Cramer's rule
You are encouraged to solve this task according to the task description, using any language you may know.
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
Given
which in matrix format is
Then the values of and can be found as follows:
- Task
Given the following system of equations:
solve for , , and , using Cramer's rule.
11l
F det(mm)
V m = copy(mm)
V result = 1.0
L(j) 0 .< m.len
V imax = j
L(i) j + 1 .< m.len
I m[i][j] > m[imax][j]
imax = i
I imax != j
swap(&m[imax], &m[j])
result = -result
I abs(m[j][j]) < 1e-12
R Float.infinity
L(i) j + 1 .< m.len
V mult = -m[i][j] / m[j][j]
L(k) 0 .< m.len
m[i][k] += mult * m[j][k]
L(i) 0 .< m.len
result *= m[i][i]
R result
F cramerSolve(aa, detA, b, col)
V a = copy(aa)
L(i) 0 .< a.len
a[i][col] = b[i]
R det(a) / detA
V A = [[2.0, -1.0, 5.0, 1.0],
[3.0, 2.0, 2.0, -6.0],
[1.0, 3.0, 3.0, -1.0],
[5.0, -2.0, -3.0, 3.0]]
V B = [-3.0, -32.0, -47.0, 49.0]
V detA = det(A)
L(i) 0 .< A.len
print(‘#3.3’.format(cramerSolve(A, detA, B, i)))- Output:
2.000 -12.000 -4.000 1.000
Ada
with Ada.Text_IO;
with Ada.Numerics.Generic_Real_Arrays;
procedure Cramers_Rules is
type Real is new Float;
-- This is the type we want to use in the matrix and vector
package Real_Arrays is
new Ada.Numerics.Generic_Real_Arrays (Real);
use Real_Arrays;
function Solve_Cramer (M : in Real_Matrix;
V : in Real_Vector)
return Real_Vector
is
Denominator : Real;
Nom_Matrix : Real_Matrix (M'Range (1),
M'Range (2));
Numerator : Real;
Result : Real_Vector (M'Range (1));
begin
if
M'Length (2) /= V'Length or
M'Length (1) /= M'Length (2)
then
raise Constraint_Error with "Dimensions does not match";
end if;
Denominator := Determinant (M);
for Col in V'Range loop
Nom_Matrix := M;
-- Substitute column
for Row in V'Range loop
Nom_Matrix (Row, Col) := V (Row);
end loop;
Numerator := Determinant (Nom_Matrix);
Result (Col) := Numerator / Denominator;
end loop;
return Result;
end Solve_Cramer;
procedure Put (V : 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 => 2);
Put (" ");
end loop;
Put ("]");
New_Line;
end Put;
M : constant Real_Matrix := ((2.0, -1.0, 5.0, 1.0),
(3.0, 2.0, 2.0, -6.0),
(1.0, 3.0, 3.0, -1.0),
(5.0, -2.0, -3.0, 3.0));
V : constant Real_Vector := (-3.0, -32.0, -47.0, 49.0);
R : constant Real_Vector := Solve_Cramer (M, V);
begin
Put (R);
end Cramers_Rules;
- Output:
[ 2.00 -12.00 -4.00 1.00 ]
ALGOL 68
Uses the non-standard DET operator available in Algol 68G.
# returns the solution of a.x = b via Cramer's rule #
# this is for REAL arrays, could define additional operators #
# for INT, COMPL, etc. #
PRIO CRAMER = 1;
OP CRAMER = ( [,]REAL a, []REAL b )[]REAL:
IF 1 UPB a /= 2 UPB a
OR 1 LWB a /= 2 LWB a
OR 1 UPB a /= UPB b
THEN
# the array sizes and bounds do not match #
print( ( "Invaid parameters to CRAMER", newline ) );
stop
ELIF REAL deta = DET a;
det a = 0
THEN
# a is singular #
print( ( "Singular matrix for CRAMER", newline ) );
stop
ELSE
# the arrays have matching bounds #
[ LWB b : UPB b ]REAL result;
FOR col FROM LWB b TO UPB b DO
# form a matrix from a with the col'th column replaced by b #
[ 1 LWB a : 1 UPB a, 2 LWB a : 2 UPB a ]REAL m := a;
m[ : , col ] := b[ : AT 1 ];
# col'th result elemet as per Cramer's rule #
result[ col ] := DET m / det a
OD;
result
FI; # CRAMER #
# test CRAMER using the matrix and column vector specified in the task #
BEGIN
[,]REAL a = ( ( 2, -1, 5, 1 )
, ( 3, 2, 2, -6 )
, ( 1, 3, 3, -1 )
, ( 5, -2, -3, 3 )
);
[]REAL b = ( -3
, -32
, -47
, 49
);
[]REAL solution = a CRAMER b;
FOR c FROM LWB solution TO UPB solution DO
print( ( " ", fixed( solution[ c ], -8, 4 ) ) )
OD;
print( ( newline ) )
END- Output:
2.0000 -12.0000 -4.0000 1.0000
C
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
typedef struct {
int n;
double **elems;
} SquareMatrix;
SquareMatrix init_square_matrix(int n, double elems[n][n]) {
SquareMatrix A = {
.n = n,
.elems = malloc(n * sizeof(double *))
};
for(int i = 0; i < n; ++i) {
A.elems[i] = malloc(n * sizeof(double));
for(int j = 0; j < n; ++j)
A.elems[i][j] = elems[i][j];
}
return A;
}
SquareMatrix copy_square_matrix(SquareMatrix src) {
SquareMatrix dest;
dest.n = src.n;
dest.elems = malloc(dest.n * sizeof(double *));
for(int i = 0; i < dest.n; ++i) {
dest.elems[i] = malloc(dest.n * sizeof(double));
for(int j = 0; j < dest.n; ++j)
dest.elems[i][j] = src.elems[i][j];
}
return dest;
}
double det(SquareMatrix A) {
double det = 1;
for(int j = 0; j < A.n; ++j) {
int i_max = j;
for(int i = j; i < A.n; ++i)
if(A.elems[i][j] > A.elems[i_max][j])
i_max = i;
if(i_max != j) {
for(int k = 0; k < A.n; ++k) {
double tmp = A.elems[i_max][k];
A.elems[i_max][k] = A.elems[j][k];
A.elems[j][k] = tmp;
}
det *= -1;
}
if(abs(A.elems[j][j]) < 1e-12) {
puts("Singular matrix!");
return NAN;
}
for(int i = j + 1; i < A.n; ++i) {
double mult = -A.elems[i][j] / A.elems[j][j];
for(int k = 0; k < A.n; ++k)
A.elems[i][k] += mult * A.elems[j][k];
}
}
for(int i = 0; i < A.n; ++i)
det *= A.elems[i][i];
return det;
}
void deinit_square_matrix(SquareMatrix A) {
for(int i = 0; i < A.n; ++i)
free(A.elems[i]);
free(A.elems);
}
double cramer_solve(SquareMatrix A, double det_A, double *b, int var) {
SquareMatrix tmp = copy_square_matrix(A);
for(int i = 0; i < tmp.n; ++i)
tmp.elems[i][var] = b[i];
double det_tmp = det(tmp);
deinit_square_matrix(tmp);
return det_tmp / det_A;
}
int main(int argc, char **argv) {
#define N 4
double elems[N][N] = {
{ 2, -1, 5, 1},
{ 3, 2, 2, -6},
{ 1, 3, 3, -1},
{ 5, -2, -3, 3}
};
SquareMatrix A = init_square_matrix(N, elems);
SquareMatrix tmp = copy_square_matrix(A);
int det_A = det(tmp);
deinit_square_matrix(tmp);
double b[] = {-3, -32, -47, 49};
for(int i = 0; i < N; ++i)
printf("%7.3lf\n", cramer_solve(A, det_A, b, i));
deinit_square_matrix(A);
return EXIT_SUCCESS;
}
- Output:
2.000 -12.000 -4.000 1.000
C#
Instead of copying a bunch of arrays, I made a class with an indexer that simply accesses the correct items in the original matrix.
using System;
using System.Collections.Generic;
using static System.Linq.Enumerable;
public static class CramersRule
{
public static void Main() {
var equations = new [] {
new [] { 2, -1, 5, 1, -3 },
new [] { 3, 2, 2, -6, -32 },
new [] { 1, 3, 3, -1, -47 },
new [] { 5, -2, -3, 3, 49 }
};
var solution = SolveCramer(equations);
Console.WriteLine(solution.DelimitWith(", "));
}
public static int[] SolveCramer(int[][] equations) {
int size = equations.Length;
if (equations.Any(eq => eq.Length != size + 1)) throw new ArgumentException($"Each equation must have {size+1} terms.");
int[,] matrix = new int[size, size];
int[] column = new int[size];
for (int r = 0; r < size; r++) {
column[r] = equations[r][size];
for (int c = 0; c < size; c++) {
matrix[r, c] = equations[r][c];
}
}
return Solve(new SubMatrix(matrix, column));
}
private static int[] Solve(SubMatrix matrix) {
int det = matrix.Det();
if (det == 0) throw new ArgumentException("The determinant is zero.");
int[] answer = new int[matrix.Size];
for (int i = 0; i < matrix.Size; i++) {
matrix.ColumnIndex = i;
answer[i] = matrix.Det() / det;
}
return answer;
}
//Extension method from library.
