Gaussian elimination: Difference between revisions
Added Wren |
m →version 3: added whitespace to the REXX section header wording. |
||
Line 5,102: | Line 5,102: | ||
This is the same as version 2, but in addition, it also shows the residuals. |
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 |
Code was added to this program version to keep a copy of the original '''A.i.k''' and '''B.#''' arrays (for calculating the |
||
<br>residuals). |
|||
⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
<lang rexx>/*REXX program solves Ax=b with Gaussian elimination and backwards substitution. */ |
<lang rexx>/*REXX program solves Ax=b with Gaussian elimination and backwards substitution. */ |
||
numeric digits 1000 /*heavy─duty decimal digits precision. */ |
numeric digits 1000 /*heavy─duty decimal digits precision. */ |
Revision as of 12:42, 30 June 2021
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
<lang 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))</lang>
- Output:
[-0.01, 1.60279, -1.6132, 1.24549, -0.49099, 0.0657607]
360 Assembly
<lang 360asm>* 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</lang>
- Output:
SOLUTION -0.00999 1.60279 -1.61322 1.24552 -0.49100 0.06576
Ada
<lang 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;</lang>
- Output:
[ -0.010000 1.602774 -1.613148 1.245437 -0.490967 0.065758 ]
ALGOL 68
File: prelude_exception.a68<lang algol68># -*- 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;
SKIP</lang>File: prelude_mat_lib.a68<lang algol68># -*- 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
);
SKIP</lang>File: prelude_gaussian_elimination.a68<lang algol68># -*- 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)
);
SKIP</lang>File: postlude_exception.a68<lang algol68># -*- 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:
stop</lang>File: test_Gaussian_elimination.a68<lang algol68>#!/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" PR</lang>Output:
( -.010000000000002, 1.602790394502130, -1.613203059905640, 1.245494121371510, -.490989719584686, .065760696175236)
C
This modifies A and b in place, which might not be quite desirable. <lang c>#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;
}</lang>
- 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. <lang csharp> 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(); } } }
} </lang> <lang csharp> 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(); } }
} </lang>
{{out}} -0.0597391027501976 1.85018966726278 -1.97278330181163 1.4697587750651 -0.553874184782179 0.0723048745759396
C++
<lang cpp>#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;
}</lang>
- Output:
-0.01 1.602790394502113 -1.61320305990556 1.245494121371436 -0.4909897195846575 0.065760696175232
Common Lisp
<lang CommonLisp> (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)) ))))
</lang>
- 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
<lang 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);
}</lang>
- Output:
[-0.01, 1.60279, -1.6132, 1.24549, -0.49099, 0.0657607]
Delphi
<lang 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.
</lang>
- Output:
1.00, 0.00, 0.00, 4.00
F#
The Function
<lang fsharp> // 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
</lang>
The Task
This task uses functionality from Continued_fraction/Arithmetic/Construct_from_rational_number#F.23 and Continued_fraction#F.23 <lang fsharp> 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))))) </lang>
- 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 <lang fortran>
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
</lang>
FreeBASIC
Gaussian elimination with pivoting. FreeBASIC version 1.05 <lang FreeBASIC>
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 </lang>
- Output:
-0.01 1.602790394502115 -1.613203059905572 1.245494121371448 -0.490989719584662 0.06576069617523256
Generic
<lang cpp> 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; } }
}</lang>
Go
Partial pivoting, no scaling
Gaussian elimination with partial pivoting by pseudocode on WP page Gaussian elimination." <lang go>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
}</lang>
- 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. <lang go>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
}</lang>
- Output:
[-0.01 1.6027903945021131 -1.6132030599055596 1.245494121371436 -0.49098971958465754 0.065760696175232]
Haskell
From scratch
<lang Haskell> 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
</lang>
- 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. <lang Haskell>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
</lang>
- 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
<lang Haskell>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
</lang>
- 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.
<lang J>
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
</lang>
Java
Naked implementation, using Java arrays instead of a matrix class.
