix_unit_matrix_to
overload matrix_sum with Real_Matrix_matrix_sum
overload matrix_sum_to with Real_Matrix_matrix_sum_to
overload matrix_difference with Real_Matrix_matrix_difference
overload matrix_difference_to with Real_Matrix_matrix_difference_to
overload matrix_product with Real_Matrix_matrix_product
overload matrix_product_to with Real_Matrix_matrix_product_to
overload scalar_product with Real_Matrix_scalar_product
overload scalar_product with Real_Matrix_scalar_product_2
overload scalar_product_to with Real_Matrix_scalar_product_to
overload scalar_multiply_and_add_to with
Real_Matrix_scalar_multiply_and_add_to
overload + with matrix_sum
overload - with matrix_difference
overload * with matrix_product
overload * with scalar_product
(*------------------------------------------------------------------*)
(* Implementation of the "little matrix library" *)
implement {tk}
Real_Matrix_make_elt (m0, n0, elt) =
Real_Matrix (matrixref_make_elt<g0float tk> (i2sz m0, i2sz n0, elt),
m0, n0, m0, n0, lam (i1, j1) => @(i1, j1))
implement {}
Real_Matrix_dimension A =
case+ A of Real_Matrix (_, m1, n1, _, _, _) => @(m1, n1)
implement {tk}
Real_Matrix_get_at (A, i1, j1) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_get_at<g0float tk> (storage, pred i0, n0, pred j0)
end
implement {tk}
Real_Matrix_set_at (A, i1, j1, x) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_set_at<g0float tk> (storage, pred i0, n0, pred j0, x)
end
implement {}
Real_Matrix_apply_index_map (A, m1, n1, index_map) =
(* This is not the most efficient way to acquire new indexing, but
it will work. It requires three closures, instead of the two
needed by our implementations of "transpose" and "block". *)
let
val+ Real_Matrix (storage, m1a, n1a, m0, n0, index_map_1a) = A
in
Real_Matrix (storage, m1, n1, m0, n0,
lam (i1, j1) =>
index_map_1a (i1a, j1a) where
{ val @(i1a, j1a) = index_map (i1, j1) })
end
implement {}
Real_Matrix_transpose A =
let
val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A
in
Real_Matrix (storage, n1, m1, m0, n0,
lam (i1, j1) => index_map (j1, i1))
end
implement {}
Real_Matrix_block (A, p0, p1, q0, q1) =
let
val+ Real_Matrix (storage, m1, n1, m0, n0, index_map) = A
in
Real_Matrix (storage, succ (p1 - p0), succ (q1 - q0), m0, n0,
lam (i1, j1) =>
index_map (p0 + pred i1, q0 + pred j1))
end
implement {tk}
Real_Matrix_copy A =
let
val @(m1, n1) = dimension A
val C = Real_Matrix_make_elt<tk> (m1, n1, A[1, 1])
val () = copy_to<tk> (C, A)
in
C
end
implement {tk}
Real_Matrix_copy_to (Dst, Src) =
let
val @(m1, n1) = dimension Src
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1
var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
Dst[i, j] := Src[i, j]
end
end
implement {tk}
Real_Matrix_fill_with_elt (A, elt) =
let
val @(m1, n1) = dimension A
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1
var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
A[i, j] := elt
end
end
implement {tk}
Real_Matrix_unit_matrix {m} m =
let
val A = Real_Matrix_make_elt<tk> (m, m, Zero)
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
A[i, i] := One;
A
end
implement {tk}
Real_Matrix_unit_matrix_to A =
let
val @(m, _) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= m + 1} .<(m + 1) - j>.
(j : int j) =>
(j := 1; j <> succ m; j := succ j)
A[i, j] := (if i = j then One else Zero)
end
end
implement {tk}
Real_Matrix_matrix_sum (A, B) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = matrix_sum_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_sum_to (C, A, B) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] + B[i, j]
end
end
implement {tk}
Real_Matrix_matrix_difference (A, B) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = matrix_difference_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_difference_to (C, A, B) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] - B[i, j]
end
end
implement {tk}
Real_Matrix_matrix_product (A, B) =
let
val @(m, n) = dimension A and @(_, p) = dimension B
val C = Real_Matrix_make_elt<tk> (m, p, NAN)
val () = matrix_product_to<tk> (C, A, B)
in
C
end
implement {tk}
Real_Matrix_matrix_product_to (C, A, B) =
let
val @(m, n) = dimension A and @(_, p) = dimension B
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
prval [p : int] EQINT () = eqint_make_gint p
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var k : intGte 1
in
for* {k : pos | k <= p + 1} .<(p + 1) - k>.
(k : int k) =>
(k := 1; k <> succ p; k := succ k)
let
var j : intGte 1
in
C[i, k] := A[i, 1] * B[1, k];
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 2; j <> succ n; j := succ j)
C[i, k] :=
multiply_and_add (A[i, j], B[j, k], C[i, k])
end
end
end
implement {tk}
Real_Matrix_scalar_product (A, r) =
let
val @(m, n) = dimension A
val C = Real_Matrix_make_elt<tk> (m, n, NAN)
val () = scalar_product_to<tk> (C, A, r)
in
C
end
implement {tk}
Real_Matrix_scalar_product_2 (r, A) =
Real_Matrix_scalar_product<tk> (A, r)
implement {tk}
Real_Matrix_scalar_product_to (C, A, r) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := A[i, j] * r
end
end
implement {tk}
Real_Matrix_scalar_multiply_and_add_to (C, A, r, B) =
let
val @(m, n) = dimension A
prval [m : int] EQINT () = eqint_make_gint m
prval [n : int] EQINT () = eqint_make_gint n
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
C[i, j] := multiply_and_add (A[i, j], r, B[i, j])
end
end
implement {tk}
Real_Matrix_fprint {m, n} (outf, A) =
let
val @(m, n) = dimension A
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
typedef FILEstar = $extype"FILE *"
extern castfn FILEref2star : FILEref -<> FILEstar
val _ = $extfcall (int, "fprintf", FILEref2star outf,
"%16.6g", A[i, j])
in
end;
fprintln! (outf)
end
end
(*------------------------------------------------------------------*)
(* Gaussian elimination *)
extern fn {tk : tkind}
Real_Matrix_gaussian_elimination :
(* Solve k systems of linear equations in n unknowns. (A special
case is if the second argument is a unit matrix. In that case,
the solution columns constitute the matrix inverse of the first
argument. See also the Gauss-Jordan matrix inversion task.) *)
{n, k : pos}
(Real_Matrix (tk, n, n), Real_Matrix (tk, n, k)) -< !refwrt >
Option (Real_Matrix (tk, n, k))
#if DO_THINGS_THAT_DO_NOT_NEED_TO_BE_DONE #then
macdef do_needless_things = true
#else
macdef do_needless_things = false
#endif
fn {tk : tkind}
needlessly_set_to_value
{n : pos}
{i, j : pos | i <= n; j <= n}
(A : Real_Matrix (tk, n, n),
i : int i,
j : int j,
x : g0float tk) :<!refwrt> void =
if do_needless_things then A[i, j] := x
implement {tk}
Real_Matrix_gaussian_elimination {n, k} (A, B) =
let
val @(n, k) = dimension B
typedef one_to_n = intBtwe (1, n)
(* Partial pivoting. *)
implement
array_tabulate$fopr<one_to_n> i =
let
val i = g1ofg0 (sz2i (succ i))
val () = assertloc ((1 <= i) * (i <= n))
in
i
end
val rows_permutation =
$effmask_all arrayref_tabulate<one_to_n> (i2sz n)
fn
index_map_A : Matrix_Index_Map (n, n, n, n) =
lam (i1, j1) => $effmask_ref
(@(i0, j1) where { val i0 = rows_permutation[i1 - 1] })
fn
index_map_B : Matrix_Index_Map (n, k, n, k) =
lam (i1, j1) => $effmask_ref
(@(i0, j1) where { val i0 = rows_permutation[i1 - 1] })
val A = apply_index_map (copy<tk> A, n, n, index_map_A)
and B = apply_index_map (copy<tk> B, n, k, index_map_B)
fn {}
exchange_rows (i1 : one_to_n,
i2 : one_to_n) :<!refwrt> void =
if i1 <> i2 then
let
val k1 = rows_permutation[pred i1]
and k2 = rows_permutation[pred i2]
in
rows_permutation[pred i1] := k2;
rows_permutation[pred i2] := k1
end
fn {}
normalize_pivot_row (j : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val pivot_val = A[j, j]
var p : intGte 1
in
needlessly_set_to_value (A, j, j, One);
for* {p : int | j + 1 <= p; p <= n + 1} .<(n + 1) - p>.
(p : int p) =>
(p := succ j; p <> succ n; p := succ p)
A[j, p] := A[j, p] / pivot_val;
for* {p : int | 1 <= p; p <= k + 1} .<(k + 1) - p>.
(p : int p) =>
(p := 1; p <> succ k; p := succ p)
B[j, p] := B[j, p] / pivot_val;
end
fn
subtract_normalized_pivot_row
(i : one_to_n, j : one_to_n) :<!refwrt> void =
let
prval [j : int] EQINT () = eqint_make_gint j
val lead_val = A[i, j]
in
if lead_val <> Zero then
let
val factor = ~lead_val
var p : intGte 1
in
needlessly_set_to_value (A, i, j, Zero);
for* {p : int | j + 1 <= p; p <= n + 1} .<(n + 1) - p>.
(p : int p) =>
(p := succ j; p <> succ n; p := succ p)
A[i, p] :=
multiply_and_add (A[j, p], factor, A[i, p]);
for* {p : int | 1 <= p; p <= k + 1} .<(k + 1) - p>.
(p : int p) =>
(p := 1; p <> succ k; p := succ p)
B[i, p] :=
multiply_and_add (B[j, p], factor, B[i, p])
end
end
fun
main_loop {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j,
success : &bool? >> bool) :<!refwrt> void =
if j = succ n then
success := true
else
let
fun
select_pivot {i : int | j <= i; i <= n + 1}
.<(n + 1) - i>.
(i : int i,
max_abs : g0float tk,
i_max_abs : intBtwe (j - 1, n))
:<!ref> intBtwe (j - 1, n) =
if i = succ n then
i_max_abs
else
let
val abs_aij = abs A[i, j]
in
if abs_aij > max_abs then
select_pivot (succ i, abs_aij, i)
else
select_pivot (succ i, max_abs, i_max_abs)
end
val i_pivot = select_pivot (j, Zero, pred j)
prval [i_pivot : int] EQINT () = eqint_make_gint i_pivot
in
if i_pivot = pred j then
success := false
else
let
var i : intGte 1
in
exchange_rows (i_pivot, j);
normalize_pivot_row (j);
(* For Gauss-Jordan elimination, we would do this,
instead of switching to back substitution:
for* {i : int | 1 <= i; i <= j}
.<j - i>.
(i : int i) =>
(i := 1; i <> j; i := succ i)
subtract_normalized_pivot_row (i, j); *)
for* {i : int | j + 1 <= i; i <= n + 1}
.<(n + 1) - i>.
(i : int i) =>
(i := succ j; i <> succ n; i := succ i)
subtract_normalized_pivot_row (i, j);
main_loop (succ j, success)
end
end
var success : bool
val () = main_loop (1, success)
in
if ~success then
None ()
else
(* Back substitution. (Doing this with block operations on rows,
as is done below, is not the most efficient way, but helps
convey what is going on.) *)
let
val bottom_row = block (B, n, n, 1, k)
(* The rows array will treat the rows of B as
submatrices. (The zeroth entry will not be used.) *)
val rows = arrayref_make_elt (i2sz (succ n), bottom_row)
var i : intGte 0
in
(* Fill in the rows array (ignoring its zeroth entry). *)
for* {i : nat | i <= n - 1} .<i>.
(i : int i) =>
(i := pred n; i <> 0; i := pred i)
rows[i] := block (B, i, i, 1, k);
(* Now do back substitution, one solution row at a time. *)
for* {i : nat | i <= n - 1} .<i>.
(i : int i) =>
(i := pred n; i <> 0; i := pred i)
let
var j : intGte 0
in
for* {j : int | i <= j; j <= n} .<j>.
(j : int j) =>
(j := n; j <> i; j := pred j)
scalar_multiply_and_add_to
(rows[i], rows[j], ~A[i, j], rows[i])
end;
(* The returned matrix will "contain" the rows_permutation
array and some extra closures. If you want a "clean"
matrix, you can use Real_Matrix_copy to get one. *)
Some B
end
end
overload gaussian_elimination with Real_Matrix_gaussian_elimination
(*------------------------------------------------------------------*)
fn {tk : tkind}
fprint_matrices_and_solutions
{n, k : pos}
(outf : FILEref,
A : Real_Matrix (tk, n, n),
B : Real_Matrix (tk, n, k)) : void =
let
typedef FILEstar = $extype"FILE *"
extern castfn FILEref2star : FILEref -<> FILEstar
macdef fmt = "%9.4f"
macdef left_bracket = "["
macdef right_bracket = " ]"
macdef product = " ✕ "
macdef equals = " = "
macdef spacing = " "
macdef msg_for_singular = " appears to be singular"
macdef print_num (x) =
ignoret ($extfcall (int, "fprintf", FILEref2star outf,
fmt, ,(x)))
val @(n, k) = dimension B
in
case+ gaussian_elimination<tk> (A, B) of
| None () =>
let
var i : intGte 1
in
for* {i : pos | i <= n + 1} .<(n + 1) - i>.
(i : int i) =>
(i := 1; i <> succ n; i := succ i)
let
var j : intGte 1
in
fprint! (outf, left_bracket);
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
print_num A[i, j];
fprint! (outf, right_bracket);
if pred i = half n then
fprint! (outf, msg_for_singular);
fprintln! (outf)
end
end
| Some X =>
let
val AX = A * X
var i : intGte 1
in
for* {i : pos | i <= n + 1} .<(n + 1) - i>.
(i : int i) =>
(i := 1; i <> succ n; i := succ i)
let
var j : intGte 1
in
fprint! (outf, left_bracket);
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
print_num A[i, j];
fprint! (outf, right_bracket);
if pred i = half n then
fprint! (outf, product)
else
fprint! (outf, spacing);
fprint! (outf, left_bracket);
for* {j : pos | j <= k + 1} .<(k + 1) - j>.
(j : int j) =>
(j := 1; j <> succ k; j := succ j)
print_num X[i, j];
fprint! (outf, right_bracket);
if pred i = half n then
fprint! (outf, equals)
else
fprint! (outf, spacing);
fprint! (outf, left_bracket);
for* {j : pos | j <= k + 1} .<(k + 1) - j>.
(j : int j) =>
(j := 1; j <> succ k; j := succ j)
print_num AX[i, j];
fprint! (outf, right_bracket);
fprintln! (outf)
end
end
end
fn {tk : tkind}
column_major_list_to_matrix
{m, n : pos}
(m : int m,
n : int n,
lst : list (g0float tk, m * n))
: Real_Matrix (tk, m, n) =
let
#define :: list_cons
prval () = mul_gte_gte_gte {m, n} ()
val A = Real_Matrix_make_elt (m, n, NAN)
val lstref : ref (List0 (g0float tk)) = ref lst
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
case- !lstref of
| hd :: tl =>
begin
A[i, j] := hd;
!lstref := tl
end
end;
A
end
fn {tk : tkind}
row_major_list_to_matrix
{m, n : pos}
(m : int m,
n : int n,
lst : list (g0float tk, m * n))
: Real_Matrix (tk, m, n) =
transpose (column_major_list_to_matrix<tk> (n, m, lst))
macdef separator = "\n"
fn
print_example
{n, k : pos}
(n : int n,
k : int k,
lst_A : list (double, n * n),
lst_B : list (double, n * k)) : void =
let
val A = row_major_list_to_matrix (n, n, lst_A)
and B = row_major_list_to_matrix (n, k, lst_B)
in
print! separator;
fprint_matrices_and_solutions<dblknd> (stdout_ref, A, B)
end
implement
main0 () =
begin
println! ("\n(The examples are printed here after ",
"rounding to 4 decimal places.)");
println! ("\nSolving Ax=b where ",
"transpose(b)=[-0.01 0.61 0.91 0.99 0.60 0.02]:");
print_example
(6, 1, $list (1.00, 0.00, 0.00, 0.00, 0.00, 0.00,
1.00, 0.63, 0.39, 0.25, 0.16, 0.10,
1.00, 1.26, 1.58, 1.98, 2.49, 3.13,
1.00, 1.88, 3.55, 6.70, 12.62, 23.80,
1.00, 2.51, 6.32, 15.88, 39.90, 100.28,
1.00, 3.14, 9.87, 31.01, 97.41, 306.02),
$list (~0.01, 0.61, 0.91, 0.99, 0.60, 0.02));
println! ("\nAn orthonormal matrix of rank 4,",
" solving for its inverse:");
print_example
(4, 4,
$list (~0.1543033499620918, ~0.1307837816808,
0.8649377296669811, 0.4593134032861146,
~0.6172133998483675, ~0.2062359634197235,
0.2410548538544755, ~0.7200047943403952,
~0.7715167498104593, 0.1911455270719389,
~0.3658314290169772, 0.4841411548150931,
0.0, ~0.950697489910433,
~0.2448318745080428, 0.190346095055507),
$list (1.0, 0.0, 0.0, 0.0,
0.0, 1.0, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0,
0.0, 0.0, 0.0, 1.0));
print! separator
end
(*------------------------------------------------------------------*)- Output:
On my computer, the following compiler options result in "fused multiply-and-add" instructions being generated. See the program text for a brief discussion of how that optimization may affect results, when the A matrix that is singular or nearly so.
$ patscc -std=gnu2x -g -O3 -march=native -DATS_MEMALLOC_GCBDW gaussian_elimination_task.dats -lgc && ./a.out (The examples are printed here after rounding to 4 decimal places.) Solving Ax=b where transpose(b)=[-0.01 0.61 0.91 0.99 0.60 0.02]: [ 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ] [ -0.0100 ] [ -0.0100 ] [ 1.0000 0.6300 0.3900 0.2500 0.1600 0.1000 ] [ 1.6028 ] [ 0.6100 ] [ 1.0000 1.2600 1.5800 1.9800 2.4900 3.1300 ] [ -1.6132 ] [ 0.9100 ] [ 1.0000 1.8800 3.5500 6.7000 12.6200 23.8000 ] ✕ [ 1.2455 ] = [ 0.9900 ] [ 1.0000 2.5100 6.3200 15.8800 39.9000 100.2800 ] [ -0.4910 ] [ 0.6000 ] [ 1.0000 3.1400 9.8700 31.0100 97.4100 306.0200 ] [ 0.0658 ] [ 0.0200 ] An orthonormal matrix of rank 4, solving for its inverse: [ -0.1543 -0.1308 0.8649 0.4593 ] [ -0.1543 -0.6172 -0.7715 -0.0000 ] [ 1.0000 0.0000 -0.0000 0.0000 ] [ -0.6172 -0.2062 0.2411 -0.7200 ] [ -0.1308 -0.2062 0.1911 -0.9507 ] [ -0.0000 1.0000 -0.0000 0.0000 ] [ -0.7715 0.1911 -0.3658 0.4841 ] ✕ [ 0.8649 0.2411 -0.3658 -0.2448 ] = [ -0.0000 -0.0000 1.0000 0.0000 ] [ 0.0000 -0.9507 -0.2448 0.1903 ] [ 0.4593 -0.7200 0.4841 0.1903 ] [ -0.0000 0.0000 0.0000 1.0000 ]
C
This modifies A and b in place, which might not be quite desirable.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define mat_elem(a, y, x, n) (a + ((y) * (n) + (x)))
void swap_row(double *a, double *b, int r1, int r2, int n)
{
double tmp, *p1, *p2;
int i;
if (r1 == r2) return;
for (i = 0; i < n; i++) {
p1 = mat_elem(a, r1, i, n);
p2 = mat_elem(a, r2, i, n);
tmp = *p1, *p1 = *p2, *p2 = tmp;
}
tmp = b[r1], b[r1] = b[r2], b[r2] = tmp;
}
void gauss_eliminate(double *a, double *b, double *x, int n)
{
#define A(y, x) (*mat_elem(a, y, x, n))
int i, j, col, row, max_row,dia;
double max, tmp;
for (dia = 0; dia < n; dia++) {
max_row = dia, max = A(dia, dia);
for (row = dia + 1; row < n; row++)
if ((tmp = fabs(A(row, dia))) > max)
max_row = row, max = tmp;
swap_row(a, b, dia, max_row, n);
for (row = dia + 1; row < n; row++) {
tmp = A(row, dia) / A(dia, dia);
for (col = dia+1; col < n; col++)
A(row, col) -= tmp * A(dia, col);
A(row, dia) = 0;
b[row] -= tmp * b[dia];
}
}
for (row = n - 1; row >= 0; row--) {
tmp = b[row];
for (j = n - 1; j > row; j--)
tmp -= x[j] * A(row, j);
x[row] = tmp / A(row, row);
}
#undef A
}
int main(void)
{
double a[] = {
1.00, 0.00, 0.00, 0.00, 0.00, 0.00,
1.00, 0.63, 0.39, 0.25, 0.16, 0.10,
1.00, 1.26, 1.58, 1.98, 2.49, 3.13,
1.00, 1.88, 3.55, 6.70, 12.62, 23.80,
1.00, 2.51, 6.32, 15.88, 39.90, 100.28,
1.00, 3.14, 9.87, 31.01, 97.41, 306.02
};
double b[] = { -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 };
double x[6];
int i;
gauss_eliminate(a, b, x, 6);
for (i = 0; i < 6; i++)
printf("%g\n", x[i]);
return 0;
}
- Output:
-0.01 1.60279 -1.6132 1.24549 -0.49099 0.0657607
C#
This modifies A and b in place, which might not be quite desirable.