static string DelimitWith<T>(this IEnumerable<T> source, string separator = " ") =>
string.Join(separator ?? " ", source ?? Empty<T>());
private class SubMatrix
{
private int[,] source;
private SubMatrix prev;
private int[] replaceColumn;
public SubMatrix(int[,] source, int[] replaceColumn) {
this.source = source;
this.replaceColumn = replaceColumn;
this.prev = null;
this.ColumnIndex = -1;
Size = replaceColumn.Length;
}
private SubMatrix(SubMatrix prev, int deletedColumnIndex = -1) {
this.source = null;
this.prev = prev;
this.ColumnIndex = deletedColumnIndex;
Size = prev.Size - 1;
}
public int ColumnIndex { get; set; }
public int Size { get; }
public int this[int row, int column] {
get {
if (source != null) return column == ColumnIndex ? replaceColumn[row] : source[row, column];
return prev[row + 1, column < ColumnIndex ? column : column + 1];
}
}
public int Det() {
if (Size == 1) return this[0, 0];
if (Size == 2) return this[0, 0] * this[1, 1] - this[0, 1] * this[1, 0];
SubMatrix m = new SubMatrix(this);
int det = 0;
int sign = 1;
for (int c = 0; c < Size; c++) {
m.ColumnIndex = c;
int d = m.Det();
det += this[0, c] * d * sign;
sign = -sign;
}
return det;
}
public void Print() {
for (int r = 0; r < Size; r++) {
Console.WriteLine(Range(0, Size).Select(c => this[r, c]).DelimitWith(", "));
}
Console.WriteLine();
}
}
}
- Output:
2, -12, -4, 1
C++
#include <algorithm>
#include <iostream>
#include <vector>
class SubMatrix {
const std::vector<std::vector<double>> *source;
std::vector<double> replaceColumn;
const SubMatrix *prev;
size_t sz;
int colIndex = -1;
public:
SubMatrix(const std::vector<std::vector<double>> &src, const std::vector<double> &rc) : source(&src), replaceColumn(rc), prev(nullptr), colIndex(-1) {
sz = replaceColumn.size();
}
SubMatrix(const SubMatrix &p) : source(nullptr), prev(&p), colIndex(-1) {
sz = p.size() - 1;
}
SubMatrix(const SubMatrix &p, int deletedColumnIndex) : source(nullptr), prev(&p), colIndex(deletedColumnIndex) {
sz = p.size() - 1;
}
int columnIndex() const {
return colIndex;
}
void columnIndex(int index) {
colIndex = index;
}
size_t size() const {
return sz;
}
double index(int row, int col) const {
if (source != nullptr) {
if (col == colIndex) {
return replaceColumn[row];
} else {
return (*source)[row][col];
}
} else {
if (col < colIndex) {
return prev->index(row + 1, col);
} else {
return prev->index(row + 1, col + 1);
}
}
}
double det() const {
if (sz == 1) {
return index(0, 0);
}
if (sz == 2) {
return index(0, 0) * index(1, 1) - index(0, 1) * index(1, 0);
}
SubMatrix m(*this);
double det = 0.0;
int sign = 1;
for (size_t c = 0; c < sz; ++c) {
m.columnIndex(c);
double d = m.det();
det += index(0, c) * d * sign;
sign = -sign;
}
return det;
}
};
std::vector<double> solve(SubMatrix &matrix) {
double det = matrix.det();
if (det == 0.0) {
throw std::runtime_error("The determinant is zero.");
}
std::vector<double> answer(matrix.size());
for (int i = 0; i < matrix.size(); ++i) {
matrix.columnIndex(i);
answer[i] = matrix.det() / det;
}
return answer;
}
std::vector<double> solveCramer(const std::vector<std::vector<double>> &equations) {
int size = equations.size();
if (std::any_of(
equations.cbegin(), equations.cend(),
[size](const std::vector<double> &a) { return a.size() != size + 1; }
)) {
throw std::runtime_error("Each equation must have the expected size.");
}
std::vector<std::vector<double>> matrix(size);
std::vector<double> column(size);
for (int r = 0; r < size; ++r) {
column[r] = equations[r][size];
matrix[r].resize(size);
for (int c = 0; c < size; ++c) {
matrix[r][c] = equations[r][c];
}
}
SubMatrix sm(matrix, column);
return solve(sm);
}
template<typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &v) {
auto it = v.cbegin();
auto end = v.cend();
os << '[';
if (it != end) {
os << *it++;
}
while (it != end) {
os << ", " << *it++;
}
return os << ']';
}
int main() {
std::vector<std::vector<double>> equations = {
{ 2, -1, 5, 1, -3},
{ 3, 2, 2, -6, -32},
{ 1, 3, 3, -1, -47},
{ 5, -2, -3, 3, 49},
};
auto solution = solveCramer(equations);
std::cout << solution << '\n';
return 0;
}
- Output:
[2, -12, -4, 1]
Common Lisp
(defun minor (m col)
(loop with dim = (1- (array-dimension m 0))
with result = (make-array (list dim dim))
for i from 1 to dim
for r = (1- i)
do (loop with c = 0
for j to dim
when (/= j col)
do (setf (aref result r c) (aref m i j))
(incf c))
finally (return result)))
(defun det (m)
(assert (= (array-rank m) 2))
(assert (= (array-dimension m 0) (array-dimension m 1)))
(let ((dim (array-dimension m 0)))
(if (= dim 1)
(aref m 0 0)
(loop for col below dim
for sign = 1 then (- sign)
sum (* sign (aref m 0 col) (det (minor m col)))))))
(defun replace-column (m col values)
(let* ((dim (array-dimension m 0))
(result (make-array (list dim dim))))
(dotimes (r dim result)
(dotimes (c dim)
(setf (aref result r c)
(if (= c col) (aref values r) (aref m r c)))))))
(defun solve (m v)
(loop with dim = (array-dimension m 0)
with det = (det m)
for col below dim
collect (/ (det (replace-column m col v)) det)))
(solve #2A((2 -1 5 1)
(3 2 2 -6)
(1 3 3 -1)
(5 -2 -3 3))
#(-3 -32 -47 49))
- Output:
(2 -12 -4 1)
D
import std.array : array, uninitializedArray;
import std.range : iota;
import std.stdio : writeln;
import std.typecons : tuple;
alias vector = double[4];
alias matrix = vector[4];
auto johnsonTrotter(int n) {
auto p = iota(n).array;
auto q = iota(n).array;
auto d = uninitializedArray!(int[])(n);
d[] = -1;
auto sign = 1;
int[][] perms;
int[] signs;
void permute(int k) {
if (k >= n) {
perms ~= p.dup;
signs ~= sign;
sign *= -1;
return;
}
permute(k + 1);
foreach (i; 0..k) {
auto z = p[q[k] + d[k]];
p[q[k]] = z;
p[q[k] + d[k]] = k;
q[z] = q[k];
q[k] += d[k];
permute(k + 1);
}
d[k] *= -1;
}
permute(0);
return tuple!("sigmas", "signs")(perms, signs);
}
auto determinant(matrix m) {
auto jt = johnsonTrotter(m.length);
auto sum = 0.0;
foreach (i,sigma; jt.sigmas) {
auto prod = 1.0;
foreach (j,s; sigma) {
prod *= m[j][s];
}
sum += jt.signs[i] * prod;
}
return sum;
}
auto cramer(matrix m, vector d) {
auto divisor = determinant(m);
auto numerators = uninitializedArray!(matrix[])(m.length);
foreach (i; 0..m.length) {
foreach (j; 0..m.length) {
foreach (k; 0..m.length) {
numerators[i][j][k] = m[j][k];
}
}
}
vector v;
foreach (i; 0..m.length) {
foreach (j; 0..m.length) {
numerators[i][j][i] = d[j];
}
}
foreach (i; 0..m.length) {
v[i] = determinant(numerators[i]) / divisor;
}
return v;
}
void main() {
matrix m = [
[2.0, -1.0, 5.0, 1.0],
[3.0, 2.0, 2.0, -6.0],
[1.0, 3.0, 3.0, -1.0],
[5.0, -2.0, -3.0, 3.0]
];
vector d = [-3.0, -32.0, -47.0, 49.0];
auto wxyz = cramer(m, d);
writeln("w = ", wxyz[0], ", x = ", wxyz[1], ", y = ", wxyz[2], ", z = ", wxyz[3]);
}
- Output:
w = 2, x = -12, y = -4, z = 1
EasyLang
proc det . a0[][] res .
res = 1
a[][] = a0[][]
n = len a[][]
for j to n
imax = j
for i = j + 1 to n
if a[i][j] > a[imax][j]
imax = i
.
.
if imax <> j
swap a[imax][] a[j][]
res = -res
.
if abs a[j][j] < 1e-12
print "Singular matrix!"
res = 0 / 0
return
.
for i = j + 1 to n
mult = -a[i][j] / a[j][j]
for k to n
a[i][k] += mult * a[j][k]
.
.
.
for i to n
res *= a[i][i]
.
.
proc cramer_solve . a0[][] deta b[] col res .
a[][] = a0[][]
for i to len a[][]
a[i][col] = b[i]
.
det a[][] d
res = d / deta
.
a[][] = [ [ 2 -1 5 1 ] [ 3 2 2 -6 ] [ 1 3 3 -1 ] [ 5 -2 -3 3 ] ]
b[] = [ -3 -32 -47 49 ]
det a[][] deta
for i to len a[][]
cramer_solve a[][] deta b[] i r
print r
.
- Output:
2.00 -12 -4 1.00
EchoLisp
(lib 'matrix)
(string-delimiter "")
(define (cramer A B (X)) ;; --> vector X
(define ∆ (determinant A))
(for/vector [(i (matrix-col-num A))]
(set! X (matrix-set-col! (array-copy A) i B))
(// (determinant X) ∆)))
(define (task)
(define A (list->array
'( 2 -1 5 1 3 2 2 -6 1 3 3 -1 5 -2 -3 3) 4 4))
(define B #(-3 -32 -47 49))
(writeln "Solving A * X = B")
(array-print A)
(writeln "B = " B)
(writeln "X = " (cramer A B)))
- Output:
(task) Solving A * X = B 2 -1 5 1 3 2 2 -6 1 3 3 -1 5 -2 -3 3 B = #( -3 -32 -47 49) X = #( 2 -12 -4 1)
Factor
USING: kernel math math.matrices.laplace prettyprint sequences ;
IN: rosetta-code.cramers-rule
: replace-col ( elt n seq -- seq' ) flip [ set-nth ] keep flip ;
: solve ( m v -- seq )
dup length <iota> [
rot [ replace-col ] keep [ determinant ] bi@ /
] 2with map ;
: cramers-rule-demo ( -- )
{
{ 2 -1 5 1 }
{ 3 2 2 -6 }
{ 1 3 3 -1 }
{ 5 -2 -3 3 }
}
{ -3 -32 -47 49 } solve . ;
MAIN: cramers-rule-demo
- Output:
{ 2 -12 -4 1 }
Fortran
In Numerical Methods That Work (Usually), in the section What not to compute, F. S. Acton remarks "...perhaps we should be glad he didn't resort to Cramer's rule (still taught as the practical method in some high schools) and solve his equations as the ratios of determinants - a process that requires labor proportional to N! if done in the schoolboy manner. The contrast with N3 can be startling!" And further on, "Having hinted darkly at my computational fundamentalism, it is probably time to commit to a public heresy by denouncing recursive calculations. I have never seen a numerical problem arising from the physical world that was best calculated by a recursive subroutine..."
Since this problem requires use of Cramer's rule, one might as well be hung for a sheep instead of a lamb, so the traditions of Old Fortran and heavy computation will be ignored and the fearsome RECURSIVE specification employed so that the determinants will be calculated recursively, all the way down to N = 1 even though the N = 2 case is easy. This requires F90 and later. Similarly, the MODULE protocol will be employed, even though there is no significant context to share. The alternative method for calculating a determinant involves generating permutations, a tiresome process.
Array passing via the modern arrangements of F90 is a source of novel difficulty to set against the slight convenience of not having to pass an additional parameter, N. Explicitly, at least. There are "secret" additional parameters when an array is being passed in the modern way, which are referred to by the new SIZE function. Anyway, for an order N square matrix, the array must be declared A(N,N), and specifically not something like A(100,100) with usage only of elements up to N = 7, say, because the locations in storage of elements in use would be quite different from those used by an array declared A(7,7). This means that the array must be re-declared for each different size usage, a tiresome and error-inviting task. One-dimensional arrays do not have this problem, but they do have to be "long enough" so B and X might as well be included. This also means that the auxiliary matrices needed within the routines have to be made the right size, and fortunately they can be declared in a way that requests this without the blather of ALLOCATE, this being a protocol introduced by Algol in the 1960s. Unfortunately, there is no scheme such as in pl/i to declare AUX "like" A, so some grotesquery results. And in the case of function DET which needs an array of order N - 1, when its recursion bottoms out with N = 1 it will have declared MINOR(0,0), a rather odd situation that fortunately evokes no complaint, and a test run in which its "value" was written out by WRITE (6,*) MINOR produced a blank line: no complaint there either, presumably because zero elements were being sent forth and so there was no improper access of ... nothing.