<lang java>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]); } }
}</lang>
JavaScript
From Numerical Recipes in C: <lang javascript>// 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]
) )</lang>
- Output:
-0.01000000000000004, 1.6027903945021095, -1.6132030599055475, 1.2454941213714232, -0.4909897195846526, 0.06576069617523138
Julia
Using built-in LAPACK-based linear solver (which employs partial-pivoted Gaussian elimination): <lang julia>x = A \ b</lang>
Klong
<lang K> 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))} </lang>
Example, matrix taken from C version:
<lang K>
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]
</lang>
Kotlin
<lang scala>// 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 } }
}</lang>
- Output:
[-0.01, 1.6027903945021138, -1.6132030599055616, 1.2454941213714392, -0.49098971958465953, 0.06576069617523238]
Lambdatalk
<lang scheme> {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] </lang>
Lobster
<lang 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()</lang>
- Output:
[-0.01, 1.602790394502, -1.613203059906, 1.245494121371, -0.490989719585, 0.065760696175]
M2000 Interpreter
Faster, with accuracy of 25 decimals <lang M2000 Interpreter> 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$
} checkit </lang>
slower with accuracy of 26 decimals <lang M2000 Interpreter> 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 </lang>
- 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
<lang Mathematica>GaussianElimination[A_?MatrixQ, b_?VectorQ] := Last /@ RowReduce[Flatten /@ Transpose[{A, b}]]</lang>
MATLAB
<lang 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 </lang>
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 whethera
is nonzero;Minus(a, b: T): T
andTimes(a, b: T): T
, which return the results of the procedures' names; andPrint(a: T)
which does what the name implies.
- Matrix interface
<lang modula3>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.</lang>
- Matrix implementation
<lang modula3>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.</lang>
- interface for the ring of integers modulo an integer
<lang modula3>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.</lang>
- 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.
<lang modula3>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.</lang>
- 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
<lang 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</lang>
- 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... <lang OCaml> 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
</lang> The solver: <lang OCaml> (* 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
</lang> Example data... <lang OCaml> 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 |] </lang> In the REPL, the solution is: <lang OCaml>
- let x = solve a b;;
val x : float array = [|-0.0100000000000000991; 1.60279039450210536; -1.61320305990553226;
1.24549412137140547; -0.490989719584644546; 0.0657606961752301433|]
</lang> Further, let's define multiplication and subtraction to check our results... <lang OCaml> 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 </lang> Now 'x' can be plugged into the equation to calculate the residual: <lang OCaml>
- 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|]
</lang>
PARI/GP
If A and B have floating-point numbers (t_REAL
s) then the following uses Gaussian elimination:
<lang parigp>matsolve(A,B)</lang>
If the entries are integers, then p-adic lifting (Dixon 1982) is used instead.
Perl
<lang 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 $x;</lang>
Math::Matrix
solve()
expects the column vector to be an extra column in the matrix, hence concat()
. Putting not just a column there but a whole identity matrix (making Nx2N) is how its invert()
is implemented. Note that solve()
doesn't notice singular matrices and still gives a return when there is in fact no solution to Ax=B.
Phix
<lang Phix>function gauss_eliminate(sequence a, b)
integer n = length(b) atom tmp for col=1 to n do integer m = col atom mx = a[m][m] for i=col+1 to n do tmp = abs(a[i][col]) if tmp>mx then {m,mx} = {i,tmp} end if end for if col!=m then {a[col],a[m]} = {a[m],a[col]} {b[col],b[m]} = {b[m],b[col]} end if for i=col+1 to n do tmp = a[i][col]/a[col][col] for j=col+1 to n do a[i][j] -= tmp*a[col][j] end for a[i][col] = 0 b[i] -= tmp*b[col] end for end for sequence x = repeat(0,n) for col=n to 1 by -1 do tmp = b[col] for j=n to col+1 by -1 do tmp -= x[j]*a[col][j] end for x[col] = tmp/a[col][col] end for return x
end function
constant a = {{1.00, 0.00, 0.00, 0.00, 0.00, 0.00},
{1.00, 0.63, 0.39, 0.25, 0.16, 0.10}, {1.00, 1.26, 1.58, 1.98, 2.49, 3.13}, {1.00, 1.88, 3.55, 6.70, 12.62, 23.80}, {1.00, 2.51, 6.32, 15.88, 39.90, 100.28}, {1.00, 3.14, 9.87, 31.01, 97.41, 306.02}}, b = {-0.01, 0.61, 0.91, 0.99, 0.60, 0.02}
pp(gauss_eliminate(a, b))</lang>
- Output:
{-0.01,1.602790395,-1.61320306,1.245494121,-0.4909897196,0.06576069618}
PHP
<lang php>function swap_rows(&$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();</lang>
- Output:
Array ( [0] => -0.01 [1] => 1.6027903945021 [2] => -1.6132030599055 [3] => 1.2454941213714 [4] => -0.49098971958463 [5] => 0.065760696175228 )
PL/I
<lang pli>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;</lang>
- 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
<lang PowerShell> 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 $b
</lang> Output:
a = 1 0 0 0 0 0 1 0.63 0.39 0.25 0.16 0.1 1 1.26 1.58 1.98 2.49 3.13 1 1.88 3.55 6.7 12.62 23.8 1 2.51 6.32 15.88 39.9 100.28 1 3.14 9.87 31.01 97.41 306.02 b = -0.01 0.61 0.91 0.99 0.6 0.02 x = -0.01 1.60279039450213 -1.6132030599056 1.24549412137148 -0.490989719584674 0.0657606961752342
Gauss-Jordan
<lang PowerShell> function gauss-jordan($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 </lang> 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
<lang 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)]]</lang>
Using numpy
<lang python3> $ 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]])
>>> </lang>
R
Here 'b' is a matrix. Partial pivoting is used, and the determinant of 'a' is returned as well.