using System;
namespace Rosetta
{
internal class Vector
{
private double[] b;
internal readonly int rows;
internal Vector(int rows)
{
this.rows = rows;
b = new double[rows];
}
internal Vector(double[] initArray)
{
b = (double[])initArray.Clone();
rows = b.Length;
}
internal Vector Clone()
{
Vector v = new Vector(b);
return v;
}
internal double this[int row]
{
get { return b[row]; }
set { b[row] = value; }
}
internal void SwapRows(int r1, int r2)
{
if (r1 == r2) return;
double tmp = b[r1];
b[r1] = b[r2];
b[r2] = tmp;
}
internal double norm(double[] weights)
{
double sum = 0;
for (int i = 0; i < rows; i++)
{
double d = b[i] * weights[i];
sum += d*d;
}
return Math.Sqrt(sum);
}
internal void print()
{
for (int i = 0; i < rows; i++)
Console.WriteLine(b[i]);
Console.WriteLine();
}
public static Vector operator-(Vector lhs, Vector rhs)
{
Vector v = new Vector(lhs.rows);
for (int i = 0; i < lhs.rows; i++)
v[i] = lhs[i] - rhs[i];
return v;
}
}
class Matrix
{
private double[] b;
internal readonly int rows, cols;
internal Matrix(int rows, int cols)
{
this.rows = rows;
this.cols = cols;
b = new double[rows * cols];
}
internal Matrix(int size)
{
this.rows = size;
this.cols = size;
b = new double[rows * cols];
for (int i = 0; i < size; i++)
this[i, i] = 1;
}
internal Matrix(int rows, int cols, double[] initArray)
{
this.rows = rows;
this.cols = cols;
b = (double[])initArray.Clone();
if (b.Length != rows * cols) throw new Exception("bad init array");
}
internal double this[int row, int col]
{
get { return b[row * cols + col]; }
set { b[row * cols + col] = value; }
}
public static Vector operator*(Matrix lhs, Vector rhs)
{
if (lhs.cols != rhs.rows) throw new Exception("I can't multiply matrix by vector");
Vector v = new Vector(lhs.rows);
for (int i = 0; i < lhs.rows; i++)
{
double sum = 0;
for (int j = 0; j < rhs.rows; j++)
sum += lhs[i,j]*rhs[j];
v[i] = sum;
}
return v;
}
internal void SwapRows(int r1, int r2)
{
if (r1 == r2) return;
int firstR1 = r1 * cols;
int firstR2 = r2 * cols;
for (int i = 0; i < cols; i++)
{
double tmp = b[firstR1 + i];
b[firstR1 + i] = b[firstR2 + i];
b[firstR2 + i] = tmp;
}
}
//with partial pivot
internal void ElimPartial(Vector B)
{
for (int diag = 0; diag < rows; diag++)
{
int max_row = diag;
double max_val = Math.Abs(this[diag, diag]);
double d;
for (int row = diag + 1; row < rows; row++)
if ((d = Math.Abs(this[row, diag])) > max_val)
{
max_row = row;
max_val = d;
}
SwapRows(diag, max_row);
B.SwapRows(diag, max_row);
double invd = 1 / this[diag, diag];
for (int col = diag; col < cols; col++)
this[diag, col] *= invd;
B[diag] *= invd;
for (int row = 0; row < rows; row++)
{
d = this[row, diag];
if (row != diag)
{
for (int col = diag; col < cols; col++)
this[row, col] -= d * this[diag, col];
B[row] -= d * B[diag];
}
}
}
}
internal void print()
{
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < cols; j++)
Console.Write(this[i,j].ToString()+" ");
Console.WriteLine();
}
}
}
}
using System;
namespace Rosetta
{
class Program
{
static void Main(string[] args)
{
Matrix A = new Matrix(6, 6,
new double[] {1.1,0.12,0.13,0.12,0.14,-0.12,
1.21,0.63,0.39,0.25,0.16,0.1,
1.03,1.26,1.58,1.98,2.49,3.13,
1.06,1.88,3.55,6.7,12.62,23.8,
1.12,2.51,6.32,15.88,39.9,100.28,
1.16,3.14,9.87,31.01,97.41,306.02});
Vector B = new Vector(new double[] { -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 });
A.ElimPartial(B);
B.print();
}
}
}
{{out}}
-0.0597391027501976
1.85018966726278
-1.97278330181163
1.4697587750651
-0.553874184782179
0.0723048745759396
C++
#include <algorithm>
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <vector>
template <typename scalar_type> class matrix {
public:
matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<scalar_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
auto i = elements_.begin();
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), i);
i += columns_;
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const scalar_type& operator()(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
scalar_type& operator()(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
private:
size_t rows_;
size_t columns_;
std::vector<scalar_type> elements_;
};
template <typename scalar_type>
void swap_rows(matrix<scalar_type>& m, size_t i, size_t j) {
size_t columns = m.columns();
for (size_t k = 0; k < columns; ++k)
std::swap(m(i, k), m(j, k));
}
template <typename scalar_type>
std::vector<scalar_type> gauss_partial(const matrix<scalar_type>& a0,
const std::vector<scalar_type>& b0) {
size_t n = a0.rows();
assert(a0.columns() == n);
assert(b0.size() == n);
// make augmented matrix
matrix<scalar_type> a(n, n + 1);
for (size_t i = 0; i < n; ++i) {
for (size_t j = 0; j < n; ++j)
a(i, j) = a0(i, j);
a(i, n) = b0[i];
}
// WP algorithm from Gaussian elimination page
// produces row echelon form
for (size_t k = 0; k < n; ++k) {
// Find pivot for column k
size_t max_index = k;
scalar_type max_value = 0;
for (size_t i = k; i < n; ++i) {
// compute scale factor = max abs in row
scalar_type scale_factor = 0;
for (size_t j = k; j < n; ++j)
scale_factor = std::max(std::abs(a(i, j)), scale_factor);
if (scale_factor == 0)
continue;
// scale the abs used to pick the pivot
scalar_type abs = std::abs(a(i, k))/scale_factor;
if (abs > max_value) {
max_index = i;
max_value = abs;
}
}
if (a(max_index, k) == 0)
throw std::runtime_error("matrix is singular");
if (k != max_index)
swap_rows(a, k, max_index);
for (size_t i = k + 1; i < n; ++i) {
scalar_type f = a(i, k)/a(k, k);
for (size_t j = k + 1; j <= n; ++j)
a(i, j) -= a(k, j) * f;
a(i, k) = 0;
}
}
// now back substitute to get result
std::vector<scalar_type> x(n);
for (size_t i = n; i-- > 0; ) {
x[i] = a(i, n);
for (size_t j = i + 1; j < n; ++j)
x[i] -= a(i, j) * x[j];
x[i] /= a(i, i);
}
return x;
}
int main() {
matrix<double> a(6, 6, {
{1.00, 0.00, 0.00, 0.00, 0.00, 0.00},
{1.00, 0.63, 0.39, 0.25, 0.16, 0.10},
{1.00, 1.26, 1.58, 1.98, 2.49, 3.13},
{1.00, 1.88, 3.55, 6.70, 12.62, 23.80},
{1.00, 2.51, 6.32, 15.88, 39.90, 100.28},
{1.00, 3.14, 9.87, 31.01, 97.41, 306.02}
});
std::vector<double> b{-0.01, 0.61, 0.91, 0.99, 0.60, 0.02};
std::vector<double> x{-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232};
std::vector<double> y(gauss_partial(a, b));
std::cout << std::setprecision(16);
const double epsilon = 1e-14;
for (size_t i = 0; i < y.size(); ++i) {
assert(std::abs(x[i] - y[i]) <= epsilon);
std::cout << y[i] << '\n';
}
return 0;
}
- Output:
-0.01 1.602790394502113 -1.61320305990556 1.245494121371436 -0.4909897195846575 0.065760696175232
Common Lisp
(defmacro mapcar-1 (fn n list)
"Maps a function of two parameters where the first one is fixed, over a list"
`(mapcar #'(lambda (l) (funcall ,fn ,n l)) ,list) )
(defun gauss (m)
(labels
((redc (m) ; Reduce to triangular form
(if (null (cdr m))
m
(cons (car m) (mapcar-1 #'cons 0 (redc (mapcar #'cdr (mapcar #'(lambda (r) (mapcar #'- (mapcar-1 #'* (caar m) r)
(mapcar-1 #'* (car r) (car m)))) (cdr m)))))) ))
(rev (m) ; Reverse each row except the last element
(reverse (mapcar #'(lambda (r) (append (reverse (butlast r)) (last r))) m)) ))
(catch 'result
(let ((m1 (redc (rev (redc m)))))
(reverse (mapcar #'(lambda (r) (let ((pivot (find-if-not #'zerop r))) (if pivot (/ (car (last r)) pivot) (throw 'result 'singular)))) m1)) ))))- Output:
(setq m1 '((1.00 0.00 0.00 0.00 0.00 0.00 -0.01)
(1.00 0.63 0.39 0.25 0.16 0.10 0.61)
(1.00 1.26 1.58 1.98 2.49 3.13 0.91)
(1.00 1.88 3.55 6.70 12.62 23.80 0.99)
(1.00 2.51 6.32 15.88 39.90 100.28 0.60)
(1.00 3.14 9.87 31.01 97.41 306.02 0.02) ))
(gauss m1)
=> (-0.009999999 1.6027923 -1.6132091 1.2455008 -0.4909925 0.06576109)
D
import std.stdio, std.math, std.algorithm, std.range, std.numeric,
std.typecons;
Tuple!(double[],"x", string,"err")
gaussPartial(in double[][] a0, in double[] b0) pure /*nothrow*/
in {
assert(a0.length == a0[0].length);
assert(a0.length == b0.length);
assert(a0.all!(row => row.length == a0[0].length));
} body {
enum eps = 1e-6;
immutable m = b0.length;
// Make augmented matrix.
//auto a = a0.zip(b0).map!(c => c[0] ~ c[1]).array; // Not mutable.
auto a = a0.zip(b0).map!(c => [] ~ c[0] ~ c[1]).array;
// Wikipedia algorithm from Gaussian elimination page,
// produces row-eschelon form.
foreach (immutable k; 0 .. a.length) {
// Find pivot for column k and swap.
a[k .. m].minPos!((x, y) => x[k] > y[k]).front.swap(a[k]);
if (a[k][k].abs < eps)
return typeof(return)(null, "singular");
// Do for all rows below pivot.
foreach (immutable i; k + 1 .. m) {
// Do for all remaining elements in current row.
a[i][k+1 .. m+1] -= a[k][k+1 .. m+1] * (a[i][k] / a[k][k]);
a[i][k] = 0; // Fill lower triangular matrix with zeros.
}
}
// End of WP algorithm. Now back substitute to get result.
auto x = new double[m];
foreach_reverse (immutable i; 0 .. m)
x[i] = (a[i][m] - a[i][i+1 .. m].dotProduct(x[i+1 .. m])) / a[i][i];
return typeof(return)(x, null);
}
void main() {
// The test case result is correct to this tolerance.
enum eps = 1e-13;
// Common RC example. Result computed with rational arithmetic
// then converted to double, and so should be about as close to
// correct as double represention allows.
immutable a = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]];
immutable b = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02];
immutable r = gaussPartial(a, b);
if (!r.err.empty)
return writeln("Error: ", r.err);
r.x.writeln;
immutable result = [-0.01, 1.602790394502114,
-1.6132030599055613, 1.2454941213714368,
-0.4909897195846576, 0.065760696175232];
foreach (immutable i, immutable xi; result)
if (abs(r.x[i] - xi) > eps)
return writeln("Out of tolerance: ", r.x[i], " ", xi);
}
- Output:
[-0.01, 1.60279, -1.6132, 1.24549, -0.49099, 0.0657607]
Delphi
program GuassianElimination;
// Modified from:
// R. Sureshkumar (10 January 1997)
// Gregory J. McRae (22 October 1997)
// http://web.mit.edu/10.001/Web/Course_Notes/Gauss_Pivoting.c
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.SysUtils;
type
TMatrix = class
private
_r, _c : integer;
data : array of TDoubleArray;
function getValue(rIndex, cIndex : integer): double;
procedure setValue(rIndex, cIndex : integer; value: double);
public
constructor Create (r, c : integer);
destructor Destroy; override;
property r : integer read _r;
property c : integer read _c;
property value[rIndex, cIndex: integer]: double read getValue write setValue; default;
end;
constructor TMatrix.Create (r, c : integer);
begin
inherited Create;
self.r := r; self.c := c;
setLength (data, r, c);
end;
destructor TMatrix.Destroy;
begin
data := nil;
inherited;
end;
function TMatrix.getValue(rIndex, cIndex: Integer): double;
begin
Result := data[rIndex-1, cIndex-1]; // 1-based array
end;
procedure TMatrix.setValue(rIndex, cIndex : integer; value: double);
begin
data[rIndex-1, cIndex-1] := value; // 1-based array
end;
// Solve A x = b
procedure gauss (A, b, x : TMatrix);
var rowx : integer;
i, j, k, n, m : integer;
amax, xfac, temp, temp1 : double;
begin
rowx := 0; // Keep count of the row interchanges
n := A.r;
for k := 1 to n - 1 do
begin
amax := abs (A[k,k]);
m := k;
// Find the row with largest pivot
for i := k + 1 to n do
begin
xfac := abs (A[i,k]);
if xfac > amax then
begin
amax := xfac;
m := i;
end;
end;
if m <> k then
begin // Row interchanges
rowx := rowx+1;
temp1 := b[k,1];
b[k,1] := b[m,1];
b[m,1] := temp1;
for j := k to n do
begin
temp := a[k,j];
a[k,j] := a[m,j];
a[m,j] := temp;
end;
end;
for i := k+1 to n do
begin
xfac := a[i, k]/a[k, k];
for j := k+1 to n do
a[i,j] := a[i,j]-xfac*a[k,j];
b[i,1] := b[i,1] - xfac*b[k,1]
end;
end;
// Back substitution
for j := 1 to n do
begin
k := n-j + 1;
x[k,1] := b[k,1];
for i := k+1 to n do
begin
x[k,1] := x[k,1] - a[k,i]*x[i,1];
end;
x[k,1] := x[k,1]/a[k,k];
end;
end;
var A, b, x : TMatrix;
begin
try
// Could have been done with simple arrays rather than a specific TMatrix class
A := TMatrix.Create (4,4);
// Note ideal but use TMatrix to define the vectors as well
b := TMatrix.Create (4,1);
x := TMatrix.Create (4,1);
A[1,1] := 2; A[1,2] := 1; A[1,3] := 0; A[1,4] := 0;
A[2,1] := 1; A[2,2] := 1; A[2,3] := 1; A[2,4] := 0;
A[3,1] := 0; A[3,2] := 1; A[3,3] := 2; A[3,4] := 1;
A[4,1] := 0; A[3,2] := 0; A[4,3] := 1; A[4,4] := 2;
b[1,1] := 2; b[2,1] := 1; b[3,1] := 4; b[4,1] := 8;
gauss (A, b, x);
writeln (x[1,1]:5:2);
writeln (x[2,1]:5:2);
writeln (x[3,1]:5:2);
writeln (x[4,1]:5:2);
readln;
except
on E: Exception do
Writeln(E.ClassName, ': ', E.Message);
end;
end.
- Output:
1.00, 0.00, 0.00, 4.00
EasyLang
proc gauss_elim . a[][] b[] x[] .
n = len a[][]
for i to n
maxr = i
maxv = abs a[i][i]
for j = i + 1 to n
if abs a[j][i] > maxv
maxr = j
maxv = abs a[j][i]
.
.
if maxr <> i
swap a[maxr][] a[i][]
swap b[maxr] b[i]
.
for j = i + 1 to n
f = a[j][i] / a[i][i]
for k = i to n
a[j][k] -= f * a[i][k]
.
b[j] -= f * b[i]
.
.
x[] = [ ]
len x[] n
for i = n downto 1
rhs = b[i]
for j = i + 1 to n
rhs -= a[i][j] * x[j]
.
x[i] = rhs / a[i][i]
.
.