With matrices, there is a problem all the way from the start in 1958. Everyone agrees that a matrix should be indexed as A(row,column) and that when written out, rows should run down the page and columns across. This is unexceptional and the F90 function MATMUL follows this usage. However, Fortran has always stored its array elements in what is called "column major" order, which is to say that element A(1,1) is followed by element A(2,1) in storage, not A(1,2). Thus, if an array is written (or read) by something like WRITE (6,*) A, consecutive elements, written along a line, will be A(1,1), A(2,1), A(3,1), ... So, subroutine SHOWMATRIX is employed to write the matrix out in the desired form, and to read the values into the array, an explicit loop is used to place them where expected rather than just READ(INF,*) A
Similarly, if instead a DATA statement were used to initialise the array for the example problem, and it looked something like
DATA A/2, -1, 5, 1
1 3, 2, 2, -6
2 1, 3, 3, -1
3 5, -2, -3, 3/
(ignoring integer/floating-point issues) thus corresponding to the layout of the example problem, there would need to be a statement A = TRANSPOSE(A) to obtain the required order.
I have never seen an explanation of why this choice was made for Fortran.
MODULE BAD IDEA !Employ Cramer's rule for solving A.x = b...
INTEGER MSG !Might as well be here.
CONTAINS !The details.
SUBROUTINE SHOWMATRIX(A) !With nice vertical bars.
DOUBLE PRECISION A(:,:) !The matrix.
INTEGER R,N !Assistants.
N = SIZE(A, DIM = 1) !Instead of passing an explicit parameter.
DO R = 1,N !Work down the rows.
WRITE (MSG,1) A(R,:) !Fling forth a row at a go.
1 FORMAT ("|",<N>F12.3,"|") !Bounded by bars.
END DO !On to the next row.
END SUBROUTINE SHOWMATRIX !Furrytran's default order is the transpose.
RECURSIVE DOUBLE PRECISION FUNCTION DET(A) !Determine the determinant.
DOUBLE PRECISION A(:,:) !The square matrix, order N.
DOUBLE PRECISION MINOR(SIZE(A,DIM=1) - 1,SIZE(A,DIM=1) - 1) !Order N - 1.
DOUBLE PRECISION D !A waystation.
INTEGER C,N !Steppers.
N = SIZE(A, DIM = 1) !Suplied via secret parameters.
IF (N .LE. 0) THEN !This really ought not happen.
STOP "DET: null array!" !But I'm endlessly suspicious.
ELSE IF (N .NE. SIZE(A, DIM = 2)) THEN !And I'd rather have a decent message
STOP "DET: not a square array!" !In place of a crashed run.
ELSE IF (N .EQ. 1) THEN !Alright, now get on with it.
DET = A(1,1) !This is really easy.
ELSE !But otherwise,
D = 0 !Here we go.
DO C = 1,N !Step along the columns of the first row.
CALL FILLMINOR(C) !Produce the auxiliary array for each column.
IF (MOD(C,2) .EQ. 0) THEN !Odd or even case?
D = D - A(1,C)*DET(MINOR) !Even: subtract.
ELSE !Otherwise,
D = D + A(1,C)*DET(MINOR) !Odd: add.
END IF !So much for that term.
END DO !On to the next.
DET = D !Declare the result.
END IF !So much for that.
CONTAINS !An assistant.
SUBROUTINE FILLMINOR(CC) !Corresponding to A(1,CC).
INTEGER CC !The column being omitted.
INTEGER R !A stepper.
DO R = 2,N !Ignoring the first row,
MINOR(R - 1,1:CC - 1) = A(R,1:CC - 1) !Copy columns 1 to CC - 1. There may be none.
MINOR(R - 1,CC:) = A(R,CC + 1:) !And from CC + 1 to N. There may be none.
END DO !On to the next row.
END SUBROUTINE FILLMINOR !Divide and conquer.
END FUNCTION DET !Rather than mess with permutations.
SUBROUTINE CRAMER(A,X,B) !Solve A.x = b, where A is a matrix...
Careful! The array must be A(N,N), and not say A(100,100) of which only up to N = 6 are in use.
DOUBLE PRECISION A(:,:) !A square matrix. I hope.
DOUBLE PRECISION X(:),B(:) !Column matrices look rather like 1-D arrays.
DOUBLE PRECISION AUX(SIZE(A,DIM=1),SIZE(A,DIM=1)) !Can't say "LIKE A", as in pl/i, alas.
DOUBLE PRECISION D !To be calculated once.
INTEGER N !The order of the square matrix. I hope.
INTEGER C !A stepper.
N = SIZE(A, DIM = 1) !Alright, what's the order of battle?
D = DET(A) !Once only.
IF (D.EQ.0) STOP "Cramer: zero determinant!" !Surely, this won't happen...
AUX = A !Prepare the assistant.
DO C = 1,N !Step across the columns.
IF (C.GT.1) AUX(1:N,C - 1) = A(1:N,C - 1) !Repair previous damage.
AUX(1:N,C) = B(1:N) !Place the current damage.
X(C) = DET(AUX)/D !The result!
END DO !On to the next column.
END SUBROUTINE CRAMER !This looks really easy!
END MODULE BAD IDEA !But actually, it is a bad idea for N > 2.
PROGRAM TEST !Try it and see.
USE BAD IDEA !Just so.
DOUBLE PRECISION, ALLOCATABLE ::A(:,:), B(:), X(:) !Custom work areas.
INTEGER N,R !Assistants..
INTEGER INF !An I/O unit.
MSG = 6 !Output.
INF = 10 !For reading test data.
OPEN (INF,FILE="Test.dat",STATUS="OLD",ACTION="READ") !As in this file..
Chew into the next problem.
10 IF (ALLOCATED(A)) DEALLOCATE(A) !First,
IF (ALLOCATED(B)) DEALLOCATE(B) !Get rid of
IF (ALLOCATED(X)) DEALLOCATE(X) !The hired help.
READ (INF,*,END = 100) N !Since there is a new order.
IF (N.LE.0) GO TO 100 !Perhaps a final order.
WRITE (MSG,11) N !Othewise, announce prior to acting.
11 FORMAT ("Order ",I0," matrix A, as follows...") !In case something goes wrong.
ALLOCATE(A(N,N)) !For instance,
ALLOCATE(B(N)) !Out of memory.
ALLOCATE(X(N)) !But otherwise, a tailored fit.
DO R = 1,N !Now read in the values for the matrix.
READ(INF,*,END=667,ERR=665) A(R,:),B(R) !One row of A at a go, followed by B's value.
END DO !In free format.
CALL SHOWMATRIX(A) !Show what we have managed to obtain.
WRITE (MSG,12) "In the scheme A.x = b, b = ",B !In case something goes wrong.
12 FORMAT (A,<N>F12.6) !How many would be too many?
CALL CRAMER(A,X,B) !The deed!
WRITE (MSG,12) " Via Cramer's rule, x = ",X !The result!
GO TO 10 !And try for another test problem.
Complaints.
665 WRITE (MSG,666) "Format error",R !I know where I came from.
666 FORMAT (A," while reading row ",I0,"!") !So I can refer to R.
GO TO 100 !So much for that.
667 WRITE (MSG,666) "End-of-file", R !Some hint as to where.
Closedown.
100 WRITE (6,*) "That was interesting." !Quite.
END !Open files are closed, allocated memory is released.
Oddly, the Compaq Visual Fortran F90/95 compiler is confused by the usage "BAD IDEA" instead of "BADIDEA" - spaces are not normally relevant in Fortran source. Anyway, file Test.dat has been filled with the numbers of the example, as follows:
4 /The order, for A.x = b. 2 -1 5 1, -3 /First row of A, b 3 2 2 -6, -32 /Second row... 1 3 3 -1, -47 third row. 5 -2 -3 3, 49 /Last row.
Fortran's free-form allows a comma, a tab, and spaces between numbers, and regards the / as starting a comment, but, because each row is read separately, once the required five (N + 1) values have been read, no further scanning of the line takes place and the next READ statement will start with a new line of input. So the / isn't needed for the third row, as shown. Omitted values lead to confusion as the input process would read additional lines to fill the required count and everything gets out of step. Echoing input very soon after it is obtained is helpful in making sense of such mistakes.
For more practical use it would probably be better to constrain the freedom somewhat, perhaps requiring that all the N + 1 values for a row appear on one input record. In such a case, the record could first be read into a text variable (from which the data would be read) so that if a problem arises the text could be printed as a part of the error message. But, this requires guessing a suitably large length for the text variable so as to accommodate the longest possible input line.
Output:
Order 4 matrix A, as follows...
| 2.000 -1.000 5.000 1.000|
| 3.000 2.000 2.000 -6.000|
| 1.000 3.000 3.000 -1.000|
| 5.000 -2.000 -3.000 3.000|
In the scheme A.x = b, b = -3.000000 -32.000000 -47.000000 49.000000
Via Cramer's rule, x = 2.000000 -12.000000 -4.000000 1.000000
That was interesting.
And at this point I suddenly noticed that the habits of Old Fortran are not so easily suppressed: all calculations are done with double precision. Curiously enough, for the specific example data, the same results are obtained if all variables are integer.