<lang R>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 <- aj, i / ai, 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, ] - ai, j * b[j, ] } } b[i, ] <- b[i, ] / ai, i det <- det * ai, 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))</lang>
- 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 $det [1] -360
Racket
<lang racket>
- lang racket
(require math/matrix) (define A
(matrix [[1.00 0.00 0.00 0.00 0.00 0.00] [1.00 0.63 0.39 0.25 0.16 0.10] [1.00 1.26 1.58 1.98 2.49 3.13] [1.00 1.88 3.55 6.70 12.62 23.80] [1.00 2.51 6.32 15.88 39.90 100.28] [1.00 3.14 9.87 31.01 97.41 306.02]]))
(define b (col-matrix [-0.01 0.61 0.91 0.99 0.60 0.02]))
(matrix-solve A b) </lang>
- Output:
<lang racket>
- <array
'#(6 1) #[-0.01 1.602790394502109 -1.613203059905556 1.2454941213714346 -0.4909897195846582 0.06576069617523222]>
</lang>
Raku
(formerly Perl 6)
Gaussian elimination results in a matrix in row echelon form. Gaussian elimination with back-substitution (also known as Gauss-Jordan elimination) results in a matrix in reduced row echelon form. That being the case, we can reuse much of the code from the Reduced row echelon form task. Raku stores and does calculations on decimal numbers within its limit of precision using Rational numbers by default, meaning the calculations are exact.
<lang perl6>sub gauss-jordan-solve (@a, @b) {
@b.kv.map: { @a[$^k].append: $^v }; @a.&rref[*]»[*-1];
}
- reduced row echelon form (Gauss-Jordan elimination)
sub rref (@m) {
return unless @m; my ($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; my $lv = @m[$r;$lead]; @m[$r] »/=» $lv; for ^$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.WHAT ~~ 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"; </lang>
- 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
<lang rexx>/* 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</lang>
- 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. <lang rexx>/*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</lang>
- 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).
<lang rexx>/*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</lang>
- 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
<lang 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]}</lang>
- Output:
-0.01 1.6027903945021135753 -1.613203059905560094 1.2454941213714351826 -0.490989719584656871 0.0657606961752318825
Rust
<lang 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!") } } }
} </lang>
Sidef
Uses the rref(A) function from Reduced row echelon form.
<lang ruby>func gauss_jordan_solve (a, b) {
var A = gather { ^b -> each {|i| take(a[i] + b[i]) } }
rref(A).map{ .last }
}
var 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 ],
]
var b = [ -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 ]
var G = gauss_jordan_solve(a, b) say G.map { "%27s" % .as_rat }.join("\n")</lang>
- Output:
-1/100 655870882787/409205648497 -660131804286/409205648497 509663229635/409205648497 -200915766608/409205648497 26909648324/409205648497
Stata
Gaussian elimination
This implementation computes also the determinant of the matrix A, as it requires only a few operations. The matrix B is overwritten with the solution of the system, and A is overwritten with garbage.