a[][] = [ [ 1.00 0.00 0.00 0.00 0.00 0.00 ] [ 1.00 0.63 0.39 0.25 0.16 0.10 ] [ 1.00 1.26 1.58 1.98 2.49 3.13 ] [ 1.00 1.88 3.55 6.70 12.62 23.80 ] [ 1.00 2.51 6.32 15.88 39.90 100.28 ] [ 1.00 3.14 9.87 31.01 97.41 306.02 ] ]
b[] = [ -0.01 0.61 0.91 0.99 0.60 0.02 ]
gauss_elim a[][] b[] x[]
print x[]
F#
The Function
// Gaussian Elimination. Nigel Galloway: February 2nd., 2019
let gelim augM=
let f=List.length augM
let fG n (g:bigint list) t=n|>List.map(fun n->List.map2(fun n g->g-n)(List.map(fun n->n*g.[t])n)(List.map(fun g->g*n.[t])g))
let rec fN i (g::e as l)=
match i with i when i=f->l|>List.mapi(fun n (g:bigint list)->(g.[f],g.[n]))
|_->fN (i+1) (fG e g i@[g])
fN 0 augM
The Task
This task uses functionality from Continued_fraction/Arithmetic/Construct_from_rational_number#F.23 and Continued_fraction#F.23
let test=[[ -6I; -18I; 13I; 6I; -6I; -15I; -2I; -9I; -231I];
[ 2I; 20I; 9I; 2I; 16I; -12I; -18I; -5I; 647I];
[ 23I; 18I; -14I; -14I; -1I; 16I; 25I; -17I; -907I];
[ -8I; -1I; -19I; 4I; 3I; -14I; 23I; 8I; 248I];
[ 25I; 20I; -6I; 15I; 0I; -10I; 9I; 17I; 1316I];
[-13I; -1I; 3I; 5I; -2I; 17I; 14I; -12I; -1080I];
[ 19I; 24I; -21I; -5I; -19I; 0I; -24I; -17I; 1006I];
[ 20I; -3I; -14I; -16I; -23I; -25I; -15I; 20I; 1496I]]
let fN (n,g)=cN2S(π(rI2cf n g))
gelim test |> List.map fN |> List.iteri(fun i n->(printfn "x[%d]=%1.14f " (i+1) (snd (Seq.pairwise n|> Seq.find(fun (n,g)->n-g < 0.0000000000001M)))))
- Output:
x[1]=12.00000000000000 x[2]=10.00000000000000 x[3]=-20.00000000000000 x[4]=22.00000000000000 x[5]=-1.00000000000000 x[6]=-20.00000000000000 x[7]=-25.00000000000000 x[8]=23.00000000000000
Fortran
Gaussian Elimination with partial pivoting using augmented matrix
program ge
real, allocatable :: a(:,:),b(:)
a = reshape( &
[1.0, 1.00, 1.00, 1.00, 1.00, 1.00, &
0.0, 0.63, 1.26, 1.88, 2.51, 3.14, &
0.0, 0.39, 1.58, 3.55, 6.32, 9.87, &
0.0, 0.25, 1.98, 6.70, 15.88, 31.01, &
0.0, 0.16, 2.49, 12.62, 39.90, 97.41, &
0.0, 0.10, 3.13, 23.80, 100.28, 306.02], [6,6] )
b = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
print'(f15.7)',solve_wbs(ge_wpp(a,b))
contains
function solve_wbs(u) result(x) ! solve with backward substitution
real :: u(:,:)
integer :: i,n
real , allocatable :: x(:)
n = size(u,1)
allocate(x(n))
forall (i=n:1:-1) x(i) = ( u(i,n+1) - sum(u(i,i+1:n)*x(i+1:n)) ) / u(i,i)
end function
function ge_wpp(a,b) result(u) ! gaussian eliminate with partial pivoting
real :: a(:,:),b(:),upi
integer :: i,j,n,p
real , allocatable :: u(:,:)
n = size(a,1)
u = reshape( [a,b], [n,n+1] )
do j=1,n
p = maxloc(abs(u(j:n,j)),1) + j-1 ! maxloc returns indices between (1,n-j+1)
if (p /= j) u([p,j],j) = u([j,p],j)
u(j+1:,j) = u(j+1:,j)/u(j,j)
do i=j+1,n+1
upi = u(p,i)
if (p /= j) u([p,j],i) = u([j,p],i)
u(j+1:n,i) = u(j+1:n,i) - upi*u(j+1:n,j)
end do
end do
end function
end program
FreeBASIC
Gaussian elimination with pivoting. FreeBASIC version 1.05
Sub GaussJordan(matrix() As Double,rhs() As Double,ans() As Double)
Dim As Long n=Ubound(matrix,1)
Redim ans(0):Redim ans(1 To n)
Dim As Double b(1 To n,1 To n),r(1 To n)
For c As Long=1 To n 'take copies
r(c)=rhs(c)
For d As Long=1 To n
b(c,d)=matrix(c,d)
Next d
Next c
#macro pivot(num)
For p1 As Long = num To n - 1
For p2 As Long = p1 + 1 To n
If Abs(b(p1,num))<Abs(b(p2,num)) Then
Swap r(p1),r(p2)
For g As Long=1 To n
Swap b(p1,g),b(p2,g)
Next g
End If
Next p2
Next p1
#endmacro
For k As Long=1 To n-1
pivot(k) 'full pivoting
For row As Long =k To n-1
If b(row+1,k)=0 Then Exit For
Var f=b(k,k)/b(row+1,k)
r(row+1)=r(row+1)*f-r(k)
For g As Long=1 To n
b((row+1),g)=b((row+1),g)*f-b(k,g)
Next g
Next row
Next k
'back substitute
For z As Long=n To 1 Step -1
ans(z)=r(z)/b(z,z)
For j As Long = n To z+1 Step -1
ans(z)=ans(z)-(b(z,j)*ans(j)/b(z,z))
Next j
Next z
End Sub
dim as double a(1 to 6,1 to 6) = { _
{1.00, 0.00, 0.00, 0.00, 0.00, 0.00}, _
{1.00, 0.63, 0.39, 0.25, 0.16, 0.10}, _
{1.00, 1.26, 1.58, 1.98, 2.49, 3.13}, _
{1.00, 1.88, 3.55, 6.70, 12.62, 23.80}, _
{1.00, 2.51, 6.32, 15.88, 39.90, 100.28}, _
{1.00, 3.14, 9.87, 31.01, 97.41, 306.02} _
}
dim as double b(1 to 6) = { -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 }
redim as double result()
GaussJordan(a(),b(),result())
for n as long=lbound(result) to ubound(result)
print result(n)
next n
sleep
- Output:
-0.01 1.602790394502115 -1.613203059905572 1.245494121371448 -0.490989719584662 0.06576069617523256
Generic
generic coordinaat
{
ecs;
uuii;
coordinaat() { ecs=+a;uuii=+a;}
coordinaat(ecs_set,uuii_set) { ecs = ecs_set; uuii=uuii_set;}
operaator<(c)
{
iph ecs < c.ecs return troo;
iph c.ecs < ecs return phals;
iph uuii < c.uuii return troo;
return phals;
}
operaator==(connpair) // eecuuols and not eecuuols deriiu phronn operaator<
{
iph this < connpair return phals;
iph connpair < this return phals;
return troo;
}
operaator!=(connpair)
{
iph this < connpair return troo;
iph connpair < this return troo;
return phals;
}
too_string()
{
return "(" + ecs.too_string() + "," + uuii.too_string() + ")";
}
print()
{
str = too_string();
str.print();
}
println()
{
str = too_string();
str.println();
}
}
generic nnaatrics
{
s; // this is a set of coordinaat/ualioo pairs.
iteraator; // this field holds an iteraator phor the nnaatrics.
nnaatrics() // no parameters required phor nnaatrics construction.
{
s = nioo set(); // create a nioo set of coordinaat/ualioo pairs.
iteraator = nul; // the iteraator is initially set to nul.
}
nnaatrics(copee) // copee the nnaatrics.
{
s = nioo set(); // create a nioo set of coordinaat/ualioo pairs.
iteraator = nul; // the iteraator is initially set to nul.
r = copee.rouus;
c = copee.cols;
i = 0;
uuiil i < r
{
j = 0;
uuiil j < c
{
this[i,j] = copee[i,j];
j++;
}
i++;
}
}
begin { get { return s.begin; } } // property: used to commence manual iteraashon.
end { get { return s.end; } } // property: used to dephiin the end itenn of iteraashon
operaator<(a) // les than operaator is corld bii the avl tree algorithnns
{ // this operaator innpliis phor instance that you could potenshalee hav sets ou nnaatricss.
iph cees < a.cees // connpair the cee sets phurst.
return troo;
els iph a.cees < cees
return phals;
els // the cee sets are eecuuol thairphor connpair nnaatrics elennents.
{
phurst1 = begin;
lahst1 = end;
phurst2 = a.begin;
lahst2 = a.end;
uuiil phurst1 != lahst1 && phurst2 != lahst2
{
iph phurst1.daata.ualioo < phurst2.daata.ualioo
return troo;
els
{
iph phurst2.daata.ualioo < phurst1.daata.ualioo
return phals;
els
{
phurst1 = phurst1.necst;
phurst2 = phurst2.necst;
}
}
}
return phals;
}
}
operaator==(connpair) // eecuuols and not eecuuols deriiu phronn operaator<
{
iph this < connpair return phals;
iph connpair < this return phals;
return troo;
}
operaator!=(connpair)
{
iph this < connpair return troo;
iph connpair < this return troo;
return phals;
}
operaator[](cee_a,cee_b) // this is the nnaatrics indexer.
{
set
{
trii { s >> nioo cee_ualioo(new coordinaat(cee_a,cee_b)); } catch {}
s << nioo cee_ualioo(new coordinaat(nioo integer(cee_a),nioo integer(cee_b)),ualioo);
}
get
{
d = s.get(nioo cee_ualioo(new coordinaat(cee_a,cee_b)));
return d.ualioo;
}
}
operaator>>(coordinaat) // this operaator reennoous an elennent phronn the nnaatrics.
{
s >> nioo cee_ualioo(coordinaat);
return this;
}
iteraat() // and this is how to iterate on the nnaatrics.
{
iph iteraator.nul()
{
iteraator = s.lepht_nnohst;
iph iteraator == s.heder
return nioo iteraator(phals,nioo nun());
els
return nioo iteraator(troo,iteraator.daata.ualioo);
}
els
{
iteraator = iteraator.necst;
iph iteraator == s.heder
{
iteraator = nul;
return nioo iteraator(phals,nioo nun());
}
els
return nioo iteraator(troo,iteraator.daata.ualioo);
}
}
couunt // this property returns a couunt ou elennents in the nnaatrics.
{
get
{
return s.couunt;
}
}
ennptee // is the nnaatrics ennptee?
{
get
{
return s.ennptee;
}
}
lahst // returns the ualioo of the lahst elennent in the nnaatrics.
{
get
{
iph ennptee
throuu "ennptee nnaatrics";
els
return s.lahst.ualioo;
}
}
too_string() // conuerts the nnaatrics too aa string
{
return s.too_string();
}
print() // prints the nnaatrics to the consohl.
{
out = too_string();
out.print();
}
println() // prints the nnaatrics as a liin too the consohl.
{
out = too_string();
out.println();
}
cees // return the set ou cees ou the nnaatrics (a set of coordinaats).
{
get
{
k = nioo set();
phor e : s k << e.cee;
return k;
}
}
operaator+(p)
{
ouut = nioo nnaatrics();
phurst1 = begin;
lahst1 = end;
phurst2 = p.begin;
lahst2 = p.end;
uuiil phurst1 != lahst1 && phurst2 != lahst2
{
ouut[phurst1.daata.cee.ecs,phurst1.daata.cee.uuii] = phurst1.daata.ualioo + phurst2.daata.ualioo;
phurst1 = phurst1.necst;
phurst2 = phurst2.necst;
}
return ouut;
}
operaator-(p)
{
ouut = nioo nnaatrics();
phurst1 = begin;
lahst1 = end;
phurst2 = p.begin;
lahst2 = p.end;
uuiil phurst1 != lahst1 && phurst2 != lahst2
{
ouut[phurst1.daata.cee.ecs,phurst1.daata.cee.uuii] = phurst1.daata.ualioo - phurst2.daata.ualioo;
phurst1 = phurst1.necst;
phurst2 = phurst2.necst;
}
return ouut;
}
rouus
{
get
{
r = +a;
phurst1 = begin;
lahst1 = end;
uuiil phurst1 != lahst1
{
iph r < phurst1.daata.cee.ecs r = phurst1.daata.cee.ecs;
phurst1 = phurst1.necst;
}
return r + +b;
}
}
cols
{
get
{
c = +a;
phurst1 = begin;
lahst1 = end;
uuiil phurst1 != lahst1
{
iph c < phurst1.daata.cee.uuii c = phurst1.daata.cee.uuii;
phurst1 = phurst1.necst;
}
return c + +b;
}
}
operaator*(o)
{
iph cols != o.rouus throw "rouus-cols nnisnnatch";
reesult = nioo nnaatrics();
rouu_couunt = rouus;
colunn_couunt = o.cols;
loop = cols;
i = +a;
uuiil i < rouu_couunt
{
g = +a;
uuiil g < colunn_couunt
{
sunn = +a.a;
h = +a;
uuiil h < loop
{
a = this[i, h];
b = o[h, g];
nn = a * b;
sunn = sunn + nn;
h++;
}
reesult[i, g] = sunn;
g++;
}
i++;
}
return reesult;
}
suuop_rouus(a, b)
{
c = cols;
i = 0;
uuiil u < cols
{
suuop = this[a, i];
this[a, i] = this[b, i];
this[b, i] = suuop;
i++;
}
}
suuop_colunns(a, b)
{
r = rouus;
i = 0;
uuiil i < rouus
{
suuopp = this[i, a];
this[i, a] = this[i, b];
this[i, b] = suuop;
i++;
}
}
transpohs
{
get
{
reesult = new nnaatrics();
r = rouus;
c = cols;
i=0;
uuiil i < r
{
g = 0;
uuiil g < c
{
reesult[g, i] = this[i, g];
g++;
}
i++;
}
return reesult;
}
}
deternninant
{
get
{
rouu_couunt = rouus;
colunn_count = cols;
if rouu_couunt != colunn_count
throw "not a scuuair nnaatrics";
if rouu_couunt == 0
throw "the nnaatrics is ennptee";
if rouu_couunt == 1
return this[0, 0];
if rouu_couunt == 2
return this[0, 0] * this[1, 1] -
this[0, 1] * this[1, 0];
temp = nioo nnaatrics();
det = 0.0;
parity = 1.0;
j = 0;
uuiil j < rouu_couunt
{
k = 0;
uuiil k < rouu_couunt-1
{
skip_col = phals;
l = 0;
uuiil l < rouu_couunt-1
{
if l == j skip_col = troo;
if skip_col
n = l + 1;
els
n = l;
temp[k, l] = this[k + 1, n];
l++;
}
k++;
}
det = det + parity * this[0, j] * temp.deeternninant;
parity = 0.0 - parity;
j++;
}
return det;
}
}
ad_rouu(a, b)
{
c = cols;
i = 0;
uuiil i < c
{
this[a, i] = this[a, i] + this[b, i];
i++;
}
}
ad_colunn(a, b)
{
c = rouus;
i = 0;
uuiil i < c
{
this[i, a] = this[i, a] + this[i, b];
i++;
}
}
subtract_rouu(a, b)
{
c = cols;
i = 0;
uuiil i < c
{
this[a, i] = this[a, i] - this[b, i];
i++;
}
}
subtract_colunn(a, b)
{
c = rouus;
i = 0;
uuiil i < c
{
this[i, a] = this[i, a] - this[i, b];
i++;
}
}
nnultiplii_rouu(rouu, scalar)
{
c = cols;
i = 0;
uuiil i < c
{
this[rouu, i] = this[rouu, i] * scalar;
i++;
}
}
nnultiplii_colunn(colunn, scalar)
{
r = rouus;
i = 0;
uuiil i < r
{
this[i, colunn] = this[i, colunn] * scalar;
i++;
}
}
diuiid_rouu(rouu, scalar)
{
c = cols;
i = 0;
uuiil i < c
{
this[rouu, i] = this[rouu, i] / scalar;
i++;
}
}
diuiid_colunn(colunn, scalar)
{
r = rouus;
i = 0;
uuiil i < r
{
this[i, colunn] = this[i, colunn] / scalar;
i++;
}
}
connbiin_rouus_ad(a,b,phactor)
{
c = cols;
i = 0;
uuiil i < c
{
this[a, i] = this[a, i] + phactor * this[b, i];
i++;
}
}
connbiin_rouus_subtract(a,b,phactor)
{
c = cols;
i = 0;
uuiil i < c
{
this[a, i] = this[a, i] - phactor * this[b, i];
i++;
}
}
connbiin_colunns_ad(a,b,phactor)
{
r = rouus;
i = 0;
uuiil i < r
{
this[i, a] = this[i, a] + phactor * this[i, b];
i++;
}
}
connbiin_colunns_subtract(a,b,phactor)
{
r = rouus;
i = 0;
uuiil i < r
{
this[i, a] = this[i, a] - phactor * this[i, b];
i++;
}
}
inuers
{
get
{
rouu_couunt = rouus;
colunn_couunt = cols;
iph rouu_couunt != colunn_couunt
throw "nnatrics not scuuair";
els iph rouu_couunt == 0
throw "ennptee nnatrics";
els iph rouu_couunt == 1
{
r = nioo nnaatrics();
r[0, 0] = 1.0 / this[0, 0];
return r;
}
gauss = nioo nnaatrics(this);
i = 0;
uuiil i < rouu_couunt
{
j = 0;
uuiil j < rouu_couunt
{
iph i == j
gauss[i, j + rouu_couunt] = 1.0;
els
gauss[i, j + rouu_couunt] = 0.0;
j++;
}
i++;
}
j = 0;
uuiil j < rouu_couunt
{
iph gauss[j, j] == 0.0
{
k = j + 1;
uuiil k < rouu_couunt
{
if gauss[k, j] != 0.0 {gauss.nnaat.suuop_rouus(j, k); break; }
k++;
}
if k == rouu_couunt throw "nnatrics is singioolar";
}
phactor = gauss[j, j];
iph phactor != 1.0 gauss.diuiid_rouu(j, phactor);
i = j+1;
uuiil i < rouu_couunt
{
gauss.connbiin_rouus_subtract(i, j, gauss[i, j]);
i++;
}
j++;
}
i = rouu_couunt - 1;
uuiil i > 0
{
k = i - 1;
uuiil k >= 0
{
gauss.connbiin_rouus_subtract(k, i, gauss[k, i]);
k--;
}
i--;
}
reesult = nioo nnaatrics();
i = 0;
uuiil i < rouu_couunt
{
j = 0;
uuiil j < rouu_couunt
{
reesult[i, j] = gauss[i, j + rouu_couunt];
j++;
}
i++;
}
return reesult;
}
}
}
Go
Partial pivoting, no scaling
Gaussian elimination with partial pivoting by pseudocode on WP page Gaussian elimination."
package main
import (
"errors"
"fmt"
"log"
"math"
)
type testCase struct {
a [][]float64
b []float64
x []float64
}
var tc = testCase{
// common RC example. Result x computed with rational arithmetic then
// converted to float64, and so should be about as close to correct as
// float64 represention allows.
a: [][]float64{
{1.00, 0.00, 0.00, 0.00, 0.00, 0.00},
{1.00, 0.63, 0.39, 0.25, 0.16, 0.10},
{1.00, 1.26, 1.58, 1.98, 2.49, 3.13},
{1.00, 1.88, 3.55, 6.70, 12.62, 23.80},
{1.00, 2.51, 6.32, 15.88, 39.90, 100.28},
{1.00, 3.14, 9.87, 31.01, 97.41, 306.02}},
b: []float64{-0.01, 0.61, 0.91, 0.99, 0.60, 0.02},
x: []float64{-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232},
}
// result from above test case turns out to be correct to this tolerance.
const ε = 1e-13
func main() {
x, err := GaussPartial(tc.a, tc.b)
if err != nil {
log.Fatal(err)
}
fmt.Println(x)
for i, xi := range x {
if math.Abs(tc.x[i]-xi) > ε {
log.Println("out of tolerance")
log.Fatal("expected", tc.x)
}
}
}
func GaussPartial(a0 [][]float64, b0 []float64) ([]float64, error) {
// make augmented matrix
m := len(b0)
a := make([][]float64, m)
for i, ai := range a0 {
row := make([]float64, m+1)
copy(row, ai)
row[m] = b0[i]
a[i] = row
}
// WP algorithm from Gaussian elimination page
// produces row-eschelon form
for k := range a {
// Find pivot for column k:
iMax := k
max := math.Abs(a[k][k])
for i := k + 1; i < m; i++ {
if abs := math.Abs(a[i][k]); abs > max {
iMax = i
max = abs
}
}
if a[iMax][k] == 0 {
return nil, errors.New("singular")
}
// swap rows(k, i_max)
a[k], a[iMax] = a[iMax], a[k]
// Do for all rows below pivot:
for i := k + 1; i < m; i++ {
// Do for all remaining elements in current row:
for j := k + 1; j <= m; j++ {
a[i][j] -= a[k][j] * (a[i][k] / a[k][k])
}
// Fill lower triangular matrix with zeros:
a[i][k] = 0
}
}
// end of WP algorithm.