FreeBASIC
Function determinant(matrix() As Double) As Double
Dim As long n=Ubound(matrix,1),sign=1
Dim As Double det=1,s=1
Dim As Double b(1 To n,1 To n)
For c As long=1 To n
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
sign=-sign
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)
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)
s=s*f
For g As long=1 To n
b((row+1),g)=(b((row+1),g)*f-b(k,g))/f
Next g
Next row
Next k
For z As long=1 To n
det=det*b(z,z)
Next z
Return sign*det
End Function
'CRAMER COLUMN SWAPS
Sub swapcolumn(m() As Double,c() As Double,_new() As Double,column As long)
Redim _new(1 To Ubound(m,1),1 To Ubound(m,1))
For x As long=1 To Ubound(m,1)
For y As long=1 To Ubound(m,1)
_new(x,y)=m(x,y)
Next y
Next x
For z As long=1 To Ubound(m,1)
_new(z,column)=c(z)
Next z
End Sub
Sub solve(mat() As Double,rhs() As Double,_out() As Double)
redim _out(Lbound(mat,1) To Ubound(mat,1))
Redim As Double result(Lbound(mat,1) To Ubound(mat,1),Lbound(mat,1) To Ubound(mat,1))
Dim As Double maindeterminant=determinant(mat())
If Abs(maindeterminant) < 1e-12 Then Print "singular":Exit Sub
For column As Long=1 To Ubound(mat,1)
swapcolumn(mat(),rhs(),result(),column)
_out(column)= determinant(result())/maindeterminant
Next
End Sub
Dim As Double MainMat(1 To ...,1 To ...)={{2,-1,5,1}, _
{3,2,2,-6}, _
{1,3,3,-1}, _
{5,-2,-3,3}}
Dim As Double rhs(1 To ...)={-3,-32,-47,49}
Redim ans() As Double
solve(MainMat(),rhs(),ans())
For n As Long=1 To Ubound(ans)
Print Csng(ans(n))
Next
Sleep
- Output:
2 -12 -4 1
Go
Library gonum:
package main
import (
"fmt"
"gonum.org/v1/gonum/mat"
)
var m = mat.NewDense(4, 4, []float64{
2, -1, 5, 1,
3, 2, 2, -6,
1, 3, 3, -1,
5, -2, -3, 3,
})
var v = []float64{-3, -32, -47, 49}
func main() {
x := make([]float64, len(v))
b := make([]float64, len(v))
d := mat.Det(m)
for c := range v {
mat.Col(b, c, m)
m.SetCol(c, v)
x[c] = mat.Det(m) / d
m.SetCol(c, b)
}
fmt.Println(x)
}
- Output:
[2 -12.000000000000007 -4.000000000000001 1.0000000000000009]
Library go.matrix:
package main
import (
"fmt"
"github.com/skelterjohn/go.matrix"
)
var m = matrix.MakeDenseMatrixStacked([][]float64{
{2, -1, 5, 1},
{3, 2, 2, -6},
{1, 3, 3, -1},
{5, -2, -3, 3},
})
var v = []float64{-3, -32, -47, 49}
func main() {
x := make([]float64, len(v))
b := make([]float64, len(v))
d := m.Det()
for c := range v {
m.BufferCol(c, b)
m.FillCol(c, v)
x[c] = m.Det() / d
m.FillCol(c, b)
}
fmt.Println(x)
}
- Output:
[2.0000000000000004 -11.999999999999998 -4 0.9999999999999999]
Groovy
class CramersRule {
static void main(String[] args) {
Matrix mat = new Matrix(Arrays.asList(2d, -1d, 5d, 1d),
Arrays.asList(3d, 2d, 2d, -6d),
Arrays.asList(1d, 3d, 3d, -1d),
Arrays.asList(5d, -2d, -3d, 3d))
List<Double> b = Arrays.asList(-3d, -32d, -47d, 49d)
println("Solution = " + cramersRule(mat, b))
}
private static List<Double> cramersRule(Matrix matrix, List<Double> b) {
double denominator = matrix.determinant()
List<Double> result = new ArrayList<>()
for (int i = 0; i < b.size(); i++) {
result.add(matrix.replaceColumn(b, i).determinant() / denominator)
}
return result
}
private static class Matrix {
private List<List<Double>> matrix
@Override
String toString() {
return matrix.toString()
}
@SafeVarargs
Matrix(List<Double>... lists) {
matrix = new ArrayList<>()
for (List<Double> list : lists) {
matrix.add(list)
}
}
Matrix(List<List<Double>> mat) {
matrix = mat
}
double determinant() {
if (matrix.size() == 1) {
return get(0, 0)
}
if (matrix.size() == 2) {
return get(0, 0) * get(1, 1) - get(0, 1) * get(1, 0)
}
double sum = 0
double sign = 1
for (int i = 0; i < matrix.size(); i++) {
sum += sign * get(0, i) * coFactor(0, i).determinant()
sign *= -1
}
return sum
}
private Matrix coFactor(int row, int col) {
List<List<Double>> mat = new ArrayList<>()
for (int i = 0; i < matrix.size(); i++) {
if (i == row) {
continue
}
List<Double> list = new ArrayList<>()
for (int j = 0; j < matrix.size(); j++) {
if (j == col) {
continue
}
list.add(get(i, j))
}
mat.add(list)
}
return new Matrix(mat)
}
private Matrix replaceColumn(List<Double> b, int column) {
List<List<Double>> mat = new ArrayList<>()
for (int row = 0; row < matrix.size(); row++) {
List<Double> list = new ArrayList<>()
for (int col = 0; col < matrix.size(); col++) {
double value = get(row, col)
if (col == column) {
value = b.get(row)
}
list.add(value)
}
mat.add(list)
}
return new Matrix(mat)
}
private double get(int row, int col) {
return matrix.get(row).get(col)
}
}
}
- Output:
Solution = [2.0, -12.0, -4.0, 1.0]
Haskell
Version 1
import Data.Matrix
solveCramer :: (Ord a, Fractional a) => Matrix a -> Matrix a -> Maybe [a]
solveCramer a y
| da == 0 = Nothing
| otherwise = Just $ map (\i -> d i / da) [1..n]
where da = detLU a
d i = detLU $ submatrix 1 n 1 n $ switchCols i (n+1) ay
ay = a <|> y
n = ncols a
task = solveCramer a y
where a = fromLists [[2,-1, 5, 1]
,[3, 2, 2,-6]
,[1, 3, 3,-1]
,[5,-2,-3, 3]]
y = fromLists [[-3], [-32], [-47], [49]]
- Output:
λ> task Just [2.0000000000000004,-11.999999999999998,-4.0,0.9999999999999999]
Version 2
We use Rational numbers for having more precision. a % b is the rational a / b.
s_permutations :: [a] -> [([a], Int)]
s_permutations = flip zip (cycle [1, -1]) . (foldl aux [[]])
where aux items x = do
(f,item) <- zip (cycle [reverse,id]) items
f (insertEv x item)
insertEv x [] = [[x]]
insertEv x l@(y:ys) = (x:l) : map (y:) (insertEv x ys)
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
matI::(Num a) => Int -> [[a]]
matI n = [ [fromIntegral.fromEnum $ i == j | i <- [1..n]] | j <- [1..n]]
elemPos::[[a]] -> Int -> Int -> a
elemPos ms i j = (ms !! i) !! j
prod:: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [Int] -> a
prod f ms = product.zipWith (f ms) [0..]
s_determinant:: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [([Int],Int)] -> a
s_determinant f ms = sum.map (\(is,s) -> fromIntegral s * prod f ms is)
elemCramerPos::Int -> Int -> [[a]] -> [[a]] -> Int -> Int -> a
elemCramerPos l k ks ms i j = if j /= l then elemPos ms i j else elemPos ks i k
solveCramer:: [[Rational]] -> [[Rational]] -> [[Rational]]
solveCramer ms ks = xs
where
xs | d /= 0 = go (reverse [0..pred.length.head $ ks])
| otherwise = []
go (u:us) = foldl glue (col u) us
glue us u = zipWith (\ys (y:_) -> y:ys) us (col u)
col k = map (\l -> [(/d) $ s_determinant (elemCramerPos l k ks) ms ps]) $ ls
ls = [0..pred.length $ ms]
ps = s_permutations ls
d = s_determinant elemPos ms ps
task::[[Rational]] -> [[Rational]] -> IO()
task a b = do
let x = solveCramer a b
let u = map (map fromRational) x
let y = mult a x
let identity = matI (length x)
let a1 = solveCramer 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 = solveCramer 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 = solveCramer 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 = [[2,-1, 5, 1]
,[3, 2, 2,-6]
,[1, 3, 3,-1]
,[5,-2,-3, 3]]
let b = [[-3], [-32], [-47], [49]]
task a b
- Output:
a = [2 % 1,(-1) % 1,5 % 1,1 % 1] [3 % 1,2 % 1,2 % 1,(-6) % 1] [1 % 1,3 % 1,3 % 1,(-1) % 1] [5 % 1,(-2) % 1,(-3) % 1,3 % 1] b = [(-3) % 1] [(-32) % 1] [(-47) % 1] [49 % 1] solve: a * x = b => x = solveCramer a b = [2 % 1] [(-12) % 1] [(-4) % 1] [1 % 1] u = fromRationaltoDouble x = [2.0] [-12.0] [-4.0] [1.0] verification: y = a * x = mult a x = [(-3) % 1] [(-32) % 1] [(-47) % 1] [49 % 1] test: y == b = True identity matrix: identity = [1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1] find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity = [4 % 171,11 % 171,10 % 171,8 % 57] [(-55) % 342,(-23) % 342,119 % 342,2 % 57] [107 % 684,(-5) % 684,11 % 684,(-7) % 114] [7 % 684,(-109) % 684,103 % 684,7 % 114] verification: h = a * a1 = mult a a1 = [1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1] test: h == identity = True z = a1 * b = mult a1 b = [2 % 1] [(-12) % 1] [(-4) % 1] [1 % 1] test: z == x = True
Version 3
import Data.List
determinant::(Fractional a, Ord a) => [[a]] -> a
determinant ls = if null ls then 0 else pivot 1 (zip ls [(0::Int)..])
where
good rs ts = (abs.head.fst $ ts) <= (abs.head.fst $ rs)
go us (vs,i) = if v == 0 then (ws,i) else (zipWith (\x y -> y - x*v) us ws,i)
where (v,ws) = (head $ vs,tail vs)
change i (ys:zs) = map (\xs -> if (==i).snd $ xs then ys else xs) zs
pivot d [] = d
pivot d zs@((_,j):ys) = if 0 == u then 0 else pivot e ws
where
e = if i == j then u*d else -u*d
((u:us),i) = foldl1 (\rs ts -> if good rs ts then rs else ts) zs
ws = map (go (map (/u) us)) $ if i == j then ys else change i zs
solveCramer::(Fractional a, Ord a) => [[a]] -> [[a]] -> [[a]]
solveCramer as bs = if 0 == d then [] else ans bs
where
d = determinant as
ans = transpose.map go.transpose
where
ms = zip [0..] (transpose as)
go us = [ (/d) $ determinant [if i /= j then vs else us | (j,vs) <- ms] | (i,_) <- ms]
matI::(Num a) => Int -> [[a]]
matI n = [ [fromIntegral.fromEnum $ i == j | i <- [1..n]] | j <- [1..n]]
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
task::[[Rational]] -> [[Rational]] -> IO()
task a b = do
let x = solveCramer a b
let u = map (map fromRational) x
let y = mult a x
let identity = matI (length x)
let a1 = solveCramer 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 = solveCramer 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 = solveCramer 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 = [[2,-1, 5, 1]
,[3, 2, 2,-6]
,[1, 3, 3,-1]
,[5,-2,-3, 3]]
let b = [[-3], [-32], [-47], [49]]
task a b
- Output:
a = [2 % 1,(-1) % 1,5 % 1,1 % 1] [3 % 1,2 % 1,2 % 1,(-6) % 1] [1 % 1,3 % 1,3 % 1,(-1) % 1] [5 % 1,(-2) % 1,(-3) % 1,3 % 1] b = [(-3) % 1] [(-32) % 1] [(-47) % 1] [49 % 1] solve: a * x = b => x = solveCramer a b = [2 % 1] [(-12) % 1] [(-4) % 1] [1 % 1] u = fromRationaltoDouble x = [2.0] [-12.0] [-4.0] [1.0] verification: y = a * x = mult a x = [(-3) % 1] [(-32) % 1] [(-47) % 1] [49 % 1] test: y == b = True identity matrix: identity = [1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1] find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity = [4 % 171,11 % 171,10 % 171,8 % 57] [(-55) % 342,(-23) % 342,119 % 342,2 % 57] [107 % 684,(-5) % 684,11 % 684,(-7) % 114] [7 % 684,(-109) % 684,103 % 684,7 % 114] verification: h = a * a1 = mult a a1 = [1 % 1,0 % 1,0 % 1,0 % 1] [0 % 1,1 % 1,0 % 1,0 % 1] [0 % 1,0 % 1,1 % 1,0 % 1] [0 % 1,0 % 1,0 % 1,1 % 1] test: h == identity = True z = a1 * b = mult a1 b = [2 % 1] [(-12) % 1] [(-4) % 1] [1 % 1] test: z == x = True
J
Implementation:
cramer=:4 :0
A=. x [ b=. y
det=. -/ .*
A %~&det (i.#A) b"_`[`]}&.|:"0 2 A
)
Task data:
A=: _&".;._2]t=: 0 :0
2 -1 5 1
3 2 2 -6
1 3 3 -1
5 -2 -3 3
)
b=: _3 _32 _47 49
Task example:
A cramer b
2 _12 _4 1
Java
Supports double data type. A more robust solution would support arbitrary precision integers, arbitrary precision decimals, arbitrary precision rationals, or even arbitrary precision algebraic numbers.