<lang stata>void gauss(real matrix a, real matrix b, real scalar det) { real scalar i,j,n,s real vector js
det = 1 n = rows(a) for (i=1; i<n; i++) { maxindex(abs(a[i::n,i]), 1, js=., .) j = js[1]+i-1 if (j!=i) { a[(i\j),i..n] = a[(j\i),i..n] b[(i\j),.] = b[(j\i),.] det = -det } for (j=i+1; j<=n; j++) { s = a[j,i]/a[i,i] a[j,i+1..n] = a[j,i+1..n]-s*a[i,i+1..n] b[j,.] = b[j,.]-s*b[i,.] } }
for (i=n; i>=1; i--) { for (j=i+1; j<=n; j++) { b[i,.] = b[i,.]-a[i,j]*b[j,.] } b[i,.] = b[i,.]/a[i,i] det = det*a[i,i] } }</lang>
LU decomposition and backsubstitution
<lang stata>void ludec(real matrix a, real matrix l, real matrix u, real vector p) { real scalar i,j,n,s real vector js
l = a n = rows(a) p = 1::n for (i=1; i<n; i++) { maxindex(abs(l[i::n,i]), 1, js=., .) j = js[1]+i-1 if (j!=i) { l[(i\j),.] = l[(j\i),.] p[(i\j)] = p[(j\i)] } for (j=i+1; j<=n; j++) { l[j,i] = s = l[j,i]/l[i,i] l[j,i+1..n] = l[j,i+1..n]-s*l[i,i+1..n] } }
u = uppertriangle(l) l = lowertriangle(l, 1) }
void luback(real matrix l, real matrix u, real vector p, real matrix y) { real scalar i,j,n
n = rows(y) y = y[p,.] for (i=1; i<=n; i++) { for (j=1; j<i; j++) { y[i,.] = y[i,.]-l[i,j]*y[j,.] } /*y[i,.] = y[i,.]/l[i,i]*/ }
for (i=n; i>=1; i--) { for (j=i+1; j<=n; j++) { y[i,.] = y[i,.]-u[i,j]*y[j,.] } y[i,.] = y[i,.]/u[i,i] } }</lang>
Example
Here we are computing the inverse of a 3x3 matrix (which happens to be a magic square), using both methods.
<lang stata>: gauss(a=(2,9,4\7,5,3\6,1,8),b=I(3),det=.)
- b
1 2 3 +----------------------------------------------+ 1 | -.1027777778 .1888888889 -.0194444444 | 2 | .1055555556 .0222222222 -.0611111111 | 3 | .0638888889 -.1444444444 .1472222222 | +----------------------------------------------+
- ludec(a=(2,9,4\7,5,3\6,1,8),l=.,u=.,p=.)
- luback(l,u,p,y=I(3))
- y
1 2 3 +----------------------------------------------+ 1 | -.1027777778 .1888888889 -.0194444444 | 2 | .1055555556 .0222222222 -.0611111111 | 3 | .0638888889 -.1444444444 .1472222222 | +----------------------------------------------+</lang>
Swift
<lang swift>func gaussEliminate(_ sys: Double) -> [Double]? {
var system = sys
let size = system.count
for i in 0..<size-1 where system[i][i] != 0 { for j in i..<size-1 { let factor = system[j + 1][i] / system[i][i]
for k in i..<size+1 { system[j + 1][k] -= factor * system[i][k] } } }
for i in (1..<size).reversed() where system[i][i] != 0 { for j in (1..<i+1).reversed() { let factor = system[j - 1][i] / system[i][i]
for k in (0..<size+1).reversed() { system[j - 1][k] -= factor * system[i][k] } } }
var solutions = [Double]()
for i in 0..<size { guard system[i][i] != 0 else { return nil }
system[i][size] /= system[i][i] system[i][i] = 1 solutions.append(system[i][size]) }
return solutions
}
let sys = [
[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]
]
guard let sols = gaussEliminate(sys) else {
fatalError("No solutions")
}
for (i, f) in sols.enumerated() {
print("X\(i + 1) = \(f)")
}</lang>
- Output:
X1 = -0.01 X2 = 1.6027903945021138 X3 = -1.613203059905563 X4 = 1.245494121371438 X5 = -0.4909897195846575 X6 = 0.065760696175232
Tcl
<lang tcl>package require math::linearalgebra
set 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}
} set b {-0.01 0.61 0.91 0.99 0.60 0.02} puts -nonewline [math::linearalgebra::show [math::linearalgebra::solveGauss $A $b] "%.2f"]</lang>
- Output:
-0.01 1.60 -1.61 1.25 -0.49 0.07
TI-83 BASIC
The rref() function performs reduced row-echelon form using Gaussian elimination
on a n*(n+1) matrix. The (n+1)th column receives the resulting vector.