// now back substitute to get result.
x := make([]float64, m)
for i := m - 1; i >= 0; i-- {
x[i] = a[i][m]
for j := i + 1; j < m; j++ {
x[i] -= a[i][j] * x[j]
}
x[i] /= a[i][i]
}
return x, nil
}
- Output:
[-0.01 1.6027903945020987 -1.613203059905494 1.245494121371364 -0.49098971958462834 0.06576069617522803]
Scaled partial pivoting
Changes from above version noted with comments. For the example data scaling does help a bit.
package main
import (
"errors"
"fmt"
"log"
"math"
)
type testCase struct {
a [][]float64
b []float64
x []float64
}
var tc = testCase{
a: [][]float64{
{1.00, 0.00, 0.00, 0.00, 0.00, 0.00},
{1.00, 0.63, 0.39, 0.25, 0.16, 0.10},
{1.00, 1.26, 1.58, 1.98, 2.49, 3.13},
{1.00, 1.88, 3.55, 6.70, 12.62, 23.80},
{1.00, 2.51, 6.32, 15.88, 39.90, 100.28},
{1.00, 3.14, 9.87, 31.01, 97.41, 306.02}},
b: []float64{-0.01, 0.61, 0.91, 0.99, 0.60, 0.02},
x: []float64{-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232},
}
// result from above test case turns out to be correct to this tolerance.
const ε = 1e-14
func main() {
x, err := GaussPartial(tc.a, tc.b)
if err != nil {
log.Fatal(err)
}
fmt.Println(x)
for i, xi := range x {
if math.Abs(tc.x[i]-xi) > ε {
log.Println("out of tolerance")
log.Fatal("expected", tc.x)
}
}
}
func GaussPartial(a0 [][]float64, b0 []float64) ([]float64, error) {
m := len(b0)
a := make([][]float64, m)
for i, ai := range a0 {
row := make([]float64, m+1)
copy(row, ai)
row[m] = b0[i]
a[i] = row
}
for k := range a {
iMax := 0
max := -1.
for i := k; i < m; i++ {
row := a[i]
// compute scale factor s = max abs in row
s := -1.
for j := k; j < m; j++ {
x := math.Abs(row[j])
if x > s {
s = x
}
}
// scale the abs used to pick the pivot.
if abs := math.Abs(row[k]) / s; abs > max {
iMax = i
max = abs
}
}
if a[iMax][k] == 0 {
return nil, errors.New("singular")
}
a[k], a[iMax] = a[iMax], a[k]
for i := k + 1; i < m; i++ {
for j := k + 1; j <= m; j++ {
a[i][j] -= a[k][j] * (a[i][k] / a[k][k])
}
a[i][k] = 0
}
}
x := make([]float64, m)
for i := m - 1; i >= 0; i-- {
x[i] = a[i][m]
for j := i + 1; j < m; j++ {
x[i] -= a[i][j] * x[j]
}
x[i] /= a[i][i]
}
return x, nil
}
- Output:
[-0.01 1.6027903945021131 -1.6132030599055596 1.245494121371436 -0.49098971958465754 0.065760696175232]
Haskell
From scratch
isMatrix xs = null xs || all ((== (length.head $ xs)).length) xs
isSquareMatrix xs = null xs || all ((== (length xs)).length) xs
mult:: Num a => [[a]] -> [[a]] -> [[a]]
mult uss vss = map ((\xs -> if null xs then [] else foldl1 (zipWith (+)) xs). zipWith (\vs u -> map (u*) vs) vss) uss
gauss::[[Double]] -> [[Double]] -> [[Double]]
gauss xs bs = map (map fromRational) $ solveGauss (toR xs) (toR bs)
where toR = map $ map toRational
solveGauss:: (Fractional a, Ord a) => [[a]] -> [[a]] -> [[a]]
solveGauss xs bs | null xs || null bs || length xs /= length bs || (not $ isSquareMatrix xs) || (not $ isMatrix bs) = []
| otherwise = uncurry solveTriangle $ triangle xs bs
solveTriangle::(Fractional a,Eq a) => [[a]] -> [[a]] -> [[a]]
solveTriangle us _ | not.null.dropWhile ((/= 0).head) $ us = []
solveTriangle ([c]:as) (b:bs) = go as bs [map (/c) b]
where
val us vs ws = let u = head us in map (/u) $ zipWith (-) vs (head $ mult [tail us] ws)
go [] _ zs = zs
go _ [] zs = zs
go (x:xs) (y:ys) zs = go xs ys $ (val x y zs):zs
triangle::(Num a, Ord a) => [[a]] -> [[a]] -> ([[a]],[[a]])
triangle xs bs = triang ([],[]) (xs,bs)
where
triang ts (_,[]) = ts
triang ts ([],_) = ts
triang (os,ps) zs = triang (us:os,cs:ps).unzip $ [(fun tus vs, fun cs es) | (v:vs,es) <- zip uss css,let fun = zipWith (\x y -> v*x - u*y)]
where ((us@(u:tus)):uss,cs:css) = bubble zs
bubble::(Num a, Ord a) => ([[a]],[[a]]) -> ([[a]],[[a]])
bubble (xs,bs) = (go xs, go bs)
where
idmax = snd.maximum.flip zip [0..].map (abs.head) $ xs
go ys = let (us,vs) = splitAt idmax ys in vs ++ us
main = do
let a = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
let b = [[-0.01], [0.61], [0.91], [0.99], [0.60], [0.02]]
mapM_ print $ gauss a b
- Output:
[-1.0e-2] [1.6027903945021098] [-1.6132030599055482] [1.245494121371424] [-0.49098971958465265] [6.576069617523134e-2]
Another example
We use Rational numbers for having more precision. a % b is the rational a / b.
mult:: Num a => [[a]] -> [[a]] -> [[a]]
mult uss vss = map ((\xs -> if null xs then [] else foldl1 (zipWith (+)) xs). zipWith (\vs u -> map (u*) vs) vss) uss
bubble::([a] -> c) -> (c -> c -> Bool) -> [[a]] -> [[b]] -> ([[a]],[[b]])
bubble _ _ [] ts = ([],ts)
bubble _ _ rs [] = (rs,[])
bubble f g (r:rs) (t:ts) = bub r t (f r) rs ts [] []
where
bub l k _ [] _ xs ys = (l:xs,k:ys)
bub l k _ _ [] xs ys = (l:xs,k:ys)
bub l k m (u:us) (v:vs) xs ys = ans
where
mu = f u
ans | g m mu = bub l k m us vs (u:xs) (v:ys)
| otherwise = bub u v mu us vs (l:xs) (k:ys)
pivot::Num a => [a] -> [a] -> [[a]] -> [[a]] -> ([[a]],[[a]])
pivot xs ks ys ls = go ys ls [] []
where
x = head xs
fun r = zipWith (\u v -> u*r - v*x)
val rs ts = let f = fun (head rs) in (tail $ f xs rs,f ks ts)
go [] _ us vs = (us,vs)
go _ [] us vs = (us,vs)
go rs ts us vs = go (tail rs) (tail ts) (es:us) (fs:vs)
where (es,fs) = val (head rs) (head ts)
triangle::(Num a,Ord a) => [[a]] -> [[a]] -> ([[a]],[[a]])
triangle as bs = go (as,bs) [] []
where
go ([],_) us vs = (us,vs)
go (_,[]) us vs = (us,vs)
go (rs,ts) us vs = ans
where
(xs:ys,ks:ls) = bubble (abs.head) (>=) rs ts
ans = go (pivot xs ks ys ls) (xs:us) (ks:vs)
solveTriangle::(Fractional a,Eq a) => [[a]] -> [[a]] -> [[a]]
solveTriangle [] _ = []
solveTriangle _ [] = []
solveTriangle as _ | not.null.dropWhile ((/= 0).head) $ as = []
solveTriangle ([c]:as) (b:bs) = go as bs [map (/c) b]
where
val us vs ws = let u = head us in map (/u) $ zipWith (-) vs (head $ mult [tail us] ws)
go [] _ zs = zs
go _ [] zs = zs
go (x:xs) (y:ys) zs = go xs ys $ (val x y zs):zs
solveGauss:: (Fractional a, Ord a) => [[a]] -> [[a]] -> [[a]]
solveGauss as bs = uncurry solveTriangle $ triangle as bs
matI::(Num a) => Int -> [[a]]
matI n = [ [fromIntegral.fromEnum $ i == j | j <- [1..n]] | i <- [1..n]]
task::[[Rational]] -> [[Rational]] -> IO()
task a b = do
let x = solveGauss a b
let u = map (map fromRational) x
let y = mult a x
let identity = matI (length x)
let a1 = solveGauss a identity
let h = mult a a1
let z = mult a1 b
putStrLn "a ="
mapM_ print a
putStrLn "b ="
mapM_ print b
putStrLn "solve: a * x = b => x = solveGauss a b ="
mapM_ print x
putStrLn "u = fromRationaltoDouble x ="
mapM_ print u
putStrLn "verification: y = a * x = mult a x ="
mapM_ print y
putStrLn $ "test: y == b = "
print $ y == b
putStrLn "identity matrix: identity ="
mapM_ print identity
putStrLn "find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveGauss a identity ="
mapM_ print a1
putStrLn "verification: h = a * a1 = mult a a1 ="
mapM_ print h
putStrLn $ "test: h == identity = "
print $ h == identity
putStrLn "z = a1 * b = mult a1 b ="
mapM_ print z
putStrLn "test: z == x ="
print $ z == x
main = do
let a = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
let b = [[-0.01], [0.61], [0.91], [0.99], [0.60], [0.02]]
task a b
- Output:
a = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [1 % 1,63 % 100,39 % 100,1 % 4,4 % 25,1 % 10] [1 % 1,63 % 50,79 % 50,99 % 50,249 % 100,313 % 100] [1 % 1,47 % 25,71 % 20,67 % 10,631 % 50,119 % 5] [1 % 1,251 % 100,158 % 25,397 % 25,399 % 10,2507 % 25] [1 % 1,157 % 50,987 % 100,3101 % 100,9741 % 100,15301 % 50] b = [(-1) % 100] [61 % 100] [91 % 100] [99 % 100] [3 % 5] [1 % 50] solve: a * x = b => x = solveGauss a b = [(-1) % 100] [655870882787 % 409205648497] [(-660131804286) % 409205648497] [509663229635 % 409205648497] [(-200915766608) % 409205648497] [26909648324 % 409205648497] u = fromRationaltoDouble x = [-1.0e-2] [1.602790394502114] [-1.6132030599055613] [1.2454941213714368] [-0.4909897195846576] [6.5760696175232e-2] verification: y = a * x = mult a x = [(-1) % 100] [61 % 100] [91 % 100] [99 % 100] [3 % 5] [1 % 50] test: y == b = True identity matrix: identity = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1] find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveGauss a identity = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [(-1373267314900) % 409205648497,2792895413400 % 409205648497,(-2539722499600) % 409205648497,1620086418000 % 409205648497,(-593562467900) % 409205648497,93570451000 % 409205648497] [1683936576500 % 409205648497,(-5515373801600) % 409205648497,7425272193600 % 409205648497,(-5318952383900) % 409205648497,2060945510400 % 409205648497,(-335828095000) % 409205648497] [(-955389934100) % 409205648497,3910562856500 % 409205648497,(-6532196158200) % 409205648497,5493636552500 % 409205648497,(-2312764532500) % 409205648497,396151215800 % 409205648497] [253880215500 % 409205648497,(-1187959549100) % 409205648497,2281116328400 % 409205648497,(-2180688584400) % 409205648497,1021846842100 % 409205648497,(-188195252500) % 409205648497] [(-25558559000) % 409205648497,131101344100 % 409205648497,(-277605537500) % 409205648497,292380217600 % 409205648497,(-151287558900) % 409205648497,30970093700 % 409205648497] verification: h = a * a1 = mult a a1 = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1] test: h == identity = True z = a1 * b = mult a1 b = [(-1) % 100] [655870882787 % 409205648497] [(-660131804286) % 409205648497] [509663229635 % 409205648497] [(-200915766608) % 409205648497] [26909648324 % 409205648497] test: z == x = True
Determinant and permutation matrix are given
mult:: Num a => [[a]] -> [[a]] -> [[a]]
mult uss vss = map ((\xs -> if null xs then [] else foldl1 (zipWith (+)) xs). zipWith (\vs u -> map (u*) vs) vss) uss
triangle::(Fractional a, Ord a) => [[a]] -> [[a]] -> (a,[(([a],[a]),Int)])
triangle as bs = pivot 1 [] $ zipWith3 (\x y i -> ((x,y),i)) as bs [(0::Int)..]
where
good rs ts = (abs.head.fst.fst $ ts) <= (abs.head.fst.fst $ rs)
go (us,vs) ((os,ps),i) = if o == 0 then ((rs,f vs ps),i) else ((f us rs,f vs ps),i)
where
(o,rs) = (head os,tail os)
f = zipWith (\x y -> y - x*o)
change i (ys:zs) = map (\xs -> if (==i).snd $ xs then ys else xs) zs
pivot d ls [] = (d,ls)
pivot d ls zs@((_,j):ys) = if u == 0 then (0,ls) else pivot e (ps:ls) ws
where
e = if i == j then u*d else -u*d
ws = map (go (map (/u) us,map (/u) vs)) $ if i == j then ys else change i zs
ps@((u:us,vs),i) = foldl1 (\rs ts -> if good rs ts then rs else ts) zs
-- ((det,sol),permutation) = gauss as bs
-- det = determinant as
-- sol is solution of: as * sol = bs
-- perm is a permutation with: (matPerm perm) * as * sol = (matPerm perm) * bs
gauss::(Fractional a,Ord a) => [[a]] -> [[a]] -> ((a,[[a]]),[Int])
gauss as bs = if 0 == det then ((0,[]),[]) else solveTriangle ms
where
(det,ms) = triangle as bs
solveTriangle ((([c],b),i):sys) = go sys [map (/c) b] [i]
where
val us vs ws = let u = head us in map (/u) $ zipWith (-) vs (head $ mult [tail us] ws)
go [] zs is = ((det,zs),is)
go (((x,y),i):sys) zs is = go sys ((val x y zs):zs) (i:is)
solveGauss::(Fractional a,Ord a) => [[a]] -> [[a]] -> [[a]]
solveGauss as = snd.fst.gauss as
matI::Num a => Int -> [[a]]
matI n = [ [fromIntegral.fromEnum $ i == j | i <- [1..n]] | j <- [1..n]]
matPerm::Num a => [Int] -> [[a]]
matPerm ns = [ [fromIntegral.fromEnum $ i == j | (j,_) <- zip [0..] ns] | i <- ns]
task::[[Rational]] -> [[Rational]] -> IO()
task a b = do
let ((d,x),perm) = gauss a b
let ps = matPerm perm
let u = map (map fromRational) x
let y = mult a x
let identity = matI (length x)
let a1 = solveGauss a identity
let h = mult a a1
let z = mult a1 b
putStrLn "d = determinant a ="
print d
putStrLn "a ="
mapM_ print a
putStrLn "b ="
mapM_ print b
putStrLn "solve: a * x = b => x = solveGauss a b ="
mapM_ print x
putStrLn "u = fromRationaltoDouble x ="
mapM_ print u
putStrLn "verification: y = a * x = mult a x ="
mapM_ print y
putStrLn $ "test: y == b = "
print $ y == b
putStrLn "ps is the permutation associated to matrix a and ps ="
mapM_ print ps
putStrLn "identity matrix: identity ="
mapM_ print identity
putStrLn "find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveGauss a identity ="
mapM_ print a1
putStrLn "verification: h = a * a1 = mult a a1 ="
mapM_ print h
putStrLn $ "test: h == identity = "
print $ h == identity
putStrLn "z = a1 * b = mult a1 b ="
mapM_ print z
putStrLn "test: z == x ="
print $ z == x
main = do
let a = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
let b = [[-0.01], [0.61], [0.91], [0.99], [0.60], [0.02]]
task a b
- Output:
d = determinant a = 409205648497 % 10000000000 a = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [1 % 1,63 % 100,39 % 100,1 % 4,4 % 25,1 % 10] [1 % 1,63 % 50,79 % 50,99 % 50,249 % 100,313 % 100] [1 % 1,47 % 25,71 % 20,67 % 10,631 % 50,119 % 5] [1 % 1,251 % 100,158 % 25,397 % 25,399 % 10,2507 % 25] [1 % 1,157 % 50,987 % 100,3101 % 100,9741 % 100,15301 % 50] b = [(-1) % 100] [61 % 100] [91 % 100] [99 % 100] [3 % 5] [1 % 50] solve: a * x = b => x = solveGauss a b = [(-1) % 100] [655870882787 % 409205648497] [(-660131804286) % 409205648497] [509663229635 % 409205648497] [(-200915766608) % 409205648497] [26909648324 % 409205648497] u = fromRationaltoDouble x = [-1.0e-2] [1.602790394502114] [-1.6132030599055613] [1.2454941213714368] [-0.4909897195846576] [6.5760696175232e-2] verification: y = a * x = mult a x = [(-1) % 100] [61 % 100] [91 % 100] [99 % 100] [3 % 5] [1 % 50] test: y == b = True ps is the permutation associated to matrix a and ps = [1,0,0,0,0,0] [0,0,0,0,0,1] [0,0,1,0,0,0] [0,0,0,0,1,0] [0,1,0,0,0,0] [0,0,0,1,0,0] identity matrix: identity = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1] find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveGauss a identity = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [(-1373267314900) % 409205648497,2792895413400 % 409205648497,(-2539722499600) % 409205648497,1620086418000 % 409205648497,(-593562467900) % 409205648497,93570451000 % 409205648497] [1683936576500 % 409205648497,(-5515373801600) % 409205648497,7425272193600 % 409205648497,(-5318952383900) % 409205648497,2060945510400 % 409205648497,(-335828095000) % 409205648497] [(-955389934100) % 409205648497,3910562856500 % 409205648497,(-6532196158200) % 409205648497,5493636552500 % 409205648497,(-2312764532500) % 409205648497,396151215800 % 409205648497] [253880215500 % 409205648497,(-1187959549100) % 409205648497,2281116328400 % 409205648497,(-2180688584400) % 409205648497,1021846842100 % 409205648497,(-188195252500) % 409205648497] [(-25558559000) % 409205648497,131101344100 % 409205648497,(-277605537500) % 409205648497,292380217600 % 409205648497,(-151287558900) % 409205648497,30970093700 % 409205648497] verification: h = a * a1 = mult a a1 = [1 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1] test: h == identity = True z = a1 * b = mult a1 b = [(-1) % 100] [655870882787 % 409205648497] [(-660131804286) % 409205648497] [509663229635 % 409205648497] [(-200915766608) % 409205648497] [26909648324 % 409205648497] test: z == x = True
J
%. , J's matrix divide verb, directly solves systems of determined and of over-determined linear equations directly. This example J session builds a noisy sine curve on the half circle, fits quintic and quadratic equations, and displays the results of evaluating these polynomials.
f=: 6j2&": NB. formatting verb
sin=: 1&o. NB. verb to evaluate circle function 1, the sine
add_noise=: ] + (* (_0.5 + 0 ?@:#~ #)) NB. AMPLITUDE add_noise SIGNAL
f RADIANS=: o.@:(%~ i.@:>:)5 NB. monadic circle function is pi times
0.00 0.63 1.26 1.88 2.51 3.14
f SINES=: sin RADIANS
0.00 0.59 0.95 0.95 0.59 0.00
f NOISY_SINES=: 0.1 add_noise SINES
_0.01 0.61 0.91 0.99 0.60 0.02
A=: (^/ i.@:#) RADIANS NB. A is the quintic coefficient matrix
NB. display the equation to solve
(f A) ; 'x' ; '=' ; f@:,. NOISY_SINES
┌────────────────────────────────────┬─┬─┬──────┐
│ 1.00 0.00 0.00 0.00 0.00 0.00│x│=│ _0.01│
│ 1.00 0.63 0.39 0.25 0.16 0.10│ │ │ 0.61│
│ 1.00 1.26 1.58 1.98 2.49 3.13│ │ │ 0.91│
│ 1.00 1.88 3.55 6.70 12.62 23.80│ │ │ 0.99│
│ 1.00 2.51 6.32 15.88 39.90100.28│ │ │ 0.60│
│ 1.00 3.14 9.87 31.01 97.41306.02│ │ │ 0.02│
└────────────────────────────────────┴─┴─┴──────┘
f QUINTIC_COEFFICIENTS=: NOISY_SINES %. A NB. %. solves the linear system
_0.01 1.71 _1.88 1.48 _0.58 0.08
quintic=: QUINTIC_COEFFICIENTS&p. NB. verb to evaluate the polynomial
NB. %. also solves the least squares fit for overdetermined system
quadratic=: (NOISY_SINES %. (^/ i.@:3:) RADIANS)&p. NB. verb to evaluate quadratic.
quadratic
_0.0200630695393961729 1.26066877804926536 _0.398275112136019516&p.
NB. The quintic is agrees with the noisy data, as it should
f@:(NOISY_SINES ,. sin ,. quadratic ,. quintic) RADIANS
_0.01 0.00 _0.02 _0.01
0.61 0.59 0.61 0.61
0.91 0.95 0.94 0.91
0.99 0.95 0.94 0.99
0.60 0.59 0.63 0.60
0.02 0.00 0.01 0.02
f MID_POINTS=: (+ -:@:(-/@:(2&{.)))RADIANS
_0.31 0.31 0.94 1.57 2.20 2.83
f@:(sin ,. quadratic ,. quintic) MID_POINTS
_0.31 _0.46 _0.79
0.31 0.34 0.38
0.81 0.81 0.77
1.00 0.98 1.00
0.81 0.83 0.86
0.31 0.36 0.27
Java
Naked implementation, using Java arrays instead of a matrix class.
import java.util.Locale;
public class GaussianElimination {
public static double solve(double[][] a, double[][] b) {
if (a == null || b == null || a.length == 0 || b.length == 0) {
throw new IllegalArgumentException("Invalid dimensions");
}
int n = b.length, p = b[0].length;
if (a.length != n || a[0].length != n) {
throw new IllegalArgumentException("Invalid dimensions");
}
double det = 1.0;
for (int i = 0; i < n - 1; i++) {
int k = i;
for (int j = i + 1; j < n; j++) {
if (Math.abs(a[j][i]) > Math.abs(a[k][i])) {
k = j;
}
}
if (k != i) {
det = -det;
for (int j = i; j < n; j++) {
double s = a[i][j];
a[i][j] = a[k][j];
a[k][j] = s;
}
for (int j = 0; j < p; j++) {
double s = b[i][j];
b[i][j] = b[k][j];
b[k][j] = s;
}
}
for (int j = i + 1; j < n; j++) {
double s = a[j][i] / a[i][i];
for (k = i + 1; k < n; k++) {
a[j][k] -= s * a[i][k];
}
for (k = 0; k < p; k++) {
b[j][k] -= s * b[i][k];
}
}
}
for (int i = n - 1; i >= 0; i--) {
for (int j = i + 1; j < n; j++) {
double s = a[i][j];
for (int k = 0; k < p; k++) {
b[i][k] -= s * b[j][k];
}
}
double s = a[i][i];
det *= s;
for (int k = 0; k < p; k++) {
b[i][k] /= s;
}
}
return det;
}
public static void main(String[] args) {
double[][] a = new double[][] {{4.0, 1.0, 0.0, 0.0, 0.0},
{1.0, 4.0, 1.0, 0.0, 0.0},
{0.0, 1.0, 4.0, 1.0, 0.0},
{0.0, 0.0, 1.0, 4.0, 1.0},
{0.0, 0.0, 0.0, 1.0, 4.0}};
double[][] b = new double[][] {{1.0 / 2.0},
{2.0 / 3.0},
{3.0 / 4.0},
{4.0 / 5.0},
{5.0 / 6.0}};
double[] x = {39.0 / 400.0,
11.0 / 100.0,
31.0 / 240.0,
37.0 / 300.0,
71.0 / 400.0};
System.out.println("det: " + solve(a, b));
for (int i = 0; i < 5; i++) {
System.out.printf(Locale.US, "%12.8f %12.4e\n", b[i][0], b[i][0] - x[i]);
}
}
}
- Output:
det: 780.0 0.09750000 0.0000e+00 0.11000000 0.0000e+00 0.12916667 0.0000e+00 0.12333333 1.3878e-17 0.17750000 2.7756e-17
JavaScript
From Numerical Recipes in C:
// Lower Upper Solver
function lusolve(A, b, update) {
var lu = ludcmp(A, update)
if (lu === undefined) return // Singular Matrix!
return lubksb(lu, b, update)
}
// Lower Upper Decomposition
function ludcmp(A, update) {
// A is a matrix that we want to decompose into Lower and Upper matrices.
var d = true
var n = A.length
var idx = new Array(n) // Output vector with row permutations from partial pivoting
var vv = new Array(n) // Scaling information
for (var i=0; i<n; i++) {
var max = 0
for (var j=0; j<n; j++) {
var temp = Math.abs(A[i][j])
if (temp > max) max = temp
}
if (max == 0) return // Singular Matrix!
vv[i] = 1 / max // Scaling
}
if (!update) { // make a copy of A
var Acpy = new Array(n)
for (var i=0; i<n; i++) {
var Ai = A[i]
Acpyi = new Array(Ai.length)
for (j=0; j<Ai.length; j+=1) Acpyi[j] = Ai[j]
Acpy[i] = Acpyi
}
A = Acpy
}
var tiny = 1e-20 // in case pivot element is zero
for (var i=0; ; i++) {
for (var j=0; j<i; j++) {
var sum = A[j][i]
for (var k=0; k<j; k++) sum -= A[j][k] * A[k][i];
A[j][i] = sum
}
var jmax = 0
var max = 0;
for (var j=i; j<n; j++) {
var sum = A[j][i]
for (var k=0; k<i; k++) sum -= A[j][k] * A[k][i];
A[j][i] = sum
var temp = vv[j] * Math.abs(sum)
if (temp >= max) {
max = temp
jmax = j
}
}
if (i <= jmax) {
for (var j=0; j<n; j++) {
var temp = A[jmax][j]
A[jmax][j] = A[i][j]
A[i][j] = temp
}
d = !d;
vv[jmax] = vv[i]
}
idx[i] = jmax;
if (i == n-1) break;
var temp = A[i][i]
if (temp == 0) A[i][i] = temp = tiny
temp = 1 / temp
for (var j=i+1; j<n; j++) A[j][i] *= temp
}
return {A:A, idx:idx, d:d}
}
// Lower Upper Back Substitution
function lubksb(lu, b, update) {
// solves the set of n linear equations A*x = b.