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public class CramersRule {
public static void main(String[] args) {
Matrix mat = new Matrix(Arrays.asList(2d, -1d, 5d, 1d),
Arrays.asList(3d, 2d, 2d, -6d),
Arrays.asList(1d, 3d, 3d, -1d),
Arrays.asList(5d, -2d, -3d, 3d));
List<Double> b = Arrays.asList(-3d, -32d, -47d, 49d);
System.out.println("Solution = " + cramersRule(mat, b));
}
private static List<Double> cramersRule(Matrix matrix, List<Double> b) {
double denominator = matrix.determinant();
List<Double> result = new ArrayList<>();
for ( int i = 0 ; i < b.size() ; i++ ) {
result.add(matrix.replaceColumn(b, i).determinant() / denominator);
}
return result;
}
private static class Matrix {
private List<List<Double>> matrix;
@Override
public String toString() {
return matrix.toString();
}
@SafeVarargs
public Matrix(List<Double> ... lists) {
matrix = new ArrayList<>();
for ( List<Double> list : lists) {
matrix.add(list);
}
}
public Matrix(List<List<Double>> mat) {
matrix = mat;
}
public double determinant() {
if ( matrix.size() == 1 ) {
return get(0, 0);
}
if ( matrix.size() == 2 ) {
return get(0, 0) * get(1, 1) - get(0, 1) * get(1, 0);
}
double sum = 0;
double sign = 1;
for ( int i = 0 ; i < matrix.size() ; i++ ) {
sum += sign * get(0, i) * coFactor(0, i).determinant();
sign *= -1;
}
return sum;
}
private Matrix coFactor(int row, int col) {
List<List<Double>> mat = new ArrayList<>();
for ( int i = 0 ; i < matrix.size() ; i++ ) {
if ( i == row ) {
continue;
}
List<Double> list = new ArrayList<>();
for ( int j = 0 ; j < matrix.size() ; j++ ) {
if ( j == col ) {
continue;
}
list.add(get(i, j));
}
mat.add(list);
}
return new Matrix(mat);
}
private Matrix replaceColumn(List<Double> b, int column) {
List<List<Double>> mat = new ArrayList<>();
for ( int row = 0 ; row < matrix.size() ; row++ ) {
List<Double> list = new ArrayList<>();
for ( int col = 0 ; col < matrix.size() ; col++ ) {
double value = get(row, col);
if ( col == column ) {
value = b.get(row);
}
list.add(value);
}
mat.add(list);
}
return new Matrix(mat);
}
private double get(int row, int col) {
return matrix.get(row).get(col);
}
}
}
- Output:
Solution = [2.0, -12.0, -4.0, 1.0]
JavaScript
var matrix = [
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3]
];
var freeTerms = [-3, -32, -47, 49];
var result = cramersRule(matrix,freeTerms);
console.log(result);
/**
* Compute Cramer's Rule
* @param {array} matrix x,y,z, etc. terms
* @param {array} freeTerms
* @return {array} solution for x,y,z, etc.
*/
function cramersRule(matrix,freeTerms) {
var det = detr(matrix),
returnArray = [],
i,
tmpMatrix;
for(i=0; i < matrix[0].length; i++) {
var tmpMatrix = insertInTerms(matrix, freeTerms,i)
returnArray.push(detr(tmpMatrix)/det)
}
return returnArray;
}
/**
* Inserts single dimensional array into
* @param {array} matrix multidimensional array to have ins inserted into
* @param {array} ins single dimensional array to be inserted vertically into matrix
* @param {array} at zero based offset for ins to be inserted into matrix
* @return {array} New multidimensional array with ins replacing the at column in matrix
*/
function insertInTerms(matrix, ins, at) {
var tmpMatrix = clone(matrix),
i;
for(i=0; i < matrix.length; i++) {
tmpMatrix[i][at] = ins[i];
}
return tmpMatrix;
}
/**
* Compute the determinate of a matrix. No protection, assumes square matrix
* function borrowed, and adapted from MIT Licensed numericjs library (www.numericjs.com)
* @param {array} m Input Matrix (multidimensional array)
* @return {number} result rounded to 2 decimal
*/
function detr(m) {
var ret = 1,
k,
A=clone(m),
n=m[0].length,
alpha;
for(var j =0; j < n-1; j++) {
k=j;
for(i=j+1;i<n;i++) { if(Math.abs(A[i][j]) > Math.abs(A[k][j])) { k = i; } }
if(k !== j) {
temp = A[k]; A[k] = A[j]; A[j] = temp;
ret *= -1;
}
Aj = A[j];
for(i=j+1;i<n;i++) {
Ai = A[i];
alpha = Ai[j]/Aj[j];
for(k=j+1;k<n-1;k+=2) {
k1 = k+1;
Ai[k] -= Aj[k]*alpha;
Ai[k1] -= Aj[k1]*alpha;
}
if(k!==n) { Ai[k] -= Aj[k]*alpha; }
}
if(Aj[j] === 0) { return 0; }
ret *= Aj[j];
}
return Math.round(ret*A[j][j]*100)/100;
}
/**
* Clone two dimensional Array using ECMAScript 5 map function and EcmaScript 3 slice
* @param {array} m Input matrix (multidimensional array) to clone
* @return {array} New matrix copy
*/
function clone(m) {
return m.map(function(a){return a.slice();});
}
- Output:
[ 2, -12, -4, 1 ]
jq
Adapted from Wren (*)
Works with gojq, the Go implementation of jq
(*) Note that cramer(a;d) as defined here assumes that d is a 1-d vector in accordance with the task description and common practice.
# The minor of the input matrix after removing the specified row and column.
# Assumptions: the input is square and the indices are hunky dory.
def minor(rowNum; colNum):
. as $in
| (length - 1) as $len
| reduce range(0; $len) as $i (null;
reduce range(0; $len) as $j (.;
if $i < rowNum and $j < colNum
then .[$i][$j] = $in[$i][$j]
elif $i >= rowNum and $j < colNum
then .[$i][$j] = $in[$i+1][$j]
elif $i < rowNum and $j >= colNum
then .[$i][$j] = $in[$i][$j+1]
else .[$i][$j] = $in[$i+1][$j+1]
end) );
# The determinant using Laplace expansion.
# Assumption: the matrix is square
def det:
. as $a
| length as $nr
| if $nr == 1 then .[0][0]
elif $nr == 2 then .[1][1] * .[0][0] - .[0][1] * .[1][0]
else reduce range(0; $nr) as $i (
{ sign: 1, sum: 0 };
($a|minor(0; $i)) as $m
| .sum += .sign * $a[0][$i] * ($m|det)
| .sign *= -1 )
| .sum
end ;
# Solve A X = D using Cramer's method
# a is assumed to be a JSON array representing the 2-d square matrix A
# d is assumed to be a JSON array representing the 1-d vector D
def cramer(a; d):
(a | length) as $n
| (a | det) as $ad
| if $ad == 0 then "matrix determinant is 0" | error
else reduce range(0; $n) as $c (null;
(reduce range(0; $n) as $r (a; .[$r][$c] = d[$r])) as $aa
| .[$c] = ($aa|det) / $ad )
end ;
def a: [
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3]
];
def d:
[ -3, -32, -47, 49 ] ;
"Solution is \(cramer(a; d))"- Output:
Solution is [2,-12,-4,1]
Julia
function cramersolve(A::Matrix, b::Vector)
return collect(begin B = copy(A); B[:, i] = b; det(B) end for i in eachindex(b)) ./ det(A)
end
A = [2 -1 5 1
3 2 2 -6
1 3 3 -1
5 -2 -3 3]
b = [-3, -32, -47, 49]
@show cramersolve(A, b)
- Output:
cramersolve(A, b) = [2.0, -12.0, -4.0, 1.0]
Note that it is entirely impractical to use Cramer's rule in this situation. It would be much better to use the built-in operator for solving linear systems. Assuming that the coefficient matrix and constant vector are defined as before, the solution vector is given by:
@show A \ b
Kotlin
As in the case of the Matrix arithmetic task, I've used the Johnson-Trotter permutations generator to assist with the calculation of the determinants for the various matrices:
// version 1.1.3
typealias Vector = DoubleArray
typealias Matrix = Array<Vector>
fun johnsonTrotter(n: Int): Pair<List<IntArray>, List<Int>> {
val p = IntArray(n) { it } // permutation
val q = IntArray(n) { it } // inverse permutation
val d = IntArray(n) { -1 } // direction = 1 or -1
var sign = 1
val perms = mutableListOf<IntArray>()
val signs = mutableListOf<Int>()
fun permute(k: Int) {
if (k >= n) {
perms.add(p.copyOf())
signs.add(sign)
sign *= -1
return
}
permute(k + 1)
for (i in 0 until k) {
val z = p[q[k] + d[k]]
p[q[k]] = z
p[q[k] + d[k]] = k
q[z] = q[k]
q[k] += d[k]
permute(k + 1)
}
d[k] *= -1
}
permute(0)
return perms to signs
}
fun determinant(m: Matrix): Double {
val (sigmas, signs) = johnsonTrotter(m.size)
var sum = 0.0
for ((i, sigma) in sigmas.withIndex()) {
var prod = 1.0
for ((j, s) in sigma.withIndex()) prod *= m[j][s]
sum += signs[i] * prod
}
return sum
}
fun cramer(m: Matrix, d: Vector): Vector {
val divisor = determinant(m)
val numerators = Array(m.size) { Matrix(m.size) { m[it].copyOf() } }
val v = Vector(m.size)
for (i in 0 until m.size) {
for (j in 0 until m.size) numerators[i][j][i] = d[j]
}
for (i in 0 until m.size) v[i] = determinant(numerators[i]) / divisor
return v
}
fun main(args: Array<String>) {
val m = arrayOf(
doubleArrayOf(2.0, -1.0, 5.0, 1.0),
doubleArrayOf(3.0, 2.0, 2.0, -6.0),
doubleArrayOf(1.0, 3.0, 3.0, -1.0),
doubleArrayOf(5.0, -2.0, -3.0, 3.0)
)
val d = doubleArrayOf(-3.0, -32.0, -47.0, 49.0)
val (w, x, y, z) = cramer(m, d)
println("w = $w, x = $x, y = $y, z = $z")
}
- Output:
w = 2.0, x = -12.0, y = -4.0, z = 1.0
Lua
local matrix = require "matrix" -- https://github.com/davidm/lua-matrix
local function cramer(mat, vec)
-- Check if matrix is quadratic
assert(#mat == #mat[1], "Matrix is not square!")