The n*n maxtrix is set to 0 and the pivots are set to 1.
The Matr>List() subroutine extracts the (n+1)th column to a list.
The matrix can be more easily entered by the matrix editor.
On TI-83 or TI-84, another way to solve this task is to use the PlySmlt2 internal apps and choose
"simult equ solver" with 6 equations and 6 unknowns.
<lang ti83b>[[ 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]]→[A]
Matr>List(rref([A]),7,L1) L1</lang>
- Output:
{-.01 1.602790395 -1.61320306 1.245494121 -.4909897196 .0657606962}
VBA
<lang vb>'Option Base 1
Private Function gauss_eliminate(a As Variant, b As Variant) As Variant
Dim n As Integer: n = UBound(b) Dim tmp As Variant, m As Integer, mx As Variant For col = 1 To n m = col mx = a(m, m) For i = col + 1 To n tmp = Abs(a(i, col)) If tmp > mx Then m = i mx = tmp End If Next i If col <> m Then For j = 1 To UBound(a, 2) tmp = a(col, j) a(col, j) = a(m, j) a(m, j) = tmp Next j tmp = b(col) b(col) = b(m) b(m) = tmp End If For i = col + 1 To n tmp = a(i, col) / a(col, col) For j = col + 1 To n a(i, j) = a(i, j) - tmp * a(col, j) Next j a(i, col) = 0 b(i) = b(i) - tmp * b(col) Next i Next col Dim x() As Variant ReDim x(n) For col = n To 1 Step -1 tmp = b(col) For j = n To col + 1 Step -1 tmp = tmp - x(j) * a(col, j) Next j x(col) = tmp / a(col, col) Next col gauss_eliminate = x
End Function Public Sub main()
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}] Dim s() As String, x() As Variant ReDim s(UBound(b)), x(UBound(b)) Debug.Print "("; x = gauss_eliminate(a, b) For i = 1 To UBound(x) s(i) = CStr(x(i)) Next i t = Join(s, ", ") Debug.Print t; ")"
End Sub</lang>
- Output:
(-0.01, 1.60279039450209, -1.61320305990548, 1.24549412137136, -0.490989719584628, 0.065760696175228)
VBScript
<lang vb>' Gaussian elimination - VBScript
const n=6 dim a(6,6),b(6),x(6),ab ab=array( 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) k=-1 for i=1 to n buf="" for j=1 to n+1 k=k+1 if j<=n then a(i,j)=ab(k) else b(i)=ab(k) end if buf=buf&right(space(8)&formatnumber(ab(k),2),8)&" " next wscript.echo buf next for j=1 to n for i=j+1 to n w=a(j,j)/a(i,j) for k=j+1 to n a(i,k)=a(j,k)-w*a(i,k) next b(i)=b(j)-w*b(i) next next x(n)=b(n)/a(n,n) for j=n-1 to 1 step -1 w=0 for i=j+1 to n w=w+a(j,i)*x(i) next x(j)=(b(j)-w)/a(j,j) next wscript.echo "solution" buf="" for i=1 to n buf=buf&right(space(8)&formatnumber(x(i),2),8)&vbcrlf next wscript.echo buf</lang>
- Output:
-0,01 1,60 -1,61 1,25 -0,49 0,07
Wren
<lang ecmascript>import "/trait" for Stepped
var 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]
]
var tb = [-0.01, 0.61, 0.91, 0.99, 0.60, 0.02]
var tx = [
-0.01, 1.602790394502114, -1.6132030599055613, 1.2454941213714368, -0.4909897195846576, 0.065760696175232
]
var EPSILON = 1e-14 // tolerance required
var gaussPartial = Fn.new { |a0, b0|