// lu is the object containing A, idx and d as determined by the routine ludcmp.
var A = lu.A
var idx = lu.idx
var n = idx.length
if (!update) { // make a copy of b
var bcpy = new Array(n)
for (var i=0; i<b.length; i+=1) bcpy[i] = b[i]
b = bcpy
}
for (var ii=-1, i=0; i<n; i++) {
var ix = idx[i]
var sum = b[ix]
b[ix] = b[i]
if (ii > -1)
for (var j=ii; j<i; j++) sum -= A[i][j] * b[j]
else if (sum)
ii = i
b[i] = sum
}
for (var i=n-1; i>=0; i--) {
var sum = b[i]
for (var j=i+1; j<n; j++) sum -= A[i][j] * b[j]
b[i] = sum / A[i][i]
}
return b // solution vector x
}
document.write(
lusolve(
[
[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]
],
[-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
)
)
- Output:
-0.01000000000000004, 1.6027903945021095, -1.6132030599055475, 1.2454941213714232, -0.4909897195846526, 0.06576069617523138
jq
Adapted from #Wren
Works with gojq, the Go implementation of jq
def ta: [
[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]
];
def tb:[-0.01, 0.61, 0.91, 0.99, 0.60, 0.02];
# Expected values:
def tx:[
-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232
];
# Input: an array or an object
def swap($i;$j):
.[$i] as $tmp | .[$i] = .[$j] | .[$j] = $tmp;
def gaussPartial(a0; b0):
(b0|length) as $m
| reduce range(0;a0|length) as $i (
{ a: [range(0;$m)|null] };
.a[$i] = a0[$i] + [b0[$i]] )
| reduce range(0; .a|length) as $k (.;
.iMax = 0
| .max = -1
| reduce range($k;$m) as $i (.;
.a[$i] as $row
# compute scale factor s = max abs in row
| .s = -1
| reduce range($k;$m) as $j (.;
($row[$j]|length) as $e
| if ($e > .s) then .s = $e else . end )
# scale the abs used to pick the pivot
| ( ($row[$k]|length) / .s) as $abs
| if $abs > .max
then .iMax = $i | .max = $abs
else .
end )
| if (.a[.iMax][$k] == 0) then "Matrix is singular." | error
else .iMax as $iMax
| .a |= swap($k; $iMax)
| reduce range($k + 1; $m) as $i (.;
reduce range($k + 1; $m + 1 ) as $j (.;
.a[$i][$j] = .a[$i][$j] - (.a[$k][$j] * .a[$i][$k] / .a[$k][$k]) )
| .a[$i][$k] = 0 )
end
)
| .x = [range(0;$m)|0]
| reduce range($m - 1; -1; -1) as $i (.;
.x[$i] = .a[$i][$m]
| reduce range($i + 1; $m) as $j (.;
.x[$i] = .x[$i] - .a[$i][$j] * .x[$j] )
| .x[$i] = .x[$i] / .a[$i][$i] )
| .x ;
def x: gaussPartial(ta; tb);
# Input: the array of values to be compared againt $target
def pointwise_check($target; $EPSILON):
. as $x
| range(0; $x|length) as $i
| select( ($target[$i] - $x[$i])|length > $EPSILON )
| "\($x[$i]) vs expected value \($target[$i])" ;
def task:
x
| ., pointwise_check(tx; 1E-14) ;
task- Output:
[-0.01,1.6027903945021138,-1.6132030599055616,1.2454941213714392,-0.49098971958465953,0.06576069617523238]
Julia
Using built-in LAPACK-based linear solver (which employs partial-pivoted Gaussian elimination):
x = A \ b
Klong
elim::{[h m];h::*m::x@>*'x;
:[2>#x;x;(,h),0,:\.f({1_x}'{x-h**x%*h}'1_m)]}
subst::{[v];v::[];
{v::v,((*x)-/:[[]~v;[];v*x@1+!#v])%x@1+#v}'||'x;|v}
gauss::{subst(elim(x))}
Example, matrix taken from C version:
gauss([[1.00 0.00 0.00 0.00 0.00 0.00 -0.01]
[1.00 0.63 0.39 0.25 0.16 0.10 0.61]
[1.00 1.26 1.58 1.98 2.49 3.13 0.91]
[1.00 1.88 3.55 6.70 12.62 23.80 0.99]
[1.00 2.51 6.32 15.88 39.90 100.28 0.60]
[1.00 3.14 9.87 31.01 97.41 306.02 0.02]]
[-0.00999999999999981
1.60279039450211414
-1.6132030599055625
1.24549412137143782
-0.490989719584658025
0.0657606961752320591]
Kotlin
// version 1.1.51
val ta = arrayOf(
doubleArrayOf(1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
doubleArrayOf(1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
doubleArrayOf(1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
doubleArrayOf(1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
doubleArrayOf(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
doubleArrayOf(1.00, 3.14, 9.87, 31.01, 97.41, 306.02)
)
val tb = doubleArrayOf(-0.01, 0.61, 0.91, 0.99, 0.60, 0.02)
val tx = doubleArrayOf(
-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232
)
const val EPSILON = 1e-14 // tolerance required
fun gaussPartial(a0: Array<DoubleArray>, b0: DoubleArray): DoubleArray {
val m = b0.size
val a = Array(m) { DoubleArray(m) }
for ((i, ai) in a0.withIndex()) {
val row = ai.copyOf(m + 1)
row[m] = b0[i]
a[i] = row
}
for (k in 0 until a.size) {
var iMax = 0
var max = -1.0
for (i in k until m) {
val row = a[i]
// compute scale factor s = max abs in row
var s = -1.0
for (j in k until m) {
val e = Math.abs(row[j])
if (e > s) s = e
}
// scale the abs used to pick the pivot
val abs = Math.abs(row[k]) / s
if (abs > max) {
iMax = i
max = abs
}
}
if (a[iMax][k] == 0.0) {
throw RuntimeException("Matrix is singular.")
}
val tmp = a[k]
a[k] = a[iMax]
a[iMax] = tmp
for (i in k + 1 until m) {
for (j in k + 1..m) {
a[i][j] -= a[k][j] * a[i][k] / a[k][k]
}
a[i][k] = 0.0
}
}
val x = DoubleArray(m)
for (i in m - 1 downTo 0) {
x[i] = a[i][m]
for (j in i + 1 until m) {
x[i] -= a[i][j] * x[j]
}
x[i] /= a[i][i]
}
return x
}
fun main(args: Array<String>) {
val x = gaussPartial(ta, tb)
println(x.asList())
for ((i, xi) in x.withIndex()) {
if (Math.abs(tx[i] - xi) > EPSILON) {
println("Out of tolerance.")
println("Expected values are ${tx.asList()}")
return
}
}
}
- Output:
[-0.01, 1.6027903945021138, -1.6132030599055616, 1.2454941213714392, -0.49098971958465953, 0.06576069617523238]
Lambdatalk
{require lib_matrix}
{M.solve
{M.new [[1.00,0.00,0.00,0.00,0.00,0.00],
[1.00,0.63,0.39,0.25,0.16,0.10],
[1.00,1.26,1.58,1.98,2.49,3.13],
[1.00,1.88,3.55,6.70,12.62,23.80],
[1.00,2.51,6.32,15.88,39.90,100.28],
[1.00,3.14,9.87,31.01,97.41,306.02]]}
[-0.01,0.61,0.91,0.99,0.60,0.02]}
->
[-0.01,1.6027903945021094,-1.613203059905548,1.245494121371424,-0.49098971958465304,0.06576069617523143]
Lobster
import std
// test case from Go version at http://rosettacode.org/wiki/Gaussian_elimination
//
let ta = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
let tb = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
let tx = [-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232]
// result from above test case turns out to be correct to this tolerance.
let ε = 1.0e-14
def GaussPartial(a0, b0) -> [float], string:
// make augmented matrix
let m = length(b0)
let a = map(m): []
for(a0) ai, i:
//let ai = a0[i]
a[i] = map(m+1) j: if j < m: ai[j] else: b0[i]
// WP algorithm from Gaussian elimination page produces row-eschelon form
var i = 0
var j = 0
for(a0) ak, k:
// Find pivot for column k:
var iMax = 0
var kmax = -1.0
i = k
while i < m:
let row = a[i]
// compute scale factor s = max abs in row
var s = -1.0
j = k
while j < m:
s = max(s, abs(row[j]))
j += 1
// scale the abs used to pick the pivot
let kabs = abs(row[k]) / s
if kabs > kmax:
iMax = i
kmax = kabs
i += 1
if a[iMax][k] == 0:
return [], "singular"
// swap rows(k, i_max)
let row = a[k]
a[k] = a[iMax]
a[iMax] = row
// Do for all rows below pivot:
i = k + 1
while i < m:
// Do for all remaining elements in current row:
j = k + 1
while j <= m:
a[i][j] -= a[k][j] * (a[i][k] / a[k][k])
j += 1
// Fill lower triangular matrix with zeros:
a[i][k] = 0
i += 1
// end of WP algorithm; now back substitute to get result
let x = map(m): 0.0
i = m - 1
while i >= 0:
x[i] = a[i][m]
j = i + 1
while j < m:
x[i] -= a[i][j] * x[j]
j += 1
x[i] /= a[i][i]
i -= 1
return x, ""
def test():
let x, err = GaussPartial(ta, tb)
if err != "":
print("Error: " + err)
return
print(x)
for(x) xi, i:
if abs(tx[i]-xi) > ε:
print("out of tolerance, expected: " + tx[i] + " got: " + xi)
test()- Output:
[-0.01, 1.602790394502, -1.613203059906, 1.245494121371, -0.490989719585, 0.065760696175]
M2000 Interpreter
Faster, with accuracy of 25 decimals
module checkit {
Dim Base 1, a(6, 6), b(6)
a(1,1)= 1.00, 0.00, 0.00, 0.00, 0.00, 0.00, 1.00, 0.63, 0.39, 0.25, 0.16, 0.10, 1.00, 1.26, 1.58, 1.98, 2.49, 3.13, 1.00, 1.88, 3.55, 6.70, 12.62, 23.80, 1.00, 2.51, 6.32, 15.88, 39.90, 100.28, 1.00, 3.14, 9.87, 31.01, 97.41, 306.02
\\ remove \\ to feed next array
\\ a(1,1)=1.1,0.12,0.13,0.12,0.14,-0.12,1.21,0.63,0.39,0.25,0.16,0.1,1.03,1.26,1.58,1.98,2.49,3.13, 1.06,1.88,3.55,6.7,12.62,23.8, 1.12,2.51,6.32,15.88,39.9,100.28,1.16,3.14,9.87,31.01,97.41,306.02
for i=1 to 6 : for j=1 to 6 : a(i,j)=val(a(i,j)->Decimal) :Next j:Next i
b(1)=-0.01, 0.61, 0.91, 0.99, 0.60, 0.02
for i=1 to 6 : b(i)=val(b(i)->Decimal) :Next i
function GaussJordan(a(), b()) {
cols=dimension(a(),1)
rows=dimension(a(),2)
\\ make augmented matrix
Dim Base 1, a(cols, rows)
\\ feed array with rationals
Dim Base 1, b(Len(b()))
for diag=1 to rows {
max_row=diag
max_val=abs(a(diag, diag))
if diag<rows Then {
for ro=diag+1 to rows {
d=abs(a(ro, diag))
if d>max_val then max_row=ro : max_val=d
}
}
\\ SwapRows diag, max_row
if diag<>max_row then {
for i=1 to cols {
swap a(diag, i), a(max_row, i)
}
swap b(diag), b(max_row)
}
invd= a(diag, diag)
if diag<=cols then {
for col=diag to cols {
a(diag, col)/=invd
}
}
b(diag)/=invd
for ro=1 to rows {
d1=a(ro,diag)
d2=d1*b(diag)
if ro<>diag Then {
for col=diag to cols {a(ro, col)-=d1*a(diag, col)}
b(ro)-=d2
}
}
}
=b()
}
Function ArrayLines$(a(), leftmargin=6, maxwidth=8,decimals$="") {
\\ defualt no set decimals, can show any number
ex$={
}
const way$=", {0:"+decimals$+":-"+str$(maxwidth,"")+"}"
if dimension(a())=1 then {
m=each(a())
while m {ex$+=format$(way$,array(m))}
Insert 3, 2 ex$=string$(" ", leftmargin)
=ex$ : Break
}
for i=1 to dimension(a(),1) {
ex1$=""
for j=1 to dimension(a(),2 ) {
ex1$+=format$(way$,a(i,j))
}
Insert 1,2 ex1$=string$(" ", leftmargin)
ex$+=ex1$+{
}
}
=ex$
}
mm=GaussJordan(a(), b())
c=each(mm)
while c {
print array(c)
}
\\ check accuracy
link mm to r()
\\ prepare output document
Document out$={Algorithm using decimals
}+"Matrix A:"+ArrayLines$(a(),,,"2")+{
}+"Vector B:"+ArrayLines$(b(),,,"2")+{
}+"Solution: "+{
}
acc=25
for i=1 to dimension(a(),1)
sum=a(1,1)-a(1,1)
For j=1 to dimension(a(),2)
sum+=r(j)*a(i,j)
next j
p$=format$("Coef. {0::-2}, rounding to {1} decimal, compare {2:-5}, solution: {3}", i, acc, round(sum-b(i),acc)=0@, r(i) )
Print p$
Out$=p$+{
}
next i
Report out$
clipboard out$
}
checkitslower with accuracy of 26 decimals
Module Checkit2 {
Dim Base 1, a(6, 6), b(6)
\\ a(1,1)= 1.00, 0.00, 0.00, 0.00, 0.00, 0.00, 1.00, 0.63, 0.39, 0.25, 0.16, 0.10, 1.00, 1.26, 1.58, 1.98, 2.49, 3.13, 1.00, 1.88, 3.55, 6.70, 12.62, 23.80, 1.00, 2.51, 6.32, 15.88, 39.90, 100.28, 1.00, 3.14, 9.87, 31.01, 97.41, 306.02
a(1,1)=1.1,0.12,0.13,0.12,0.14,-0.12,1.21,0.63,0.39,0.25,0.16,0.1,1.03,1.26,1.58,1.98,2.49,3.13, 1.06,1.88,3.55,6.7,12.62,23.8, 1.12,2.51,6.32,15.88,39.9,100.28,1.16,3.14,9.87,31.01,97.41,306.02
for i=1 to 6 : for j=1 to 6 : a(i,j)=val(a(i,j)->Decimal) :Next j:Next i
b(1)=-0.01, 0.61, 0.91, 0.99, 0.60, 0.02
for i=1 to 6 : b(i)=val(b(i)->Decimal) :Next i
\\ modules/function to use rational nymbers
Module Global subd(m as array, n as array) { ' change m
link m to m()
link n to n()
if m(0)=0 then return m, 0:=-n(0), 1:=n(1) : exit
if n(0)=0 then exit
return m, 0:=m(0)*(n(1)/m(1))-n(0), 1:=n(1)
}
Function Global Inv(m as array){
link m to m()
if m(0)=0@ then =m : exit
=(m(1), m(0))
}
Function Global mul(m as array, n as array){' nothing change
link m to m()
link n to n()
if n(0)=0 or n(1)=0 then =(0@,0@) : exit
=((m(0)/n(1))*n(0),m(1))
}
Module Global mul(m as array, n as array) { ' change m
link m to m()
link n to n()
if n(0)=0 or n(1)=0 then m=(0@,0@) : exit
return m, 0:=(m(0)/n(1))*n(0)
}
Function Global Res(m as array) {
link m to m()
if m(0)=0@ then =0@: exit
=m(0)/m(1)
}
\\ GaussJordan get arrays byvalue
function GaussJordan(a(), b()) {
Function copypointer(m) { Dim a() : a()=m:=a()}
\\ we can use : def copypointer(a())=a(0),a(1)
cols=dimension(a(),1)
rows=dimension(a(),2)
Dim Base 1, a(cols, rows)
for i=1 to cols : for j=1 to rows : a(i, j)=(a(i, j), 1@) : next j : next i
def d as decimal
for j=1 to rows : b(j)=(b(j), 1@) : next j
for diag=1 to rows {
max_row=diag
max_val=abs(Res(a(diag, diag)))
if diag<rows Then {
for ro=diag+1 to rows {
d=abs(Res(a(ro, diag)))
if d>max_val then max_row=ro : max_val=d
}
}
\\ SwapRows diag, max_row
if diag<>max_row then {
for i=1 to cols {
swap a(diag, i), a(max_row, i)
}
swap b(diag), b(max_row)
}
invd= Inv(a(diag, diag))
if diag<=cols then {
for col=diag to cols {
mul a(diag, col), invd
}
}
mul b(diag), invd
for ro=1 to rows {
\\ work also d1=(a(ro,diag)(0), a(ro,diag)(1))
d1=copypointer(a(ro, diag))
if ro<>diag Then {
for col=diag to cols {subd a(ro, col), mul(d1, a(diag, col))}
subd b(ro), mul(d1, b(diag))
}
}
}
dim base 1, ans(len(b()))
for i=1 to cols {
ans(i)=res(b(i)) \\ : Print b(i) ' print pairs
}
=ans()
}
Function ArrayLines$(a(), leftmargin=6, maxwidth=8,decimals$="") {
\\ defualt no set decimals, can show any number
ex$={
}
const way$=", {0:"+decimals$+":-"+str$(maxwidth,"")+"}"
if dimension(a())=1 then {
m=each(a())
while m {ex$+=format$(way$,array(m))}
Insert 3, 2 ex$=string$(" ", leftmargin)
=ex$ : Break
}
for i=1 to dimension(a(),1) {
ex1$=""
for j=1 to dimension(a(),2 ) {
ex1$+=format$(way$,a(i,j))
}
Insert 1,2 ex1$=string$(" ", leftmargin)
ex$+=ex1$+{
}
}
=ex$
}
mm=GaussJordan(a(), b())
c=each(mm)
while c {
print array(c)
}
\\ check accuracy
link mm to r()
for i=1 to dimension(a(),1)
sum=a(1,1)-a(1,1)
For j=1 to dimension(a(),2)
sum+=r(j)*a(i,j)
next j
Print round(sum-b(i),26), b(i)
next i
\\ check accuracy
Document out$={Algorithm using pair of decimals as rational numbers
}+"Matrix A:"+ArrayLines$(a(),,,"2")+{
}+"Vector B:"+ArrayLines$(b(),,,"2")+{
}+"Solution: "+{
}
acc=26
for i=1 to dimension(a(),1)
sum=a(1,1)-a(1,1)
For j=1 to dimension(a(),2)
sum+=r(j)*a(i,j)
next j
p$=format$("Coef. {0::-2}, rounding to {1} decimal, compare {2:-5}, solution: {3}", i, acc, round(sum-b(i),acc)=0@, r(i) )
Print p$
Out$=p$+{
}
next i
Report out$
clipboard out$
}
Checkit2- Output:
Algorithm using decimals
Matrix A:
1,10, 0,12, 0,13, 0,12, 0,14, -0,12
1,21, 0,63, 0,39, 0,25, 0,16, 0,10
1,03, 1,26, 1,58, 1,98, 2,49, 3,13
1,06, 1,88, 3,55, 6,70, 12,62, 23,80
1,12, 2,51, 6,32, 15,88, 39,90, 100,28
1,16, 3,14, 9,87, 31,01, 97,41, 306,02
Vector B:
-0,01, 0,61, 0,91, 0,99, 0,60, 0,02
Solution:
Coef. 1, rounding to 26 decimal, compare True, solution: -0,0597391027501962649904316335
Coef. 2, rounding to 26 decimal, compare True, solution: 1,8501896672627829700670299288
Coef. 3, rounding to 26 decimal, compare True, solution: -1,9727833018116428175300387318
Coef. 4, rounding to 26 decimal, compare True, solution: 1,4697587750651240151384675034
Coef. 5, rounding to 26 decimal, compare True, solution: -0,5538741847821888403564152897
Coef. 6, rounding to 26 decimal, compare True, solution: 0,0723048745759411900531809852
Algorithm using pair of decimals as rational numbers
Matrix A:
1,10, 0,12, 0,13, 0,12, 0,14, -0,12
1,21, 0,63, 0,39, 0,25, 0,16, 0,10
1,03, 1,26, 1,58, 1,98, 2,49, 3,13
1,06, 1,88, 3,55, 6,70, 12,62, 23,80
1,12, 2,51, 6,32, 15,88, 39,90, 100,28
1,16, 3,14, 9,87, 31,01, 97,41, 306,02
Vector B:
-0,01, 0,61, 0,91, 0,99, 0,60, 0,02
Solution:
Coef. 1, rounding to 26 decimal, compare True, solution: -0,0597391027501962649904316335
Coef. 2, rounding to 26 decimal, compare True, solution: 1,8501896672627829700670299288
Coef. 3, rounding to 26 decimal, compare True, solution: -1,9727833018116428175300387317
Coef. 4, rounding to 26 decimal, compare True, solution: 1,4697587750651240151384675034
Coef. 5, rounding to 26 decimal, compare True, solution: -0,5538741847821888403564152897
Coef. 6, rounding to 26 decimal, compare True, solution: 0,0723048745759411900531809852
Algorithm using decimals
Matrix A:
1,00, 0,00, 0,00, 0,00, 0,00, 0,00
1,00, 0,63, 0,39, 0,25, 0,16, 0,10
1,00, 1,26, 1,58, 1,98, 2,49, 3,13
1,00, 1,88, 3,55, 6,70, 12,62, 23,80
1,00, 2,51, 6,32, 15,88, 39,90, 100,28
1,00, 3,14, 9,87, 31,01, 97,41, 306,02
Vector B:
-0,01, 0,61, 0,91, 0,99, 0,60, 0,02
Solution:
Coef. 1, rounding to 25 decimal, compare True, solution: -0,01
Coef. 2, rounding to 25 decimal, compare True, solution: 1,6027903945021139442641548525
Coef. 3, rounding to 25 decimal, compare True, solution: -1,6132030599055614189052834829
Coef. 4, rounding to 25 decimal, compare True, solution: 1,2454941213714367443882298102
Coef. 5, rounding to 25 decimal, compare True, solution: -0,4909897195846576129526569211
Coef. 6, rounding to 25 decimal, compare True, solution: 0,0657606961752320046201065486
Algorithm using pair of decimals as rational numbers
Matrix A:
1,00, 0,00, 0,00, 0,00, 0,00, 0,00
1,00, 0,63, 0,39, 0,25, 0,16, 0,10
1,00, 1,26, 1,58, 1,98, 2,49, 3,13
1,00, 1,88, 3,55, 6,70, 12,62, 23,80
1,00, 2,51, 6,32, 15,88, 39,90, 100,28
1,00, 3,14, 9,87, 31,01, 97,41, 306,02
Vector B:
-0,01, 0,61, 0,91, 0,99, 0,60, 0,02
Solution:
Coef. 1, rounding to 26 decimal, compare True, solution: -0,01
Coef. 2, rounding to 26 decimal, compare True, solution: 1,6027903945021139442641548522
Coef. 3, rounding to 26 decimal, compare True, solution: -1,6132030599055614189052834817
Coef. 4, rounding to 26 decimal, compare True, solution: 1,2454941213714367443882298085
Coef. 5, rounding to 26 decimal, compare True, solution: -0,4909897195846576129526569203
Coef. 6, rounding to 26 decimal, compare True, solution: 0,0657606961752320046201065485
Mathematica / Wolfram Language
GaussianElimination[A_?MatrixQ, b_?VectorQ] := Last /@ RowReduce[Flatten /@ Transpose[{A, b}]]
MATLAB
function [ x ] = GaussElim( A, b)
% Ensures A is n by n
sz = size(A);
if sz(1)~=sz(2)
fprintf('A is not n by n\n');
clear x;
return;
end
n = sz(1);
% Ensures b is n x 1.