-- Check if vector has the same size of the matrix
assert(#mat == #vec, "Vector has not the same size of the matrix!")
local size = #mat
local main_det = matrix.det(mat)
local aux_mats = {}
local dets = {}
local result = {}
for i = 1, size do
-- Construct the auxiliary matrixes
aux_mats[i] = matrix.copy(mat)
for j = 1, size do
aux_mats[i][j][i] = vec[j]
end
-- Calculate the auxiliary determinants
dets[i] = matrix.det(aux_mats[i])
-- Calculate results
result[i] = dets[i]/main_det
end
return result
end
-----------------------------------------------
local A = {{ 2, -1, 5, 1},
{ 3, 2, 2, -6},
{ 1, 3, 3, -1},
{ 5, -2, -3, 3}}
local b = {-3, -32, -47, 49}
local result = cramer(A, b)
print("Result: " .. table.concat(result, ", "))
- Output:
Result: 2, -12, -4, 1
Maple
with(LinearAlgebra):
cramer:=proc(A,B)
local n,d,X,V,i;
n:=upperbound(A,2);
d:=Determinant(A);
X:=Vector(n,0);
for i from 1 to n do
V:=A(1..-1,i);
A(1..-1,i):=B;
X[i]:=Determinant(A)/d;
A(1..-1,i):=V;
od;
X;
end:
A:=Matrix([[2,-1,5,1],[3,2,2,-6],[1,3,3,-1],[5,-2,-3,3]]):
B:=Vector([-3,-32,-47,49]):
printf("%a",cramer(A,B));- Output:
Vector(4, [2,-12,-4,1])
Mathematica /Wolfram Language
crule[m_, b_] := Module[{d = Det[m], a},
Table[a = m; a[[All, k]] = b; Det[a]/d, {k, Length[m]}]]
crule[{
{2, -1, 5, 1},
{3, 2, 2, -6},
{1, 3, 3, -1},
{5, -2, -3, 3}
} , {-3, -32, -47, 49}]
- Output:
{2,-12,-4,1}
Maxima
(%i1) eqns: [ 2*w-x+5*y+z=-3, 3*w+2*x+2*y-6*z=-32, w+3*x+3*y-z=-47, 5*w-2*x-3*y+3*z=49];
(%o1) [z + 5 y - x + 2 w = - 3, (- 6 z) + 2 y + 2 x + 3 w = - 32,
(- z) + 3 y + 3 x + w = - 47, 3 z - 3 y - 2 x + 5 w = 49]
(%i2) A: augcoefmatrix (eqns, [w,x,y,z]);
[ 2 - 1 5 1 3 ]
[ ]
[ 3 2 2 - 6 32 ]
(%o2) [ ]
[ 1 3 3 - 1 47 ]
[ ]
[ 5 - 2 - 3 3 - 49 ]
(%i3) C: coefmatrix(eqns, [w,x,y,z]);
[ 2 - 1 5 1 ]
[ ]
[ 3 2 2 - 6 ]
(%o3) [ ]
[ 1 3 3 - 1 ]
[ ]
[ 5 - 2 - 3 3 ]
(%i4) c[n]:= (-1)^(n+1) * determinant (submatrix (A,n))/determinant (C);
n + 1
(- 1) determinant(submatrix(A, n))
(%o4) c := ---------------------------------------
n determinant(C)
(%i5) makelist (c[n],n,1,4);
(%o5) [2, - 12, - 4, 1]
(%i6) linsolve(eqns, [w,x,y,z]);
(%o6) [w = 2, x = - 12, y = - 4, z = 1]
Nim
type
SquareMatrix[N: static Positive] = array[N, array[N, float]]
Vector[N: static Positive] = array[N, float]
####################################################################################################
# Templates.
template `[]`(m: SquareMatrix; i, j: Natural): float =
## Allow to get value of an element using m[i, j] syntax.
m[i][j]
template `[]=`(m: var SquareMatrix; i, j: Natural; val: float) =
## Allow to set value of an element using m[i, j] syntax.
m[i][j] = val
#---------------------------------------------------------------------------------------------------
func det(m: SquareMatrix): float =
## Return the determinant of matrix "m".
var m = m
result = 1
for j in 0..m.high:
var imax = j
for i in (j + 1)..m.high:
if m[i, j] > m[imax, j]:
imax = i
if imax != j:
swap m[iMax], m[j]
result = -result
if abs(m[j, j]) < 1e-12:
return NaN
for i in (j + 1)..m.high:
let mult = -m[i, j] / m[j, j]
for k in 0..m.high:
m[i, k] += mult * m[j, k]
for i in 0..m.high:
result *= m[i, i]
#---------------------------------------------------------------------------------------------------
func cramerSolve(a: SquareMatrix; detA: float; b: Vector; col: Natural): float =
## Apply Cramer rule on matrix "a", using vector "b" to replace column "col".
when a.N != b.N:
{.error: "incompatible matrix and vector sizes".}
else:
var a = a
for i in 0..a.high:
a[i, col] = b[i]
result = det(a) / detA
#———————————————————————————————————————————————————————————————————————————————————————————————————
import strformat
const
A: SquareMatrix[4] = [[2.0, -1.0, 5.0, 1.0],
[3.0, 2.0, 2.0, -6.0],
[1.0, 3.0, 3.0, -1.0],
[5.0, -2.0, -3.0, 3.0]]
B: Vector[4] = [-3.0, -32.0, -47.0, 49.0]
let detA = det(A)
if detA == NaN:
echo "Singular matrix!"
quit(QuitFailure)
for i in 0..A.high:
echo &"{cramerSolve(A, detA, B, i):7.3f}"
- Output:
2.000 -12.000 -4.000 1.000
PARI/GP
M = [2,-1,5,1;3,2,2,-6;1,3,3,-1;5,-2,-3,3];
V = Col([-3,-32,-47,49]);
matadjoint(M) * V / matdet(M)Output:
[2, -12, -4, 1]~
Perl
use Math::Matrix;
sub cramers_rule {
my ($A, $terms) = @_;
my @solutions;
my $det = $A->determinant;
foreach my $i (0 .. $#{$A}) {
my $Ai = $A->clone;
foreach my $j (0 .. $#{$terms}) {
$Ai->[$j][$i] = $terms->[$j];
}
push @solutions, $Ai->determinant / $det;
}
@solutions;
}
my $matrix = Math::Matrix->new(
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3],
);
my $free_terms = [-3, -32, -47, 49];
my ($w, $x, $y, $z) = cramers_rule($matrix, $free_terms);
print "w = $w\n";
print "x = $x\n";
print "y = $y\n";
print "z = $z\n";
- Output:
w = 2 x = -12 y = -4 z = 1
Phix
The copy-on-write semantics of Phix really shine here; because there is no explicit return/re-assign, you can treat parameters as a private workspace, confident in the knowledge that the updated version will be quietly discarded; all the copying and freeing of the C version is automatic/unnecessary here.
UPDATE: For the next release, "with js" (or "with javascript_semantics") diables said copy-on-write semantics, so this now needs a couple of deep_copy() calls.
requires("0.8.4") with javascript_semantics constant inf = 1e300*1e300, nan = -(inf/inf) function det(sequence a) atom res = 1 a = deep_copy(a) integer l = length(a) for j=1 to l do integer i_max = j for i=j+1 to l do if a[i][j] > a[i_max][j] then i_max = i end if end for if i_max != j then sequence aim = a[i_max] a[i_max] = a[j] a[j] = aim res *= -1 end if if abs(a[j][j]) < 1e-12 then puts(1,"Singular matrix!") return nan end if for i=j+1 to l do atom mult = -a[i][j] / a[j][j] for k=1 to l do a[i][k] += mult * a[j][k] end for end for end for for i=1 to l do res *= a[i][i] end for return res end function function cramer_solve(sequence a, atom det_a, sequence b, integer v) a = deep_copy(a) for i=1 to length(a) do a[i][v] = b[i] end for return det(a)/det_a end function sequence a = {{2,-1, 5, 1}, {3, 2, 2,-6}, {1, 3, 3,-1}, {5,-2,-3, 3}}, b = {-3,-32,-47,49} integer det_a = det(a) for i=1 to length(a) do printf(1, "%7.3f\n", cramer_solve(a, det_a, b, i)) end for
- Output:
2.000 -12.000 -4.000 1.000
Prolog
removeElement([_|Tail], 0, Tail).
removeElement([Head|Tail], J, [Head|X]) :-
J_2 is J - 1,
removeElement(Tail, J_2, X).
removeColumn([], _, []).
removeColumn([Matrix_head|Matrix_tail], J, [X|Y]) :-
removeElement(Matrix_head, J, X),
removeColumn(Matrix_tail, J, Y).
removeRow([_|Matrix_tail], 0, Matrix_tail).
removeRow([Matrix_head|Matrix_tail], I, [Matrix_head|X]) :-
I_2 is I - 1,
removeRow(Matrix_tail, I_2, X).
cofactor(Matrix, I, J, X) :-
removeRow(Matrix, I, Matrix_2),
removeColumn(Matrix_2, J, Matrix_3),
det(Matrix_3, Y),
X is (-1) ** (I + J) * Y.
det_summand(_, _, [], 0).
det_summand(Matrix, J, B, X) :-
B = [B_head|B_tail],
cofactor(Matrix, 0, J, Z),
J_2 is J + 1,
det_summand(Matrix, J_2, B_tail, Y),
X is B_head * Z + Y.
det([[X]], X).
det(Matrix, X) :-
Matrix = [Matrix_head|_],
det_summand(Matrix, 0, Matrix_head, X).
replaceElement([_|Tail], 0, New, [New|Tail]).
replaceElement([Head|Tail], J, New, [Head|Y]) :-
J_2 is J - 1,
replaceElement(Tail, J_2, New, Y).
replaceColumn([], _, _, []).
replaceColumn([Matrix_head|Matrix_tail], J, [Column_head|Column_tail], [X|Y]) :-
replaceElement(Matrix_head, J, Column_head, X),
replaceColumn(Matrix_tail, J, Column_tail, Y).
cramerElements(_, B, L, []) :- length(B, L).
cramerElements(A, B, J, [X_J|Others]) :-
replaceColumn(A, J, B, A_J),
det(A_J, Det_A_J),
det(A, Det_A),
X_J is Det_A_J / Det_A,
J_2 is J + 1,
cramerElements(A, B, J_2, Others).
cramer(A, B, X) :- cramerElements(A, B, 0, X).
results(X) :-
A = [
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3]
],
B = [-3, -32, -47, 49],
cramer(A, B, X).