var m = b0.count var a = List.filled(m, null) var i = 0 for (ai in a0) { var row = ai.toList row.add(b0[i]) a[i] = row i = i + 1 } for (k in 0...a.count) { var iMax = 0 var max = -1 for (i in Stepped.ascend(k...m)) { var row = a[i] // compute scale factor s = max abs in row var s = -1 for (j in Stepped.ascend(k...m)) { var e = row[j].abs if (e > s) s = e } // scale the abs used to pick the pivot var abs = row[k].abs / s if (abs > max) { iMax = i max = abs } } if (a[iMax][k] == 0) Fiber.abort("Matrix is singular.") a.swap(k, iMax) for (i in Stepped.ascend(k + 1...m)) { for (j in Stepped.ascend(k + 1..m)) { a[i][j] = a[i][j] - a[k][j] * a[i][k] / a[k][k] } a[i][k] = 0 } } var x = List.filled(m, 0) for (i in Stepped.descend(m - 1..0)) { x[i] = a[i][m] for (j in Stepped.ascend(i + 1...m)) { x[i] = x[i] - a[i][j] * x[j] } x[i] = x[i] / a[i][i] } return x
}
var x = gaussPartial.call(ta, tb) System.print(x) var i = 0 for (xi in x) {
if ((tx[i] - xi).abs > EPSILON) { System.print("Out of tolerance.") System.print("Expected values are %(tx)") return } i = i + 1
}</lang>
- Output:
[-0.01, 1.6027903945021, -1.6132030599056, 1.2454941213714, -0.49098971958466, 0.065760696175232]
zkl
Using the GNU Scientific Library: <lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library) a:=GSL.Matrix(6,6).set(
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:=GSL.VectorFromData(-0.01, 0.61, 0.91, 0.99, 0.60, 0.02); x:=a.AxEQb(b); x.format(8,5).println();</lang>
- Output:
-0.01000, 1.60279,-1.61320, 1.24549,-0.49099, 0.06576
Or, using lists:
<lang zkl>fcn gaussEliminate(a,b){ // modifies a&b --> vector
n:=b.len(); foreach dia in ([0..n-1]){ maxRow:=dia; max:=a[dia][dia]; foreach row in ([dia+1 .. n-1]){ if((tmp:=a[row][dia].abs()) > max){ maxRow=row; max=tmp; } } a.swap(dia,maxRow); b.swap(dia,maxRow); // swap rows foreach row in ([dia+1 .. n-1]){ ar:=a[row]; ad:=a[dia]; tmp:=ar[dia] / ad[dia];
foreach col in ([dia+1 .. n-1]){ ar[col]-=tmp*ad[col]; } ar[dia]=0.0; b[row]-=tmp*b[dia];
} } x:=(0).pump(n,List().write); // -->list filled with garbage foreach row in ([n-1 .. 0,-1]){ tmp:=b[row]; ar:=a[row]; foreach j in ([n-1 .. row+1,-1]){ tmp-=x[j]*ar[j]; } x[row]=tmp/a[row][row]; } x
}</lang> <lang zkl>a:=List( List(1.00, 0.00, 0.00, 0.00, 0.00, 0.00,),
List(1.00, 0.63, 0.39, 0.25, 0.16, 0.10,), List(1.00, 1.26, 1.58, 1.98, 2.49, 3.13,), List(1.00, 1.88, 3.55, 6.70, 12.62, 23.80,), List(1.00, 2.51, 6.32, 15.88, 39.90, 100.28,), List(1.00, 3.14, 9.87, 31.01, 97.41, 306.02) );
b:=List( -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 ); gaussEliminate(a,b).println();</lang>
- Output:
L(-0.01,1.60279,-1.6132,1.24549,-0.49099,0.0657607)
- Programming Tasks
- Solutions by Programming Task
- Matrices
- 11l
- 360 Assembly
- Ada
- ALGOL 68
- C
- C sharp
- C++
- Common Lisp
- D
- Delphi
- F Sharp
- Fortran
- FreeBASIC
- Generic
- Go
- Haskell
- J
- Java
- JavaScript
- Julia
- Klong
- Kotlin
- Lambdatalk
- Lobster
- M2000 Interpreter
- Mathematica
- Wolfram Language
- MATLAB
- Modula-3
- Nim
- OCaml
- PARI/GP
- Perl
- Math::Matrix
- Phix
- PHP
- PL/I
- PowerShell
- Python
- R
- Racket
- Raku
- REXX
- Ruby
- Rust
- Sidef
- Stata
- Swift
- Tcl
- Tcllib
- TI-83 BASIC
- VBA
- VBScript
- Wren
- Wren-trait
- Zkl