if n~=sz(1)
fprintf('b is not 1 by n.\n');
return
end
x = zeros(n,1);
aug = [A b];
tempmatrix = aug;
for i=2:sz(1)
% Find maximum of row and divide by the maximum
tempmatrix(1,:) = tempmatrix(1,:)/max(tempmatrix(1,:));
% Finds the maximum in column
temp = find(abs(tempmatrix) - max(abs(tempmatrix(:,1))));
if length(temp)>2
for j=1:length(temp)-1
if j~=temp(j)
maxi = j; %maxi = column number of maximum
break;
end
end
else % length(temp)==2
maxi=1;
end
% Row swap if maxi is not 1
if maxi~=1
temp = tempmatrix(maxi,:);
tempmatrix(maxi,:) = tempmatrix(1,:);
tempmatrix(1,:) = temp;
end
% Row reducing
for j=2:length(tempmatrix)-1
tempmatrix(j,:) = tempmatrix(j,:)-tempmatrix(j,1)/tempmatrix(1,1)*tempmatrix(1,:);
if tempmatrix(j,j)==0 || isnan(tempmatrix(j,j)) || abs(tempmatrix(j,j))==Inf
fprintf('Error: Matrix is singular.\n');
clear x;
return
end
end
aug(i-1:end,i-1:end) = tempmatrix;
% Decrease matrix size
tempmatrix = tempmatrix(2:end,2:end);
end
% Backwards Substitution
x(end) = aug(end,end)/aug(end,end-1);
for i=n-1:-1:1
x(i) = (aug(i,end)-dot(aug(i,1:end-1),x))/aug(i,i);
end
end
Modula-3
This implementation defines a generic Matrix type so that the code can be used with different types. As a bonus, we implemented it to work with rings rather than fields, and tested it on two rings: the ring of integers and the ring of integers modulo 46. We include the interface of a ring modulo 46 below; the project's m3makefile (not included) is set up to automatically generates an interface and module for a matrix over each ring.
- requirements of the generic type
The Matrix needs its generic type to implement the following:
- It must have a type
T, as per Modula-3 convention. - It must have procedures
Nonzero(a: T): BOOLEAN, which indicates whetherais nonzero;Minus(a, b: T): TandTimes(a, b: T): T, which return the results of the procedures' names; andPrint(a: T)which does what the name implies.
- Matrix interface
GENERIC INTERFACE Matrix(RingElem);
(*
"RingElem" must export the following:
- a type T;
- procedures
+ "Nonzero(a: T): BOOLEAN", which indicates whether "a" is nonzero;
+ "Minus(a, b: T): T" and "Times(a, b: T): T",
which return the results you'd guess from the procedures' names; and
+ "Print(a: T)", which does what the name implies.
*)
TYPE
T <: Public;
Public = OBJECT
METHODS
init(READONLY data: ARRAY OF ARRAY OF RingElem.T): T;
(* use this to copy the entries in "data"; returns "self" *)
initDimensions(m, n: CARDINAL): T;
(* use this for an mxn matrix of random entries *)
num_rows(): CARDINAL;
(* returns the number of rows in "self" *)
num_cols(): CARDINAL;
(* returns the number of columns in "self" *)
entries(): REF ARRAY OF ARRAY OF RingElem.T;
(* returns the entries in "self" *)
triangularize();
(*
Performs Gaussian elimination in the context of a ring.
We can add scalar multiples of rows,
and we can swap rows, but we may lack multiplicative inverses,
so we cannot necessarily obtain 1 as a row's first entry.
*)
END;
PROCEDURE PrintMatrix(m: T);
(* prints the matrix row-by-row; sorry, no special padding to line up columns *)
END Matrix.
- Matrix implementation
GENERIC MODULE Matrix(RingElem);
IMPORT IO;
TYPE
REVEAL T = Public BRANDED OBJECT
rows, cols: CARDINAL;
data: REF ARRAY OF ARRAY OF RingElem.T;
OVERRIDES
init := Init;
initDimensions := InitDimensions;
num_rows := Rows;
num_cols := Columns;
entries := Entries;
triangularize := Triangularize;
END;
PROCEDURE Init(self: T; READONLY d: ARRAY OF ARRAY OF RingElem.T): T =
BEGIN
self.rows := NUMBER(d);
self.cols := NUMBER(d[0]);
self.data := NEW(REF ARRAY OF ARRAY OF RingElem.T, self.rows, self.cols);
FOR i := FIRST(d) TO LAST(d) DO
FOR j := FIRST(d[0]) TO LAST(d[0]) DO
self.data[i-FIRST(d)][j-FIRST(d[0])] := d[i][j];
END;
END;
RETURN self;
END Init;
PROCEDURE InitDimensions(self: T; r, c: CARDINAL): T =
BEGIN
self.rows := r;
self.cols := c;
self.data := NEW(REF ARRAY OF ARRAY OF RingElem.T, r, c);
RETURN self;
END InitDimensions;
PROCEDURE Rows(self: T): CARDINAL =
BEGIN
RETURN self.rows;
END Rows;
PROCEDURE Columns(self: T): CARDINAL =
BEGIN
RETURN self.cols;
END Columns;
PROCEDURE Entries(self: T): REF ARRAY OF ARRAY OF RingElem.T =
BEGIN
RETURN self.data;
END Entries;
PROCEDURE SwapRows(VAR data: ARRAY OF ARRAY OF RingElem.T; i, j: CARDINAL) =
(* swaps rows i and j of data *)
VAR
a: RingElem.T;
BEGIN
WITH Ai = data[i], Aj = data[j], m = FIRST(data[0]), n = LAST(data[0]) DO
FOR k := m TO n DO
a := Ai[k];
Ai[k] := Aj[k];
Aj[k] := a;
END;
END;
END SwapRows;
PROCEDURE PivotExists(
VAR data: ARRAY OF ARRAY OF RingElem.T;
r: CARDINAL;
VAR i: CARDINAL;
j: CARDINAL
): BOOLEAN =
(*
Returns true iff column j of data has a pivot in some row at or after r.
The row with a pivot is stored in i.
*)
VAR
searching := TRUE;
result := LAST(data) + 1;
BEGIN
i := r;
WHILE searching AND i <= LAST(data) DO
IF RingElem.Nonzero(data[i,j]) THEN
searching := FALSE;
result := i;
ELSE
INC(i);
END;
END;
RETURN NOT searching;
END PivotExists;
PROCEDURE Pivot(VAR data: ARRAY OF ARRAY OF RingElem.T; i, j, k: CARDINAL) =
(*
Pivots on row i, column j to eliminate row k, column j.
*)
BEGIN
WITH n = LAST(data[0]), Ai = data[i], Ak = data[k] DO
VAR a := Ai[j]; b := Ak[j];
BEGIN
FOR l := j TO n DO
IF RingElem.Nonzero(Ai[l]) THEN
Ak[l] := RingElem.Minus(
RingElem.Times(Ak[l], a),
RingElem.Times(Ai[l], b)
);
ELSE
Ak[l] := RingElem.Times(Ak[l], a);
END;
END;
END;
END;
END Pivot;
PROCEDURE Triangularize(self: T) =
VAR
i: CARDINAL;
r := FIRST(self.data[0]);
BEGIN
WITH data = self.data, m = FIRST(data[0]), n = LAST(data[0]) DO
FOR j := m TO n DO
IF PivotExists(data^, r, i, j) THEN
IF i # j THEN
SwapRows(data^, i, r);
END;
FOR k := r + 1 TO LAST(data^) DO
IF RingElem.Nonzero(data[k][j]) THEN
Pivot(data^, r, j, k);
END;
END;
INC(r);
END;
END;
END;
END Triangularize;
PROCEDURE PrintMatrix(self: T) =
BEGIN
WITH data = self.data DO
FOR i := FIRST(data^) TO LAST(data^) DO
IO.Put("[ ");
WITH Ai = data[i] DO
FOR j := FIRST(Ai) TO LAST(Ai) DO
RingElem.Print(Ai[j]);
IF j # LAST(Ai) THEN
IO.PutChar(' ');
END;
END;
END;
IO.Put(" ]\n");
END;
END;
END PrintMatrix;
BEGIN
END Matrix.
- interface for the ring of integers modulo an integer
INTERFACE ModularRing;
(*
Implements arithmetic modulo a nonzero integer.
Assertions check that the modulus is nonzero.
*)
TYPE
T = RECORD
value, modulus: CARDINAL;
END;
PROCEDURE Init(VAR a: T; value: INTEGER; modulus: CARDINAL);
(* initializes a to the given value and modulus *)
PROCEDURE Nonzero(n: T): BOOLEAN;
PROCEDURE Plus(a, b: T): T;
PROCEDURE Minus(a, b: T): T;
PROCEDURE Times(a, b: T): T;
PROCEDURE Print(a: T; withModulus := FALSE);
(*
when "withModulus" is "TRUE",
this adds after "a" the letter "m",
followed by the modulus
*)
END ModularRing.
- test implementation
It's fairly easy to initialize an array of types in Modula-3, but it can get cumbersome with structured types, so we wrote a procedure to convert an integer matrix to a matrix of integers modulo a number.
MODULE GaussianElimination EXPORTS Main;
IMPORT IO, ModularRing AS MR, IntMatrix AS IM, ModMatrix AS MM;
CONST
(* data to set up the matrices *)
A1 = ARRAY OF INTEGER { 2, 1, 0 };
A2 = ARRAY OF INTEGER { 1, 2, 0 };
A3 = ARRAY OF INTEGER { 0, 3, 0 };
A = ARRAY OF ARRAY OF INTEGER { A1, A2, A3 };
B1 = ARRAY OF INTEGER { 4, 8, 0, -4, 0 };
B2 = ARRAY OF INTEGER { -3, -6, 0, 9, 0 };
B3 = ARRAY OF INTEGER { 1, 3, 5, 7, 2 };
B4 = ARRAY OF INTEGER { 7, 5, 3, 1, 2 };
B = ARRAY OF ARRAY OF INTEGER { B1, B2, B3, B4 };
PROCEDURE IntToModArray(READONLY A: IM.T; VAR B: MM.T; mod: CARDINAL) =
(*
copies a two-dimensional array of integers
to a two-dimension array of integers modulo "mod"
*)
BEGIN
B := NEW(MM.T).initDimensions(A.num_rows(), A.num_cols());
WITH Adata = A.entries(), Bdata = B.entries() DO
FOR i := FIRST(Adata^) TO LAST(Adata^) DO
WITH Ai = Adata[i], Bi = Bdata[i] DO
FOR j := FIRST(Ai) TO LAST(Ai) DO
MR.Init(Bi[j], Ai[j], mod);
END;
END;
END;
END;
END IntToModArray;
VAR
M: IM.T;
N: MM.T;
BEGIN
(* triangularize the data in A *)
M := NEW(IM.T).init(A);
IO.Put("Initial A:\n");
IM.PrintMatrix(M);
IO.PutChar('\n');
M.triangularize();
IO.Put("Final A:\n");
IM.PrintMatrix(M);
IO.PutChar('\n');
IO.PutChar('\n');
(* triangularize the data in B, all computations modulo 46 *)
M := NEW(IM.T).init(B);
IntToModArray(M, N, 46);
IO.Put("Initial B:\n");
MM.PrintMatrix(N);
IO.PutChar('\n');
N.triangularize();
IO.Put("Final B:\n");
MM.PrintMatrix(N);
IO.PutChar('\n');
END GaussianElimination.
- Output:
Initial A: [ 2 1 0 ] [ 1 2 0 ] [ 0 3 0 ] Final A: [ 2 1 0 ] [ 0 3 0 ] [ 0 0 0 ] Initial B: [ 4 8 0 42 0 ] [ 43 40 0 9 0 ] [ 1 3 5 7 2 ] [ 7 5 3 1 2 ] Final B: [ 4 8 0 42 0 ] [ 0 4 20 32 8 ] [ 0 0 32 38 44 ] [ 0 0 0 24 0 ]
Nim
const Eps = 1e-14 # Tolerance required.
type
Vector[N: static Positive] = array[N, float]
Matrix[M, N: static Positive] = array[M, Vector[N]]
SquareMatrix[N: static Positive] = Matrix[N, N]
func gaussPartialScaled(a: SquareMatrix; b: Vector): Vector =
doAssert a.N == b.N, "matrix and vector have incompatible dimensions"
const N = a.N
var m: Matrix[N, N + 1]
for i, row in a:
m[i][0..<N] = row
m[i][N] = b[i]
for k in 0..<N:
var imax = 0
var vmax = -1.0
for i in k..<N:
# Compute scale factor s = max abs in row.
var s = -1.0
for j in k..N:
let e = abs(m[i][j])
if e > s: s = e
# Scale the abs used to pick the pivot.
let val = abs(m[i][k]) / s
if val > vmax:
imax = i
vmax = val
if m[imax][k] == 0:
raise newException(ValueError, "matrix is singular")
swap m[imax], m[k]
for i in (k + 1)..<N:
for j in (k + 1)..N:
m[i][j] -= m[k][j] * m[i][k] / m[k][k]
m[i][k] = 0
for i in countdown(N - 1, 0):
result[i] = m[i][N]
for j in (i + 1)..<N:
result[i] -= m[i][j] * result[j]
result[i] /= m[i][i]
#———————————————————————————————————————————————————————————————————————————————————————————————————
let a: SquareMatrix[6] = [[1.00, 0.00, 0.00, 0.00, 0.00, 0.00],
[1.00, 0.63, 0.39, 0.25, 0.16, 0.10],
[1.00, 1.26, 1.58, 1.98, 2.49, 3.13],
[1.00, 1.88, 3.55, 6.70, 12.62, 23.80],
[1.00, 2.51, 6.32, 15.88, 39.90, 100.28],
[1.00, 3.14, 9.87, 31.01, 97.41, 306.02]]
let b: Vector[6] = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
let refx: Vector[6] = [-0.01, 1.602790394502114, -1.6132030599055613,
1.2454941213714368, -0.4909897195846576, 0.065760696175232]
let x = gaussPartialScaled(a, b)
echo x
for i, xi in x:
if abs(xi - refx[i]) > Eps:
echo "Out of tolerance."
echo "Expected values are ", refx
break
- Output:
[-0.01, 1.602790394502114, -1.613203059905562, 1.245494121371439, -0.4909897195846595, 0.06576069617523238]
OCaml
The OCaml stdlib is fairly lean, so these stand-alone solutions often need to include support functions which would be part of a codebase, like these...