- Output:
| ?- results(X). X = [2.0,-12.0,-4.0,1.0] ? yes
Python
def det(m,n):
if n==1: return m[0][0]
z=0
for r in range(n):
k=m[:]
del k[r]
z+=m[r][0]*(-1)**r*det([p[1:]for p in k],n-1)
return z
w=len(t)
d=det(h,w)
if d==0:r=[]
else:r=[det([r[0:i]+[s]+r[i+1:]for r,s in zip(h,t)],w)/d for i in range(w)]
print(r)
Racket
#lang racket
(require math/matrix)
(define sys
(matrix [[2 -1 5 1]
[3 2 2 -6]
[1 3 3 -1]
[5 -2 -3 3]]))
(define soln
(col-matrix [-3 -32 -47 49]))
(define (matrix-set-column M new-col idx)
(matrix-augment (list-set (matrix-cols M) idx new-col)))
(define (cramers-rule M soln)
(let ([denom (matrix-determinant M)]
[nvars (matrix-num-cols M)])
(letrec ([roots (λ (position)
(if (>= position nvars)
'()
(cons (/ (matrix-determinant
(matrix-set-column M soln position))
denom)
(roots (add1 position)))))])
(map cons '(w x y z) (roots 0)))))
(cramers-rule sys soln)
- Output:
'((w . 2) (x . -12) (y . -4) (z . 1))
Raku
(formerly Perl 6)
sub det(@matrix) {
my @a = @matrix.map: { [|$_] };
my $sign = 1;
my $pivot = 1;
for ^@a -> \k {
my @r = (k+1 ..^ @a);
my $previous-pivot = $pivot;
if 0 == ($pivot = @a[k;k]) {
(my \s = @r.first: { @a[$_;k] != 0 }) // return 0;
(@a[s], @a[k]) = (@a[k], @a[s]);
my $pivot = @a[k;k];
$sign = -$sign;
}
for @r X @r -> (\i,\j) {
((@a[i;j] ×= $pivot) -= @a[i;k]×@a[k;j]) /= $previous-pivot;
}
}
$sign × $pivot
}
sub cramers_rule(@A, @terms) {
gather for ^@A -> \i {
my @Ai = @A.map: { [|$_] };
for ^@terms -> \j {
@Ai[j;i] = @terms[j];
}
take det(@Ai);
} »/» det(@A);
}
my @matrix = (
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3],
);
my @free_terms = <-3 -32 -47 49>;
my ($w, $x, $y, $z) = cramers_rule(@matrix, @free_terms);
("w = $w", "x = $x", "y = $y", "z = $z").join("\n").say;
- Output:
w = 2 x = -12 y = -4 z = 1
REXX
version 1
/* REXX Use Cramer's rule to compute solutions of given linear equations */
Numeric Digits 20
names='w x y z'
M=' 2 -1 5 1',
' 3 2 2 -6',
' 1 3 3 -1',
' 5 -2 -3 3'
v=' -3',
'-32',
'-47',
' 49'
Call mk_mat(m) /* M -> a.i.j */
Do j=1 To dim /* Show the input */
ol=''
Do i=1 To dim
ol=ol format(a.i.j,6)
End
ol=ol format(word(v,j),6)
Say ol
End
Say copies('-',35)
d=det(m) /* denominator determinant */
Do k=1 To dim /* construct nominator matrix */
Do j=1 To dim
Do i=1 To dim
If i=k Then
b.i.j=word(v,j)
Else
b.i.j=a.i.j
End
End
Call show_b
d.k=det(mk_str()) /* numerator determinant */
Say word(names,k) '=' d.k/d /* compute value of variable k */
End
Exit
mk_mat: Procedure Expose a. dim /* Turn list into matrix a.i.j */
Parse Arg list
dim=sqrt(words(list))
k=0
Do j=1 To dim
Do i=1 To dim
k=k+1
a.i.j=word(list,k)
End
End
Return
mk_str: Procedure Expose b. dim /* Turn matrix b.i.j into list */
str=''
Do j=1 To dim
Do i=1 To dim
str=str b.i.j
End
End
Return str
show_b: Procedure Expose b. dim /* show numerator matrix */
do j=1 To dim
ol=''
Do i=1 To dim
ol=ol format(b.i.j,6)
end
Call dbg ol
end
Return
det: Procedure /* compute determinant */
Parse Arg list
n=words(list)
call dbg 'det:' list
do dim=1 To 10
If dim**2=n Then Leave
End
call dbg 'dim='dim
If dim=2 Then Do
det=word(list,1)*word(list,4)-word(list,2)*word(list,3)
call dbg 'det=>'det
Return det
End
k=0
Do j=1 To dim
Do i=1 To dim
k=k+1
a.i.j=word(list,k)
End
End
Do j=1 To dim
ol=j
Do i=1 To dim
ol=ol format(a.i.j,6)
End
call dbg ol
End
det=0
Do i=1 To dim
ol=''
Do j=2 To dim
Do ii=1 To dim
If ii<>i Then
ol=ol a.ii.j
End
End
call dbg 'i='i 'ol='ol
If i//2 Then
det=det+a.i.1*det(ol)
Else
det=det-a.i.1*det(ol)
End
Call dbg 'det=>>>'det
Return det
sqrt: Procedure
/* REXX ***************************************************************
* EXEC to calculate the square root of a = 2 with high precision
**********************************************************************/
Parse Arg x,prec
If prec<9 Then prec=9
prec1=2*prec
eps=10**(-prec1)
k = 1
Numeric Digits 3
r0= x
r = 1
Do i=1 By 1 Until r=r0 | (abs(r*r-x)<eps)
r0 = r
r = (r + x/r) / 2
k = min(prec1,2*k)
Numeric Digits (k + 5)
End
Numeric Digits prec
r=r+0
Return r
dbg: Return
- Output:
2 -1 5 1 -3
3 2 2 -6 -32
1 3 3 -1 -47
5 -2 -3 3 49
-----------------------------------
w = 2
x = -12
y = -4
z = 1
version 2
The REXX version is based on the REXX version 1 program with the following improvements:
- aligns all output
- shows the values of the linear equations
- uses a PARSE for finding some matrix elements
- allows larger matrices to be used
- finds the widest decimal numbers for better formatting instead of assuming six digits
- use true variable names, doesn't assume that there are only four variables
- uses exact comparisons where appropriate
- added a check to see if the matrix had all its elements specified, added an error message
- uses a faster form of DO loops
- elided dead code and superfluous statements
- elided the need for a high precision sqrt function
- eschewed the use of a variable name with a function with the same name (bad practice)
- eschewed the deprecated use of: call func(args) syntax
- automatically used the minimum width when showing the matrix elements and equation values
/*REXX program uses Cramer's rule to find and display solution of given linear equations*/
values= '-3 -32 -47 49' /*values of each matrix row of numbers.*/
variables= substr('abcdefghijklmnopqrstuvwxyz', 27 - words(values) ) /*variable names.*/
call makeM ' 2 -1 5 1 3 2 2 -6 1 3 3 -1 5 -2 -3 3'
do y=1 for sz; $= /*display the matrix (linear equations)*/
do x=1 for sz; $= $ right(psign(@.x.y), w)'*'substr(variables, x, 1)
end /*y*/ /* [↑] right─justify matrix elements.*/
pad= left('', length($) - 2); say $ ' = ' right( word(values, y), wv)
end /*x*/ /* [↑] obtain value of the equation. */
say; say
do k=1 for sz /*construct the nominator matrix. */
do j=1 for sz
do i=1 for sz; if i==k then !.i.j= word(values, j)
else !.i.j= @.i.j
end /*i*/
end /*j*/
say pad substr(variables,k,1) ' = ' right(det(makeL())/det(mat), digits()+2)
end /*k*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
makeL: $=; do x=1 for sz; do y=1 for sz; $= $ !.x.y; end; end; return $ /*matrix─►list*/
mSize: arg _; do sz=0 for 1e3; if sz*sz==_ then return; end; say 'error,bad matrix';exit 9
psign: parse arg num; if left(num, 1)\=='-' & x>1 then return "+"num; return num
/*──────────────────────────────────────────────────────────────────────────────────────*/
det: procedure; parse arg a b c d 1 nums; call mSize words(nums); _= 0
if sz==2 then return a*d - b*c
do j=1 for sz
do i=1 for sz; _= _ + 1; @.i.j= word(nums, _)
end /*i*/
end
aa= 0
do i=1 for sz; odd= i//2; $=
do j=2 for sz-1
do k=1 for sz; if k\==i then $= $ @.k.j
end /*k*/
end /*j*/
aa= aa - (-1 ** odd) * @.i.1 * det($)
end; /*i*/; return aa
/*──────────────────────────────────────────────────────────────────────────────────────*/
makeM: procedure expose @. values mat sz w wv; parse arg mat; call mSize words(mat)
#= 0; wv= 0; w= 0
do j=1 for sz; wv= max(wv, length( word( values, j) ) )
do k=1 for sz; #= #+1; @.k.j= word(mat, #); w= max(w, length(@.k.j))
end /*k*/
end; /*j*/; w= w + 1; return
- output when using the internal default inputs:
2*w -1*x +5*y +1*z = -3
3*w +2*x +2*y -6*z = -32
1*w +3*x +3*y -1*z = -47
5*w -2*x -3*y +3*z = 49
w = 2
x = -12
y = -4
z = 1
RPL
Explicit Cramer's rule
≪ IF OVER DET
THEN
LAST ROT DUP SIZE 1 GET → vect det mtx dim
≪ 1 dim FOR c
mtx
1 dim FOR r
r c 2 →LIST vect r GET PUT
NEXT
DET det /
NEXT
dim →ARRY
≫
END
≫
‘CRAMR’ STO
[[ 2 -1 5 1 ][ 3 2 2 -6 ][ 1 3 3 -1 ][ 5 -2 -3 3 ]] [ -3 -32 -47 49 ] CRAMR
- Output:
1: [ 2 -12 -4 1 ]
Implicit Cramer's rule
RPL use Cramer's rule for its built-in equation system resolution feature, performed by the / instruction.
