How to resolve the algorithm Cramer's rule step by step in the Prolog programming language

Published on 12 May 2024 09:40 PM
#Prolog

How to resolve the algorithm Cramer's rule step by step in the Prolog programming language

Table of Contents

Problem Statement

Given

which in matrix format is

Then the values of

x , y

{\displaystyle x,y}

and

z

{\displaystyle z}

can be found as follows:

Given the following system of equations:

solve for

w

{\displaystyle w}

,

x

{\displaystyle x}

,

y

{\displaystyle y}

and

z

{\displaystyle z}

, using Cramer's rule.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Cramer's rule step by step in the Prolog programming language

Source code in the prolog programming language

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).

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