1. Matlab solution Ax=b
Only the simplest way is written, where b needs to be a column vector, and semicolons are used to separate elements;
Ax = b
where the matrix A(nxn) and the column vector b(n) are known; the solution of the system of equations is stored in x, written in matlab:
> x=A\b
octave:7> A=[1,2; 1.0001, 2;] A = 1.0000 2.0000 1.0001 2.0000 octave:8> b=[3; 3.0001;] b = 3.0000 3.0001 octave:9> x=A\b x = 1.0000 1.0000 octave:10> b-A*x ans = 0 0 octave:11>
2. Ax=b iterative algorithm example
2.1 Principle
The principle of the iterative method is to create a convergent sequence of n-dimensional infinite terms. The convergence limit point x satisfies the equation Ax = b;
There are some techniques for using matrix A to create a convergent sequence. In the example here, since A is strictly dominated by the pivot element, it is easier to create convergence and the diagonal elements are directly removed;
For detailed principles, please refer to the numerical calculation textbook. It uses the induced norm of the matrix, spectral radius and other related theorems, and it is not complicated;
In order to express the principle here, I used the algorithm implementation in matrix language. The general principle is as follows. The principle is not written in a standardized way, so you can just skip it:
/**********************************
*
* ******Ax = b**********
* x = Bx + f
* x = -D'(L + U)x + D’b
*
* Dx = b – (L + U)x
*************************The following is the iterative process
* X1=(L + U) * X0
* X1=b-X1
* Dx = X1 ***(D here is a diagonal matrix, so you can get x by directly doing scalar division)
* x = D^-1 * X1
*X0 = x
*
*******************************/
The implementation of matrix language has two advantages. One is to facilitate traditional theoretical analysis, such as convergence analysis, and the other is to facilitate GPU parallel implementation;
2.2 Source code implementation
Here is the simplest Jacobi iteration method to find the solution of Ax=b;
gcc can be directly compiled and run;
jacobi.cpp
Source code:
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#define NA 3
void test_sgemv(int op, int N, float a, float *AA, int lda, float b, float *yy);
void sgemv(int op, int N, float a, float *A, int lda, float b, float *y); // y = a*op(A)*X + b*y
void print_matrix(float *A, int N, int lda);
void print_vector(float *V, int N);
void set_LpU(float *LpU, float *A, int N);
void set_D(float *D, float *A, int N);
void Saxpy(int n, float alpha, float *x, float beta, float *y); //y = a*x + b*y
void x_div_y(int N, float* x, float* y);//x[i] = x[i]/y[i];
void Scopy(int n, float* x, float* y); // x -> y
void Saxpy(int n, float alpha, float *x, float beta, float *y) //y = a*x + b*y
{
for(int i=0; i<n; i + + ){
y[i] = alpha*x[i] + beta*y[i];
}
}
void x_div_y(int N, float* x, float* y){//x[i] = x[i]/y[i];
for(int i=0; i<N; i + + )
x[i] = x[i]/y[i];
}
void Scopy(int n, float* x, float* y)// x -> y
{
for(int i=0; i<n; i + + )
y[i] = x[i];
}
void Jacobi_Ax_b(float *A, float *b, float *X0, float *Xk, int N, float eps, int iter){
float *LpU = nullptr;
//float *X0 = nullptr;
//float *X1 = nullptr;
float *D = nullptr;
LpU = (float*)malloc( N*N*sizeof(float));
//X0 = (float*)malloc( N*sizeof(float));