[ -3 -32 -47 49 ] [[ 2 -1 5 1 ][ 3 2 2 -6 ][ 1 3 3 -1 ][ 5 -2 -3 3 ]] /
- Output:
1: [ 2 -12 -4 1 ]
Ruby
require 'matrix'
def cramers_rule(a, terms)
raise ArgumentError, " Matrix not square" unless a.square?
cols = a.to_a.transpose
cols.each_index.map do |i|
c = cols.dup
c[i] = terms
Matrix.columns(c).det / a.det
end
end
matrix = Matrix[
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3],
]
vector = [-3, -32, -47, 49]
puts cramers_rule(matrix, vector)
- Output:
2 -12 -4 1
Rust
use std::ops::{Index, IndexMut};
fn main() {
let m = matrix(
vec![
2., -1., 5., 1., 3., 2., 2., -6., 1., 3., 3., -1., 5., -2., -3., 3.,
],
4,
);
let mm = m.solve(&vec![-3., -32., -47., 49.]);
println!("{:?}", mm);
}
#[derive(Clone)]
struct Matrix {
elts: Vec<f64>,
dim: usize,
}
impl Matrix {
// Compute determinant using cofactor method
// Using Gaussian elimination would have been more efficient, but it also solves the linear
// system, so…
fn det(&self) -> f64 {
match self.dim {
0 => 0.,
1 => self[0][0],
2 => self[0][0] * self[1][1] - self[0][1] * self[1][0],
d => {
let mut acc = 0.;
let mut signature = 1.;
for k in 0..d {
acc += signature * self[0][k] * self.comatrix(0, k).det();
signature *= -1.
}
acc
}
}
}
// Solve linear systems using Cramer's method
fn solve(&self, target: &Vec<f64>) -> Vec<f64> {
let mut solution: Vec<f64> = vec![0.; self.dim];
let denominator = self.det();
for j in 0..self.dim {
let mut mm = self.clone();
for i in 0..self.dim {
mm[i][j] = target[i]
}
solution[j] = mm.det() / denominator
}
solution
}
// Compute the cofactor matrix for determinant computations
fn comatrix(&self, k: usize, l: usize) -> Matrix {
let mut v: Vec<f64> = vec![];
for i in 0..self.dim {
for j in 0..self.dim {
if i != k && j != l {
v.push(self[i][j])
}
}
}
matrix(v, self.dim - 1)
}
}
fn matrix(elts: Vec<f64>, dim: usize) -> Matrix {
assert_eq!(elts.len(), dim * dim);
Matrix { elts, dim }
}
impl Index<usize> for Matrix {
type Output = [f64];
fn index(&self, i: usize) -> &Self::Output {
let m = self.dim;
&self.elts[m * i..m * (i + 1)]
}
}
impl IndexMut<usize> for Matrix {
fn index_mut(&mut self, i: usize) -> &mut Self::Output {
let m = self.dim;
&mut self.elts[m * i..m * (i + 1)]
}
}
Which outputs:
[2.0, -12.0, -4.0, 1.0]
Sidef
func cramers_rule(A, terms) {
gather {
for i in ^A {
var Ai = A.map{.map{_}}
for j in ^terms {
Ai[j][i] = terms[j]
}
take(Ai.det)
}
} »/» A.det
}
var matrix = [
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3],
]
var free_terms = [-3, -32, -47, 49]
var (w, x, y, z) = cramers_rule(matrix, free_terms)...
say "w = #{w}"
say "x = #{x}"
say "y = #{y}"
say "z = #{z}"
- Output:
w = 2 x = -12 y = -4 z = 1
Tcl
package require math::linearalgebra
namespace path ::math::linearalgebra
# Setting matrix to variable A and size to n
set A [list { 2 -1 5 1} { 3 2 2 -6} { 1 3 3 -1} { 5 -2 -3 3}]
set n [llength $A]
# Setting right side of equation
set right {-3 -32 -47 49}
# Calculating determinant of A
set detA [det $A]
# Apply Cramer's rule
for {set i 0} {$i < $n} {incr i} {
set tmp $A ;# copy A to tmp
setcol tmp $i $right ;# replace column i with right side vector
set detTmp [det $tmp] ;# calculate determinant of tmp
set v [expr $detTmp / $detA] ;# divide two determinants
puts [format "%0.4f" $v] ;# format and display result
}
- Output:
2.0000 -12.0000 -4.0000 1.0000
VBA
Sub CramersRule()
OrigM = [{2, -1, 5, 1; 3,2,2,-6;1,3,3,-1;5,-2,-3,3}]
OrigD = [{-3;-32;-47;49}]
MatrixSize = UBound(OrigM)
DetOrigM = WorksheetFunction.MDeterm(OrigM)
For i = 1 To MatrixSize
ChangeM = OrigM
For j = 1 To MatrixSize
ChangeM(j, i) = OrigD(j, 1)
Next j
DetChangeM = WorksheetFunction.MDeterm(ChangeM)
Debug.Print i & ": " & DetChangeM / DetOrigM
Next i
End Sub- Output:
1: 2 2: -12 3: -4 4: 1
Visual Basic .NET
Imports System.Runtime.CompilerServices
Imports System.Linq.Enumerable
Module Module1
<Extension()>
Function DelimitWith(Of T)(source As IEnumerable(Of T), Optional seperator As String = " ") As String
Return String.Join(seperator, source)
End Function
Private Class SubMatrix
Private ReadOnly source As Integer(,)
Private ReadOnly prev As SubMatrix
Private ReadOnly replaceColumn As Integer()
Public Sub New(source As Integer(,), replaceColumn As Integer())
Me.source = source
Me.replaceColumn = replaceColumn
prev = Nothing
ColumnIndex = -1
Size = replaceColumn.Length
End Sub
Public Sub New(prev As SubMatrix, Optional deletedColumnIndex As Integer = -1)
source = Nothing
replaceColumn = Nothing
Me.prev = prev
ColumnIndex = deletedColumnIndex
Size = prev.Size - 1
End Sub
Public Property ColumnIndex As Integer
Public ReadOnly Property Size As Integer
Default Public ReadOnly Property Index(row As Integer, column As Integer) As Integer
Get
If Not IsNothing(source) Then
Return If(column = ColumnIndex, replaceColumn(row), source(row, column))
Else
Return prev(row + 1, If(column < ColumnIndex, column, column + 1))
End If
End Get
End Property
Public Function Det() As Integer
If Size = 1 Then Return Me(0, 0)
If Size = 2 Then Return Me(0, 0) * Me(1, 1) - Me(0, 1) * Me(1, 0)
Dim m As New SubMatrix(Me)
Dim detVal = 0
Dim sign = 1
For c = 0 To Size - 1
m.ColumnIndex = c
Dim d = m.Det()
detVal += Me(0, c) * d * sign
sign = -sign
Next
Return detVal
End Function
Public Sub Print()
For r = 0 To Size - 1
Dim rl = r
Console.WriteLine(Range(0, Size).Select(Function(c) Me(rl, c)).DelimitWith(", "))
Next
Console.WriteLine()
End Sub
End Class
Private Function Solve(matrix As SubMatrix) As Integer()
Dim det = matrix.Det()
If det = 0 Then Throw New ArgumentException("The determinant is zero.")
Dim answer(matrix.Size - 1) As Integer
For i = 0 To matrix.Size - 1
matrix.ColumnIndex = i
answer(i) = matrix.Det() / det
Next
Return answer
End Function
Public Function SolveCramer(equations As Integer()()) As Integer()
Dim size = equations.Length
If equations.Any(Function(eq) eq.Length <> size + 1) Then Throw New ArgumentException($"Each equation must have {size + 1} terms.")
Dim matrix(size - 1, size - 1) As Integer
Dim column(size - 1) As Integer
For r = 0 To size - 1
column(r) = equations(r)(size)
For c = 0 To size - 1
matrix(r, c) = equations(r)(c)
Next
Next
Return Solve(New SubMatrix(matrix, column))
End Function
Sub Main()
Dim equations = {
({2, -1, 5, 1, -3}),
({3, 2, 2, -6, -32}),
({1, 3, 3, -1, -47}),
({5, -2, -3, 3, 49})
}
Dim solution = SolveCramer(equations)
Console.WriteLine(solution.DelimitWith(", "))
End Sub
End Module
- Output:
2, -12, -4, 1
Wren
import "./matrix" for Matrix
var cramer = Fn.new { |a, d|
var n = a.numRows
var x = List.filled(n, 0)
var ad = a.det
for (c in 0...n) {
var aa = a.copy()
for (r in 0...n) aa[r, c] = d[r, 0]
x[c] = aa.det/ad
}
return x
}
var a = Matrix.new([
[2, -1, 5, 1],
[3, 2, 2, -6],
[1, 3, 3, -1],
[5, -2, -3, 3]
])
var d = Matrix.new([
[- 3],
[-32],
[-47],
[ 49]
])
var x = cramer.call(a, d)
System.print("Solution is %(x)")
- Output:
Solution is [2, -12, -4, 1]
XPL0
func Det(A, N); \Return value of determinate A, order N
int A, N;
int B, Sum, I, K, L, Term;
[if N = 1 then return A(0, 0);
B:= Reserve((N-1)*4\IntSize\);
Sum:= 0;
for I:= 0 to N-1 do
[L:= 0;
for K:= 0 to N-1 do
if K # I then
[B(L):= @A(K, 1); L:= L+1];
Term:= A(I, 0) * Det(B, N-1);
if I & 1 then Term:= -Term;
Sum:= Sum + Term;
];
return Sum;
];
real D;
[D:= float(Det([[2,-1,5,1], [3,2,2,-6], [1,3,3,-1], [5,-2,-3,3]], 4));
RlOut(0, float(Det([[-3,-1,5,1], [-32,2,2,-6], [-47,3,3,-1], [49,-2,-3,3]], 4)) / D);
RlOut(0, float(Det([[2,-3,5,1], [3,-32,2,-6], [1,-47,3,-1], [5,49,-3,3]], 4)) / D);
RlOut(0, float(Det([[2,-1,-3,1], [3,2,-32,-6], [1,3,-47,-1], [5,-2,49,3]], 4)) / D);
RlOut(0, float(Det([[2,-1,5,-3], [3,2,2,-32], [1,3,3,-47], [5,-2,-3,49]], 4)) / D);
]- Output:
2.00000 -12.00000 -4.00000 1.00000
zkl
Using the GNU Scientific Library, we define the values:
var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library)
A:=GSL.Matrix(4,4).set(2,-1, 5, 1,
3, 2, 2,-6,
1, 3, 3,-1,
5,-2,-3, 3);
b:=GSL.Vector(4).set(-3,-32,-47,49);First, just let GSL solve:
A.AxEQb(b).format().println();Actually implement Cramer's rule:
fcn cramersRule(A,b){
b.len().pump(GSL.Vector(b.len()),'wrap(i){ // put calculations into new Vector
A.copy().setColumn(i,b).det();
}).close()/A.det();
}
cramersRule(A,b).format().println();- Output:
2.00,-12.00,-4.00,1.00