module Array = struct
include Array
(* Computes: f a.(0) + f a.(1) + ... where + is 'g'. *)
let foldmap g f a =
let n = Array.length a in
let rec aux acc i =
if i >= n then acc else aux (g acc (f a.(i))) (succ i)
in aux (f a.(0)) 1
(* like the stdlib fold_left, but also provides index to f *)
let foldi_left f x a =
let r = ref x in
for i = 0 to length a - 1 do
r := f i !r (unsafe_get a i)
done;
!r
end
let foldmap_range g f (a,b) =
let rec aux acc n =
let n = succ n in
if n > b then acc else aux (g acc (f n)) n
in aux (f a) a
let fold_range f init (a,b) =
let rec aux acc n =
if n > b then acc else aux (f acc n) (succ n)
in aux init a
The solver:
(* Some less-general support functions for 'solve'. *)
let swap_elem m i j = let x = m.(i) in m.(i) <- m.(j); m.(j) <- x
let maxtup a b = if (snd a) > (snd b) then a else b
let augmented_matrix m b =
Array.(init (length m) ( fun i -> append m.(i) [|b.(i)|] ))
(* Solve Ax=b for x, using gaussian elimination with scaled partial pivot,
* and then back-substitution of the resulting row-echelon matrix. *)
let solve m b =
let n = Array.length m in
let n' = pred n in (* last index = n-1 *)
let s = Array.(map (foldmap max abs_float) m) in (* scaling vector *)
let a = augmented_matrix m b in
for k = 0 to pred n' do
(* Scaled partial pivot, to preserve precision *)
let pair i = (i, abs_float a.(i).(k) /. s.(i)) in
let i_max,v = foldmap_range maxtup pair (k,n') in
if v < epsilon_float then failwith "Matrix is singular.";
swap_elem a k i_max;
swap_elem s k i_max;
(* Eliminate one column *)
for i = succ k to n' do
let tmp = a.(i).(k) /. a.(k).(k) in
for j = succ k to n do
a.(i).(j) <- a.(i).(j) -. tmp *. a.(k).(j);
done
done
done;
(* Backward substitution; 'b' is in the 'nth' column of 'a' *)
let x = Array.copy b in (* just a fresh array of the right size and type *)
for i = n' downto 0 do
let minus_dprod t j = t -. x.(j) *. a.(i).(j) in
x.(i) <- fold_range minus_dprod a.(i).(n) (i+1,n') /. a.(i).(i);
done;
x
Example data...
let a =
[| [| 1.00; 0.00; 0.00; 0.00; 0.00; 0.00 |];
[| 1.00; 0.63; 0.39; 0.25; 0.16; 0.10 |];
[| 1.00; 1.26; 1.58; 1.98; 2.49; 3.13 |];
[| 1.00; 1.88; 3.55; 6.70; 12.62; 23.80 |];
[| 1.00; 2.51; 6.32; 15.88; 39.90; 100.28 |];
[| 1.00; 3.14; 9.87; 31.01; 97.41; 306.02 |] |]
let b = [| -0.01; 0.61; 0.91; 0.99; 0.60; 0.02 |]
In the REPL, the solution is:
# let x = solve a b;;
val x : float array =
[|-0.0100000000000000991; 1.60279039450210536; -1.61320305990553226;
1.24549412137140547; -0.490989719584644546; 0.0657606961752301433|]
Further, let's define multiplication and subtraction to check our results...
let mul m v =
Array.mapi (fun i u ->
Array.foldi_left (fun j sum uj ->
sum +. uj *. v.(j)
) 0. u
) m
let sub u v = Array.mapi (fun i e -> e -. v.(i)) u
Now 'x' can be plugged into the equation to calculate the residual:
# let residual = sub b (mul a x);;
val residual : float array =
[|9.8879238130678e-17; 1.11022302462515654e-16; 2.22044604925031308e-16;
8.88178419700125232e-16; -5.5511151231257827e-16; 4.26741975090294545e-16|]
PARI/GP
If A and B have floating-point numbers (t_REALs) then the following uses Gaussian elimination:
matsolve(A,B)If the entries are integers, then p-adic lifting (Dixon 1982) is used instead.
Perl
use Math::Matrix;
my $a = Math::Matrix->new([0,1,0],
[0,0,1],
[2,0,1]);
my $b = Math::Matrix->new([1],
[2],
[4]);
my $x = $a->concat($b)->solve;
print $x;
Math::Matrix solve() expects the column vector to be an extra column in the matrix, hence concat(). Putting not just a column there but a whole identity matrix (making Nx2N) is how its invert() is implemented. Note that solve() doesn't notice singular matrices and still gives a return when there is in fact no solution to Ax=B.
Phix
with javascript_semantics function gauss_eliminate(sequence a, b) {a, b} = deep_copy({a,b}) integer n = length(b) atom tmp for col=1 to n do integer m = col atom mx = a[m][m] for i=col+1 to n do tmp = abs(a[i][col]) if tmp>mx then {m,mx} = {i,tmp} end if end for if col!=m then {a[col],a[m]} = {a[m],a[col]} {b[col],b[m]} = {b[m],b[col]} end if for i=col+1 to n do tmp = a[i][col]/a[col][col] for j=col+1 to n do a[i][j] -= tmp*a[col][j] end for a[i][col] = 0 b[i] -= tmp*b[col] end for end for sequence x = repeat(0,n) for col=n to 1 by -1 do tmp = b[col] for j=n to col+1 by -1 do tmp -= x[j]*a[col][j] end for x[col] = tmp/a[col][col] end for return x end function constant a = {{1.00, 0.00, 0.00, 0.00, 0.00, 0.00}, {1.00, 0.63, 0.39, 0.25, 0.16, 0.10}, {1.00, 1.26, 1.58, 1.98, 2.49, 3.13}, {1.00, 1.88, 3.55, 6.70, 12.62, 23.80}, {1.00, 2.51, 6.32, 15.88, 39.90, 100.28}, {1.00, 3.14, 9.87, 31.01, 97.41, 306.02}}, b = {-0.01, 0.61, 0.91, 0.99, 0.60, 0.02} pp(gauss_eliminate(a, b))
- Output:
{-0.01,1.602790395,-1.61320306,1.245494121,-0.4909897196,0.06576069618}
PHP
function swap_rows(&$a, &$b, $r1, $r2)
{
if ($r1 == $r2) return;
$tmp = $a[$r1];
$a[$r1] = $a[$r2];
$a[$r2] = $tmp;
$tmp = $b[$r1];
$b[$r1] = $b[$r2];
$b[$r2] = $tmp;
}
function gauss_eliminate($A, $b, $N)
{
for ($col = 0; $col < $N; $col++)
{
$j = $col;
$max = $A[$j][$j];
for ($i = $col + 1; $i < $N; $i++)
{
$tmp = abs($A[$i][$col]);
if ($tmp > $max)
{
$j = $i;
$max = $tmp;
}
}
swap_rows($A, $b, $col, $j);
for ($i = $col + 1; $i < $N; $i++)
{
$tmp = $A[$i][$col] / $A[$col][$col];
for ($j = $col + 1; $j < $N; $j++)
{
$A[$i][$j] -= $tmp * $A[$col][$j];
}
$A[$i][$col] = 0;
$b[$i] -= $tmp * $b[$col];
}
}
$x = array();
for ($col = $N - 1; $col >= 0; $col--)
{
$tmp = $b[$col];
for ($j = $N - 1; $j > $col; $j--)
{
$tmp -= $x[$j] * $A[$col][$j];
}
$x[$col] = $tmp / $A[$col][$col];
}
return $x;
}
function test_gauss()
{
$a = array(
array(1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
array(1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
array(1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
array(1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
array(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
array(1.00, 3.14, 9.87, 31.01, 97.41, 306.02)
);
$b = array( -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 );
$x = gauss_eliminate($a, $b, 6);
ksort($x);
print_r($x);
}
test_gauss();
- Output:
Array
(
[0] => -0.01
[1] => 1.6027903945021
[2] => -1.6132030599055
[3] => 1.2454941213714
[4] => -0.49098971958463
[5] => 0.065760696175228
)
PL/I
Solve: procedure options (main); /* 11 January 2014 */
declare n fixed binary;
put ('Program to solve n simultaneous equations of the form Ax = b. Please type n:' );
get (n);
begin;
declare (A(n, n), b(n), x(n)) float(18);
declare (SA(n,n), Sb(n)) float (18);
declare i fixed binary;
put skip list ('Please type A:');
get (a);
put skip list ('Please type the right-hand sides, b:');
get (b);
SA = A; Sb = b;
put skip list ('The equations are:');
do i = 1 to n;
put skip edit (A(i,*), b(i)) (f(5), x(1));
end;
call Gauss_elimination (A, b);
call Backward_substitution (A, b, x);
put skip list ('Solutions:'); put skip data (x);
/* Check solutions: */
put skip list ('Residuals:');
do i = 1 to n;
put skip list (sum(SA(i,*) * x(*)) - Sb(i));
end;
end;
Gauss_elimination: procedure (A, b) options (reorder); /* Triangularise */
declare (A(*,*), b(*)) float(18);
declare n fixed binary initial (hbound(A, 1));
declare (i, j, k) fixed binary;
declare t float(18);
do j = 1 to n;
do i = j+1 to n; /* For each of the rows beneath the current (pivot) row. */
t = A(j,j) / A(i,j);
do k = j+1 to n; /* Subtract a multiple of row i from row j. */
A(i,k) = A(j,k) - t*A(i,k);
end;
b(i) = b(j) - t*b(i); /* ... and the right-hand side. */
end;
end;
end Gauss_elimination;
Backward_substitution: procedure (A, b, x) options (reorder);
declare (A(*,*), b(*), x(*)) float(18);
declare t float(18);
declare n fixed binary initial (hbound(A, 1));
declare (i, j) fixed binary;
x(n) = b(n) / a(n,n);
do j = n-1 to 1 by -1;
t = 0;
do i = j+1 to n;
t = t + a(j,i)*x(i);
end;
x(j) = (b(j) - t) / a(j,j);
end;
end Backward_substitution;
end Solve;- Output:
Program to solve n simultaneous equations of the form Ax = b. Please type n:
Please type A:
Please type the right-hand sides, b:
The equations are:
1 2 3 14
2 1 3 13
3 -2 -1 -4
Solutions:
X(1)= 1.00000000000000000E+0000 X(2)= 2.00000000000000000E+0000
X(3)= 3.00000000000000000E+0000;
Residuals:
0.00000000000000000E+0000
0.00000000000000000E+0000
0.00000000000000000E+0000
PowerShell
Gauss
function gauss($a,$b) {
$n = $a.count
for ($k = 0; $k -lt $n; $k++) {
$lmax, $max = $k, [Math]::Abs($a[$k][$k])
for ($l = $k+1; $l -lt $n; $l++) {
$tmp = [Math]::Abs($a[$l][$k])
if($max -lt $tmp) {
$max, $lmax = $tmp, $l
}
}
if ($k -ne $lmax) {
$a[$k], $a[$lmax] = $a[$lmax], $a[$k]
$b[$k], $b[$lmax] = $b[$lmax], $b[$k]
}
$akk = $a[$k][$k]
for ($i = $k+1; $i -lt $n; $i++){
$aik = $a[$i][$k]
for ($j = $k; $j -lt $n; $j++) {
$a[$i][$j] = $a[$i][$j]*$akk - $a[$k][$j]*$aik
}
$b[$i] = $b[$i]*$akk - $b[$k]*$aik
}
}
for ($i = $n-1; $i -ge 0; $i--) {
for ($j = $i+1; $j -lt $n; $j++) {
$b[$i] -= $b[$j]*$a[$i][$j]
}
$b[$i] = $b[$i]/$a[$i][$i]
}
$b
}
function show($a) {
if($a) {
0..($a.Count - 1) | foreach{ if($a[$_]){"$($a[$_][0..($a[$_].count -1)])"}else{""} }
}
}
$a =(
@(1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
@(1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
@(1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
@(1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
@(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
@(1.00, 3.14, 9.87, 31.01, 97.41, 306.02)
)
"a ="
show $a
""
$b = @(-0.01, 0.61, 0.91, 0.99, 0.60, 0.02)
"b ="
$b
""
"x ="
gauss $a $b
Output:
a = 1 0 0 0 0 0 1 0.63 0.39 0.25 0.16 0.1 1 1.26 1.58 1.98 2.49 3.13 1 1.88 3.55 6.7 12.62 23.8 1 2.51 6.32 15.88 39.9 100.28 1 3.14 9.87 31.01 97.41 306.02 b = -0.01 0.61 0.91 0.99 0.6 0.02 x = -0.01 1.60279039450213 -1.6132030599056 1.24549412137148 -0.490989719584674 0.0657606961752342
Gauss-Jordan
function gauss-jordan($a,$b) {
$n = $a.count
for ($k = 0; $k -lt $n; $k++) {
$lmax, $max = $k, [Math]::Abs($a[$k][$k])
for ($l = $k+1; $l -lt $n; $l++) {
$tmp = [Math]::Abs($a[$l][$k])
if($max -lt $tmp) {
$max, $lmax = $tmp, $l
}
}
if ($k -ne $lmax) {
$a[$k], $a[$lmax] = $a[$lmax], $a[$k]
$b[$k], $b[$lmax] = $b[$lmax], $b[$k]
}
$akk = $a[$k][$k]
for ($j = $k; $j -lt $n; $j++) {$a[$k][$j] /= $akk}
$b[$k] /= $akk
for ($i = 0; $i -lt $n; $i++){
if ($i -ne $k) {
$aik = $a[$i][$k]
for ($j = $k; $j -lt $n; $j++) {
$a[$i][$j] = $a[$i][$j] - $a[$k][$j]*$aik
}
$b[$i] = $b[$i] - $b[$k]*$aik
}
}
}
$b
}
function show($a) {
if($a) {
0..($a.Count - 1) | foreach{ if($a[$_]){"$($a[$_][0..($a[$_].count -1)])"}else{""} }
}
}
$a =(
@(1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
@(1.00, 0.63, 0.39, 0.25, 0.16, 0.10),
@(1.00, 1.26, 1.58, 1.98, 2.49, 3.13),
@(1.00, 1.88, 3.55, 6.70, 12.62, 23.80),
@(1.00, 2.51, 6.32, 15.88, 39.90, 100.28),
@(1.00, 3.14, 9.87, 31.01, 97.41, 306.02)
)
"a ="
show $a
""
$b = @(-0.01, 0.61, 0.91, 0.99, 0.60, 0.02)
"b ="
$b
""
"x ="
gauss-jordan $a $b
Output:
a = 1 0 0 0 0 0 1 0.63 0.39 0.25 0.16 0.1 1 1.26 1.58 1.98 2.49 3.13 1 1.88 3.55 6.7 12.62 23.8 1 2.51 6.32 15.88 39.9 100.28 1 3.14 9.87 31.01 97.41 306.02 b = -0.01 0.61 0.91 0.99 0.6 0.02 x = -0.01 1.60279039450211 -1.61320305990556 1.24549412137144 -0.490989719584659 0.0657606961752323
Python
# The 'gauss' function takes two matrices, 'a' and 'b', with 'a' square, and it return the determinant of 'a' and a matrix 'x' such that a*x = b.
# If 'b' is the identity, then 'x' is the inverse of 'a'.
import copy
from fractions import Fraction
def gauss(a, b):
a = copy.deepcopy(a)
b = copy.deepcopy(b)
n = len(a)
p = len(b[0])
det = 1
for i in range(n - 1):
k = i
for j in range(i + 1, n):
if abs(a[j][i]) > abs(a[k][i]):
k = j
if k != i:
a[i], a[k] = a[k], a[i]
b[i], b[k] = b[k], b[i]
det = -det
for j in range(i + 1, n):
t = a[j][i]/a[i][i]
for k in range(i + 1, n):
a[j][k] -= t*a[i][k]
for k in range(p):
b[j][k] -= t*b[i][k]
for i in range(n - 1, -1, -1):
for j in range(i + 1, n):
t = a[i][j]
for k in range(p):
b[i][k] -= t*b[j][k]
t = 1/a[i][i]
det *= a[i][i]
for j in range(p):
b[i][j] *= t
return det, b
def zeromat(p, q):
return [[0]*q for i in range(p)]
def matmul(a, b):
n, p = len(a), len(a[0])
p1, q = len(b), len(b[0])
if p != p1:
raise ValueError("Incompatible dimensions")
c = zeromat(n, q)
for i in range(n):
for j in range(q):
c[i][j] = sum(a[i][k]*b[k][j] for k in range(p))
return c
def mapmat(f, a):
return [list(map(f, v)) for v in a]
def ratmat(a):
return mapmat(Fraction, a)
# As an example, compute the determinant and inverse of 3x3 magic square
a = [[2, 9, 4], [7, 5, 3], [6, 1, 8]]
b = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
det, c = gauss(a, b)
det
-360.0
c
[[-0.10277777777777776, 0.18888888888888888, -0.019444444444444438],
[0.10555555555555554, 0.02222222222222223, -0.061111111111111116],
[0.0638888888888889, -0.14444444444444446, 0.14722222222222223]]
# Check product
matmul(a, c)
[[1.0, 0.0, 0.0], [5.551115123125783e-17, 1.0, 0.0],
[1.1102230246251565e-16, -2.220446049250313e-16, 1.0]]
# Same with fractions, so the result is exact
det, c = gauss(ratmat(a), ratmat(b))
det
Fraction(-360, 1)
c
[[Fraction(-37, 360), Fraction(17, 90), Fraction(-7, 360)],
[Fraction(19, 180), Fraction(1, 45), Fraction(-11, 180)],
[Fraction(23, 360), Fraction(-13, 90), Fraction(53, 360)]]
matmul(a, c)
[[Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)],
[Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)],
[Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)]]
Using numpy
$ python3
Python 3.6.0 |Anaconda custom (64-bit)| (default, Dec 23 2016, 12:22:00)
[GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> # https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.solve.html
>>> import numpy.linalg
>>> a = [[2, 9, 4], [7, 5, 3], [6, 1, 8]]
>>> b = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> numpy.linalg.solve(a,b)
array([[-0.10277778, 0.18888889, -0.01944444],
[ 0.10555556, 0.02222222, -0.06111111],
[ 0.06388889, -0.14444444, 0.14722222]])
>>>
R
Here 'b' is a matrix. Partial pivoting is used, and the determinant of 'a' is returned as well.
gauss <- function(a, b) {
n <- nrow(a)
det <- 1
for (i in seq_len(n - 1)) {
j <- which.max(a[i:n, i]) + i - 1
if (j != i) {
a[c(i, j), i:n] <- a[c(j, i), i:n]
b[c(i, j), ] <- b[c(j, i), ]
det <- -det
}
k <- seq(i + 1, n)
for (j in k) {
s <- a[[j, i]] / a[[i, i]]
a[j, k] <- a[j, k] - s * a[i, k]
b[j, ] <- b[j, ] - s * b[i, ]
}
}
for (i in seq(n, 1)) {
if (i < n) {
for (j in seq(i + 1, n)) {
b[i, ] <- b[i, ] - a[[i, j]] * b[j, ]
}
}
b[i, ] <- b[i, ] / a[[i, i]]
det <- det * a[[i, i]]
}
list(x=b, det=det)
}
a <- matrix(c(2, 9, 4, 7, 5, 3, 6, 1, 8), 3, 3, byrow=T)
gauss(a, diag(3))
- Output:
$x
[,1] [,2] [,3]
[1,] -0.10277778 0.18888889 -0.01944444
[2,] 0.10555556 0.02222222 -0.06111111
[3,] 0.06388889 -0.14444444 0.14722222
$det
[1] -360
Racket
#lang racket
(require math/matrix)
(define A
(matrix [[1.00 0.00 0.00 0.00 0.00 0.00]
[1.00 0.63 0.39 0.25 0.16 0.10]
[1.00 1.26 1.58 1.98 2.49 3.13]
[1.00 1.88 3.55 6.70 12.62 23.80]
[1.00 2.51 6.32 15.88 39.90 100.28]
[1.00 3.14 9.87 31.01 97.41 306.02]]))
(define b (col-matrix [-0.01 0.61 0.91 0.99 0.60 0.02]))
(matrix-solve A b)
- Output:
#<array
'#(6 1)
#[-0.01
1.602790394502109
-1.613203059905556
1.2454941213714346
-0.4909897195846582
0.06576069617523222]>
Raku
(formerly Perl 6)
Gaussian elimination results in a matrix in row echelon form. Gaussian elimination with back-substitution (also known as Gauss-Jordan elimination) results in a matrix in reduced row echelon form. That being the case, we can reuse much of the code from the Reduced row echelon form task. Raku stores and does calculations on decimal numbers within its limit of precision using Rational numbers by default, meaning the calculations are exact.