//X1 = (float*)malloc( N*sizeof(float));
D = (float*)malloc( N*sizeof(float));
//1.0 set LpU, set D, X0 = 0;
set_LpU(LpU, A, N); printf("LpU =\\
"); print_matrix(LpU, N, N);
set_D(D, A, N); printf("D =\\
"); print_vector(D, N);
memset(X0, 0, N*sizeof(float)); printf("X0 =\\
"); print_vector(Xk, N);
//2.0 loop
/* S1 X1=LU* x0
* S2 X1=b-X1
* S3 Dx = X1
* S4 X1 = X1/D
*/
for(int it=0; it<iter; it + + ){
//void sgemv(int op, int N, float a, float *A, int lda, float b, float *y);
// S1 X1=LU* X0
sgemv(0, N, 1.0f, LpU, N, 0.0f, Xk);
//void Saxpy(int n, float alpha, float *x, float beta, float *y); //y = a*x + b*y
// S2 X1=b-X1
Saxpy(N, 1.0f, b, -1.0f, Xk);
//void x_div_y(int N, float* x, float* y){//x[i] = x[i]/y[i];
// S3 DXk + 1 = Xk
x_div_y(N, Xk, D);
}
//Scopy(N, X1, xk);
}
void set_LpU(float *LpU, float *A, int N){
for(int i=0; i<N; i + + ){
for(int j=0; j<N; j + + ){
LpU[i + j*N] = A[i + j*N];
if(i == j)
LpU[i + i*N] = 0.0f;
}
}
}
void set_D(float *D, float *A, int N){
for(int i=0; i<N; i + + )
D[i] = A[i + i*N];
}
void sgemv(int op, int N, float a, float *A, int lda, float b, float *y){
// y = a*op(A)*X + b*y
float* y0 = nullptr;
y0 = (float*)malloc(N*sizeof(float));
memcpy(y0, y, N*sizeof(float));
for(int i=0; i<N; i + + ){
float sigma = 0.0f;
for(int j=0; j<N; j + + ){
sigma + = A[i + j*lda]*y0[j];
}
y[i] = a*sigma + b*y[i];
}
free(y0);
}
void print_matrix(float *A, int N, int lda){
printf("\\
");
for(int i=0; i<N; i + + ){
for(int j=0; j<N; j + + ){
printf("%8.3f ", A[i + j*lda]);
}
printf("\\
");
}
printf("\\
");
}
void print_vector(float *V, int N){
printf("\\
");
for(int i=0; i<N; i + + ){
printf("%f ", V[i]);
}
printf("\\
");
}
//void sgemv(int op, int N, float a, float *A, int lda, float b, float *y){
void test_sgemv(int op, int N, float a, float *AA, int lda, float b, float *yy){
float *y = nullptr;
float *A = nullptr;
y = (float*)malloc(N*sizeof(float));
A = (float*)malloc(N*N*sizeof(float));
memcpy(y, yy, N*sizeof(float));
memcpy(A, AA, N*N*sizeof(float));
printf("\\
A =\\
");
print_matrix(A, N, N);
printf("\\
y =\\
");
print_vector(y, N);
sgemv(1, N, a, A, lda, b, y);
printf("\\
y=Ax + y=\\
");
print_vector(y, N);
free(y);
free(A);
}
int main(){
float A[NA*NA] =
{
10, -1, -1, -1, 10, -1, -2, -2, 5
// 10, 2, 1, 3, -10, 3, 1, 3, 10// column major
/*
10, 3, 1,
2, -10, 3,
1, 3, 10*/
};
float b[NA]={
7.2, 8.3, 4.2
//14, -5, 14
};
// void test_sgemv(int op, int N, float a, float *AA, int lda, float b, float *yy){
//test_sgemv(1, NA, 1.0f, A, NA, 1.0f, b);
float* Ah = nullptr;
float* bh = nullptr;
int N = NA;
Ah = (float*)malloc(N*N*sizeof(float));
bh = (float*)malloc(N*sizeof(float));
memcpy(Ah, A, N*N*sizeof(float));
memcpy(bh, b, N*sizeof(float));
printf("Ah =\\
");
print_matrix(Ah, N, N);
float *x0 = nullptr;
float *x1 = nullptr;
x0 = (float*)malloc(N*sizeof(float));
x1 = (float*)malloc(N*sizeof(float));
float eps = 1.0e-7;
int iter = 15;
Jacobi_Ax_b(Ah, bh, x0, x1, N, eps, iter);
print_vector(x1, N);
return 0;
}
Compile:
Makefile
Jacobi: hello_Jacobi.cpp
g + + -g $< -o $@
.PHONY: clean
clean:
-rm -rf Jacobi
As a reminder, the g++ line is preceded by a tab space, which is the syntax requirement of the Makefile, and the -rm is also preceded;
run:
3.3 matlab to verify the results:
