###### tags: `code`
# c14086028-homework-7-2
###### tags `use'c' to find a vector ,which is in C(A), is most approach to vector b`

#include<stdio.h>
#include<stdlib.h>
#include<math.h>
#define N 8
void initial(double [N][N]);
void matrixmulvector(double [N][N], double [N]);
double computernorm2(double [N]);
double linearlyapproach(double [N][N],double [N][N],double [N],double [N]);
int main() {
	double A[N][N] = { {} };
    double LD[N][N] = { {} };
    double D2[N][N] = { {} };
    double x[N] = { 1,1,1,1,1,1,1,1 };
    double x1[N] = { 1,1,1,1,1,1,1,1 };
	double b[N] = { 5,10,12,16,20,24,28,31 };
    double k = 0;
    initial(A);
    /* give the value of entries of L , U and D then print they */

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            if (j == i) {
                D2[i][j] = (double)1/A[i][j];
            }
            else {
                D2[i][j] = 0;
            }
        }
    }
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            if (j != i) {
                LD[i][j] = -A[i][j];
            }
            else {
                LD[i][j] = 0;
            }
        }
    }   
    double a = 0;
    matrixmulvector(A, x);
    double l2[N] = {};
    for (int i = 0; i < N; i++) {
        l2[i] = b[i] - x[i];
    }
    a=computernorm2(l2);
    printf("%lf\n", a);
    
    while (a>0.0000001) {
        linearlyapproach(D2,LD,x1,b);
        for (int i = 0; i < N; i++) {
            x[i] = x1[i];
        }
        matrixmulvector(A, x);
        for (int i = 0; i < N; i++) {
            l2[i] = 0.0;
            l2[i] = b[i] - x[i];
        }
        a = computernorm2(l2);
        printf("%lf\n", a);
    }
    
    
    printf("the x whitch is belng to C(A) is max approach to b is \n");

    for (int i = 0; i < N; i++) {
        printf("%lf\n",x1[i]);
    }
    printf("the distance is %lf\n", a);

    system("pause");
    return 0;
}
void initial(double A[N][N]) {
    for (int i = 0; i < N; i++) {
        A[i][i] = 3;
        if ((i + 1) < N) {
            A[i][i + 1] = 1;
            A[i + 1][i] = 1;
        }
    }
}double linearlyapproach(double D2[N][N], double LD[N][N], double x[N], double b[N]) {
    double li = 0;
    double g[N] = {0,0,0,0,0,0,0,0};

    matrixmulvector(LD, x);

    for (int i = 0; i < N; i++) {
        g[i] = x[i] + b[i];
    }

    matrixmulvector(D2, g);

    for (int i = 0; i < N; i++) {
        x[i] = g[i];
    }
    return 0;
}

void matrixmulvector(double A[N][N], double x[N]) {
    double k[N] = { 0,0,0,0,0,0,0,0 };
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            k[i] += A[i][j] * x[j];
        }
    }

    for (int i = 0; i < N; i++) {
        x[i] = k[i];
    }
    
}
double computernorm2(double x[N]) {
    double l = 0;
    for (int i = 0; i < N; i++) {
        l += x[i] * x[i];
    }
    return sqrt((double)l);
}