sub gauss-jordan-solve (@a, @b) {
@b.kv.map: { @a[$^k].append: $^v };
@a.&rref[*]»[*-1];
}
# reduced row echelon form
sub rref (@m) {
my ($lead, $rows, $cols) = 0, @m, @m[0];
for ^$rows -> $r {
$lead < $cols or return @m;
my $i = $r;
until @m[$i;$lead] {
++$i == $rows or next;
$i = $r;
++$lead == $cols and return @m;
}
@m[$i, $r] = @m[$r, $i] if $r != $i;
@m[$r] »/=» $ = @m[$r;$lead];
for ^$rows -> $n {
next if $n == $r;
@m[$n] »-=» @m[$r] »×» (@m[$n;$lead] // 0);
}
++$lead;
}
@m
}
sub rat-or-int ($num) {
return $num unless $num ~~ Rat;
return $num.narrow if $num.narrow ~~ Int;
$num.nude.join: '/';
}
sub say-it ($message, @array, $fmt = " %8s") {
say "\n$message";
$_».&rat-or-int.fmt($fmt).put for @array;
}
my @a = (
[ 1.00, 0.00, 0.00, 0.00, 0.00, 0.00 ],
[ 1.00, 0.63, 0.39, 0.25, 0.16, 0.10 ],
[ 1.00, 1.26, 1.58, 1.98, 2.49, 3.13 ],
[ 1.00, 1.88, 3.55, 6.70, 12.62, 23.80 ],
[ 1.00, 2.51, 6.32, 15.88, 39.90, 100.28 ],
[ 1.00, 3.14, 9.87, 31.01, 97.41, 306.02 ],
);
my @b = ( -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 );
say-it 'A matrix:', @a, "%6.2f";
say-it 'or, A in exact rationals:', @a;
say-it 'B matrix:', @b, "%6.2f";
say-it 'or, B in exact rationals:', @b;
say-it 'x matrix:', (my @gj = gauss-jordan-solve @a, @b), "%16.12f";
say-it 'or, x in exact rationals:', @gj, "%28s";
- Output:
A matrix:
1.00 0.00 0.00 0.00 0.00 0.00
1.00 0.63 0.39 0.25 0.16 0.10
1.00 1.26 1.58 1.98 2.49 3.13
1.00 1.88 3.55 6.70 12.62 23.80
1.00 2.51 6.32 15.88 39.90 100.28
1.00 3.14 9.87 31.01 97.41 306.02
or, A in exact rationals:
1 0 0 0 0 0
1 63/100 39/100 1/4 4/25 1/10
1 63/50 79/50 99/50 249/100 313/100
1 47/25 71/20 67/10 631/50 119/5
1 251/100 158/25 397/25 399/10 2507/25
1 157/50 987/100 3101/100 9741/100 15301/50
B matrix:
-0.01
0.61
0.91
0.99
0.60
0.02
or, B in exact rationals:
-1/100
61/100
91/100
99/100
3/5
1/50
x matrix:
-0.010000000000
1.602790394502
-1.613203059906
1.245494121371
-0.490989719585
0.065760696175
or, x in exact rationals:
-1/100
655870882787/409205648497
-660131804286/409205648497
509663229635/409205648497
-200915766608/409205648497
26909648324/409205648497
REXX
version 1
/* REXX ---------------------------------------------------------------
* 07.08.2014 Walter Pachl translated from PL/I)
* improved to get integer results for, e.g. this input:
-6 -18 13 6 -6 -15 -2 -9 -231
2 20 9 2 16 -12 -18 -5 647
23 18 -14 -14 -1 16 25 -17 -907
-8 -1 -19 4 3 -14 23 8 248
25 20 -6 15 0 -10 9 17 1316
-13 -1 3 5 -2 17 14 -12 -1080
19 24 -21 -5 -19 0 -24 -17 1006
20 -3 -14 -16 -23 -25 -15 20 1496
*--------------------------------------------------------------------*/
Numeric Digits 20
Parse Arg t
n=3
Parse Value '1 2 3 14' With a.1.1 a.1.2 a.1.3 b.1
Parse Value '2 1 3 13' With a.2.1 a.2.2 a.2.3 b.2
Parse Value '3 -2 -1 -4' With a.3.1 a.3.2 a.3.3 b.3
If t=6 Then Do
n=6
Parse Value '1.00 0.00 0.00 0.00 0.00 0.00 ' With a.1.1 a.1.2 a.1.3 a.1.4 a.1.5 a.1.6 .
Parse Value '1.00 0.63 0.39 0.25 0.16 0.10 ' With a.2.1 a.2.2 a.2.3 a.2.4 a.2.5 a.2.6 .
Parse Value '1.00 1.26 1.58 1.98 2.49 3.13 ' With a.3.1 a.3.2 a.3.3 a.3.4 a.3.5 a.3.6 .
Parse Value '1.00 1.88 3.55 6.70 12.62 23.80 ' With a.4.1 a.4.2 a.4.3 a.4.4 a.4.5 a.4.6 .
Parse Value '1.00 2.51 6.32 15.88 39.90 100.28' With a.5.1 a.5.2 a.5.3 a.5.4 a.5.5 a.5.6 .
Parse Value '1.00 3.14 9.87 31.01 97.41 306.02' With a.6.1 a.6.2 a.6.3 a.6.4 a.6.5 a.6.6 .
Parse Value '-0.01 0.61 0.91 0.99 0.60 0.02' With b.1 b.2 b.3 b.4 b.5 b.6 .
End
Do i=1 To n
Do j=1 To n
sa.i.j=a.i.j
End
sb.i=b.i
End
Say 'The equations are:'
do i = 1 to n;
ol=''
Do j=1 To n
ol=ol format(a.i.j,4,4)
End
ol=ol' 'format(b.i,4,4)
Say ol
end
call Gauss_elimination
call Backward_substitution
Say 'Solutions:'
Do i=1 To n
Say 'x('i')='||x.i
End
/* Check solutions: */
Say 'Residuals:'
do i = 1 to n
res=0
Do j=1 To n
res=res+(sa.i.j*x.j)
End
res=res-sb.i
Say 'res('i')='res
End
Exit
Gauss_elimination:
Do j=1 to n-1
ma=a.j.j
Do ja=j+1 To n
mb=a.ja.j
Do i=1 To n
new=a.j.i*mb-a.ja.i*ma
a.ja.i=new
End
b.ja=b.j*mb-b.ja*ma
End
End
Return
Backward_substitution:
x.n = b.n / a.n.n
do j = n-1 to 1 by -1
t = 0
do i = j+1 to n
t = t + a.j.i*x.i
end
x.j = (b.j - t) / a.j.j
end
Return
- Output:
The equations are:
1.0000 2.0000 3.0000 14.0000
2.0000 1.0000 3.0000 13.0000
3.0000 -2.0000 -1.0000 -4.0000
Solutions:
x(1)=1
x(2)=2
x(3)=3
Residuals:
res(1)=0
res(2)=0
res(3)=0
and with test data from PHP
The equations are:
1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0100
1.0000 0.6300 0.3900 0.2500 0.1600 0.1000 0.6100
1.0000 1.2600 1.5800 1.9800 2.4900 3.1300 0.9100
1.0000 1.8800 3.5500 6.7000 12.6200 23.8000 0.9900
1.0000 2.5100 6.3200 15.8800 39.9000 100.2800 0.6000
1.0000 3.1400 9.8700 31.0100 97.4100 306.0200 0.0200
Solutions:
x(1)=-0.01
x(2)=1.6027903945021139463
x(3)=-1.6132030599055614262
x(4)=1.2454941213714367527
x(5)=-0.49098971958465761669
x(6)=0.065760696175232005188
Residuals:
res(1)=0
res(2)=0.00000000000000000001
res(3)=-0.00000000000000000016
res(4)=0
res(5)=-0.0000000000000000017
res(6)=0.000000000000000001
version 2
(Data was placed into a file instead of placing the data into the REXX program.)
Programming note: with the large precision (numeric digits 1000), the residuals were insignificant.
Only 8 (fractional) decimal digits were used for the output display.
/*REXX program solves Ax=b with Gaussian elimination and backwards substitution. */
numeric digits 1000 /*heavy─duty decimal digits precision. */
parse arg iFID . /*obtain optional argument from the CL.*/
if iFID=='' | iFID=="," then iFID= 'GAUSS_E.DAT' /*Not specified? Then use the default.*/
do rec=1 while lines(iFID) \== 0 /*read the equation sets. */
#= 0 /*the number of equations (so far). */
do $=1 while lines(iFID) \== 0 /*process the equation. */
z= linein(iFID); if z='' then leave /*Is this a blank line? end─of─data.*/
if $==1 then do; say; say center(' equations ', 75, "▓"); say
end /* [↑] if 1st equation, then show hdr.*/
say z /*display an equation to the terminal. */
if left(space(z), 1)=='*' then iterate /*Is this a comment? Then ignore it.*/
#= # + 1; n= words(z) - 1 /*assign equation #; calculate # items.*/
do e=1 for n; a.#.e= word(z, e)
end /*e*/ /* [↑] process A numbers. */
b.#= word(z, n + 1) /* ◄─── " B " */
end /*$*/
if #\==0 then call Gauss_elim /*Not zero? Then display the results. */
end /*rec*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Gauss_elim: say; do j=1 for n; jp= j + 1
do i=jp to n; _= a.j.j / a.i.j
do k=jp to n; a.i.k= a.j.k - _ * a.i.k
end /*k*/
b.i= b.j - _ * b.i
end /*i*/
end /*j*/
x.n= b.n / a.n.n
do j=n-1 to 1 by -1; _= 0
do i=j+1 to n; _= _ + a.j.i * x.i
end /*i*/
x.j= (b.j - _) / a.j.j
end /*j*/ /* [↑] uses backwards substitution. */
numeric digits /*for the display, only use 8 digits. */
say center('solution', 75, "═"); say /*a title line for articulated output. */
do o=1 for n; say right('x['o"] = ", 38) left('', x.o>=0) x.o/1
end /*o*/
return
- input file : GAUSS_E.DAT
* a1 a2 a3 b
* ─── ─── ─── ───
1 2 3 14
2 1 3 13
3 -2 -1 -4
* a1 a2 a3 a4 a5 a6 b
* ─────── ─────── ─────── ─────── ─────── ─────── ───────
1 0 0 0 0 0 -0.01
1 0.63 0.39 0.25 0.16 0.10 0.61
1 1.26 1.58 1.98 2.49 3.13 0.91
1 1.88 3.55 6.70 12.62 23.80 0.99
1 2.51 6.32 15.88 39.90 100.28 0.60
1 3.14 9.87 31.01 97.41 306.02 0.02
- output when using the default input file:
▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ equations ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
* a1 a2 a3 b
* ─── ─── ─── ───
1 2 3 14
2 1 3 13
3 -2 -1 -4
═════════════════════════════════solution══════════════════════════════════
x[1] = 1
x[2] = 2
x[3] = 3
▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ equations ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
* a1 a2 a3 a4 a5 a6 b
* ─────── ─────── ─────── ─────── ─────── ─────── ───────
1 0 0 0 0 0 -0.01
1 0.63 0.39 0.25 0.16 0.10 0.61
1 1.26 1.58 1.98 2.49 3.13 0.91
1 1.88 3.55 6.70 12.62 23.80 0.99
1 2.51 6.32 15.88 39.90 100.28 0.60
1 3.14 9.87 31.01 97.41 306.02 0.02
═════════════════════════════════solution══════════════════════════════════
x[1] = -0.01
x[2] = 1.6027904
x[3] = -1.6132031
x[4] = 1.2454941
x[5] = -0.49098972
x[6] = 0.065760696
version 3
This is the same as version 2, but in addition, it also shows the residuals.
Code was added to this program version to keep a copy of the original A.i.k and B.# arrays (for calculating the
residuals).
Also added was the rounding the residual numbers to zero if the number of significant decimal digits was ≤ 5% of
the number of significant fractional decimal digits (in this case, 5% of 1,000 digits for the decimal fraction).
/*REXX program solves Ax=b with Gaussian elimination and backwards substitution. */
numeric digits 1000 /*heavy─duty decimal digits precision. */
parse arg iFID . /*obtain optional argument from the CL.*/
if iFID=='' | iFID=="," then iFID= 'GAUSS_E.DAT' /*Not specified? Then use the default.*/
pad= left('', 23) /*used for indenting residual numbers. */
do rec=1 while lines(iFID) \== 0 /*read the equation sets. */
#=0 /*the number of equations (so far). */
do $=1 while lines(iFID) \== 0 /*process the equation. */
z= linein(iFID); if z='' then leave /*Is this a blank line? end─of─data.*/
if $==1 then do; say; say center(' equations ', 75, "▓"); say
end /* [↑] if 1st equation, then show hdr.*/
say z /*display an equation to the terminal. */
if left(space(z), 1)=='*' then iterate /*Is this a comment? Then ignore it.*/
#= # + 1; n= words(z) - 1 /*assign equation #; calculate # items.*/
do e=1 for n; a.#.e= word(z, e); oa.#.e= a.#.e
end /*e*/ /* [↑] process A numbers; save orig.*/
b.#= word(z, n+1); ob.#=b.# /* ◄─── " B " " " */
end /*$*/
if #\==0 then call Gauss_elim /*Not zero? Then display the results. */
say
do i=1 for n; r=0 /*display the residuals to the terminal*/
do j=1 for n; r=r + oa.i.j * x.j /* ┌───◄ don't display a fraction if */
end /*j*/ /* ↓ res ≤ 5% of significant digs.*/
r= format(r-ob.i, , digits() - digits() * 0.05 % 1 , 0) / 1 /*should be tiny*/
say pad 'residual['right(i, length(n) )"] = " left('', r>=0) r /*right justify.*/
end /*i*/
end /*rec*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Gauss_elim: say; do j=1 for n; jp= j + 1
do i=jp to n; _= a.j.j / a.i.j
do k=jp to n; a.i.k= a.j.k - _ * a.i.k
end /*k*/
b.i= b.j - _ * b.i
end /*i*/
end /*j*/
x.n= b.n / a.n.n
do j=n-1 to 1 by -1; _= 0
do i=j+1 to n; _= _ + a.j.i * x.i
end /*i*/
x.j= (b.j - _) / a.j.j
end /*j*/ /* [↑] uses backwards substitution. */
numeric digits /*for the display, only use 8 digits. */
say center('solution', 75, "═"); say /*a title line for articulated output. */
do o=1 for n; say right('x['o"] = ", 38) left('', x.o>=0) x.o/1
end /*o*/
return
- output when using the same default input file as for version 2:
▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ equations ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
* a1 a2 a3 b
* ─── ─── ─── ───
1 2 3 14
2 1 3 13
3 -2 -1 -4
═════════════════════════════════solution══════════════════════════════════
x[1] = 1
x[2] = 2
x[3] = 3
residual[1] = 0
residual[2] = 0
residual[3] = 0
▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ equations ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
* a1 a2 a3 a4 a5 a6 b
* ─────── ─────── ─────── ─────── ─────── ─────── ───────
1 0 0 0 0 0 -0.01
1 0.63 0.39 0.25 0.16 0.10 0.61
1 1.26 1.58 1.98 2.49 3.13 0.91
1 1.88 3.55 6.70 12.62 23.80 0.99
1 2.51 6.32 15.88 39.90 100.28 0.60
1 3.14 9.87 31.01 97.41 306.02 0.02
═════════════════════════════════solution══════════════════════════════════
x[1] = -0.01
x[2] = 1.6027904
x[3] = -1.6132031
x[4] = 1.2454941
x[5] = -0.49098972
x[6] = 0.065760696
residual[1] = 0
residual[2] = 0
residual[3] = 0
residual[4] = 0
residual[5] = 0
residual[6] = 0
Ruby
require 'bigdecimal/ludcmp'
include LUSolve
BigDecimal::limit(30)
a = [1.00, 0.00, 0.00, 0.00, 0.00, 0.00,
1.00, 0.63, 0.39, 0.25, 0.16, 0.10,
1.00, 1.26, 1.58, 1.98, 2.49, 3.13,
1.00, 1.88, 3.55, 6.70, 12.62, 23.80,
1.00, 2.51, 6.32, 15.88, 39.90, 100.28,
1.00, 3.14, 9.87, 31.01, 97.41, 306.02].map{|i|BigDecimal(i,16)}
b = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02].map{|i|BigDecimal(i,16)}
n = 6
zero = BigDecimal("0.0")
one = BigDecimal("1.0")
lusolve(a, b, ludecomp(a, n, zero,one), zero).each{|v| puts v.to_s('F')[0..20]}
- Output:
-0.01 1.6027903945021135753 -1.613203059905560094 1.2454941213714351826 -0.490989719584656871 0.0657606961752318825
Rust
// using a Vec<f32> might be a better idea
// for now, let us create a fixed size array
// of size:
const SIZE: usize = 6;
pub fn eliminate(mut system: [[f32; SIZE+1]; SIZE]) -> Option<Vec<f32>> {
// produce the row reduced echelon form
//
// for every row...
for i in 0..SIZE-1 {
// for every column in that row...
for j in i..SIZE-1 {
if system[i][i] == 0f32 {
continue;
} else {
// reduce every element under that element to 0
let factor = system[j + 1][i] as f32 / system[i][i] as f32;
for k in i..SIZE+1 {
// potential optimization: set every element to zero, instead of subtracting
// i think subtraction helps showcase the process better
system[j + 1][k] -= factor * system[i][k] as f32;
}
}
}
}
// produce gaussian eliminated array
//
// the process follows a similar pattern
// but this one reduces the upper triangular
// elements
for i in (1..SIZE).rev() {
if system[i][i] == 0f32 {
continue;
} else {
for j in (1..i+1).rev() {
let factor = system[j - 1][i] as f32 / system[i][i] as f32;
for k in (0..SIZE+1).rev() {
system[j - 1][k] -= factor * system[i][k] as f32;
}
}
}
}
// produce solutions through back substitution
let mut solutions: Vec<f32> = vec![];
for i in 0..SIZE {
if system[i][i] == 0f32 {
return None;
}
else {
system[i][SIZE] /= system[i][i] as f32;
system[i][i] = 1f32;
println!("X{} = {}", i + 1, system[i][SIZE]);
solutions.push(system[i][SIZE])
}
}
return Some(solutions);
}
#[cfg(test)]
mod tests {
use super::*;
// sample run of the program
#[test]
fn eliminate_seven_by_six() {
let system: [[f32; SIZE +1]; SIZE] = [
[1.00 , 0.00 , 0.00 , 0.00 , 0.00 , 0.00 , -0.01 ] ,
[1.00 , 0.63 , 0.39 , 0.25 , 0.16 , 0.10 , 0.61 ] ,
[1.00 , 1.26 , 1.58 , 1.98 , 2.49 , 3.13 , 0.91 ] ,
[1.00 , 1.88 , 3.55 , 6.70 , 12.62 , 23.80 , 0.99 ] ,
[1.00 , 2.51 , 6.32 , 15.88 , 39.90 , 100.28 , 0.60 ] ,
[1.00 , 3.14 , 9.87 , 31.01 , 97.41 , 306.02 , 0.02 ]
] ;
let solutions = eliminate(system).unwrap();
assert_eq!(6, solutions.len());
let assert_solns = vec![-0.01, 1.60278, -1.61320, 1.24549, -0.49098, 0.06576];
for (ans, key) in solutions.iter().zip(assert_solns.iter()) {
if (ans - key).abs() > 1E-4 { panic!("Test Failed!") }
}
}
}
Scala
object GaussianElimination {
def solve(a: Array[Array[Double]], b: Array[Array[Double]]): Double = {
if (a == null || b == null || a.length == 0 || b.length == 0) {
throw new IllegalArgumentException("Invalid dimensions")
}
val n = b.length
val p = b(0).length
if (a.length != n || a(0).length != n) {
throw new IllegalArgumentException("Invalid dimensions")
}
var det = 1.0
for (i <- 0 until n - 1) {
var k = i
for (j <- i + 1 until n) {
if (Math.abs(a(j)(i)) > Math.abs(a(k)(i))) {
k = j
}
}
if (k != i) {
det = -det
for (j <- i until n) {
val s = a(i)(j)
a(i)(j) = a(k)(j)
a(k)(j) = s
}
for (j <- 0 until p) {
val s = b(i)(j)
b(i)(j) = b(k)(j)
b(k)(j) = s
}
}
for (j <- i + 1 until n) {
val s = a(j)(i) / a(i)(i)
for (k <- i + 1 until n) {
a(j)(k) -= s * a(i)(k)
}
for (k <- 0 until p) {
b(j)(k) -= s * b(i)(k)
}
}
}
for (i <- n - 1 to 0 by -1) {
for (j <- i + 1 until n) {
val s = a(i)(j)
for (k <- 0 until p) {
b(i)(k) -= s * b(j)(k)
}
}
val s = a(i)(i)
det *= s
for (k <- 0 until p) {
b(i)(k) /= s
}
}
det
}
def main(args: Array[String]): Unit = {
val a = Array(
Array(4.0, 1.0, 0.0, 0.0, 0.0),
Array(1.0, 4.0, 1.0, 0.0, 0.0),
Array(0.0, 1.0, 4.0, 1.0, 0.0),
Array(0.0, 0.0, 1.0, 4.0, 1.0),
Array(0.0, 0.0, 0.0,