svd奇异值分解 c语言实现

SVD（奇异值分解）是一种矩阵分解方法，可以将矩阵分解为三个矩阵的乘积。在C语言中，可以使用以下代码实现SVD奇异值分解：

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define TOLERANCE 1.0e-10 // 精度

void svd(int m, int n, double **a, double **u, double *w, double **v) {
    int flag, i, its, j, jj, k, l, nm;
    double c, f, h, s, x, y, z;
    double anorm = 0.0, g = 0.0, scale = 0.0;
    double *rv1;

    if (m < n) {
        fprintf(stderr, "SVD failed: m < n\n");
        return;
    }

    rv1 = (double *) malloc(n * sizeof(double));
    for (i = 0; i < n; i++) {
        l = i + 1;
        rv1[i] = scale * g;
        g = s = scale = 0.0;
        if (i < m) {
            for (k = i; k < m; k++) {
                scale += fabs(a[k][i]);
            }
            if (scale) {
                for (k = i; k < m; k++) {
                    a[k][i] /= scale;
                    s += a[k][i] * a[k][i];
                }
                f = a[i][i];
                g = -SIGN(sqrt(s), f);
                h = f * g - s;
                a[i][i] = f - g;
                for (j = l; j < n; j++) {
                    for (s = 0.0, k = i; k < m; k++) {
                        s += a[k][i] * a[k][j];
                    }
                    f = s / h;
                    for (k = i; k < m; k++) {
                        a[k][j] += f * a[k][i];
                    }
                }
                for (k = i; k < m; k++) {
                    a[k][i] *= scale;
                }
            }
        }
        w[i] = scale * g;
        g = s = scale = 0.0;
        if (i < m && i != n - 1) {
            for (k = l; k < n; k++) {
                scale += fabs(a[i][k]);
            }
            if (scale) {
                for (k = l; k < n; k++) {
                    a[i][k] /= scale;
                    s += a[i][k] * a[i][k];
                }
                f = a[i][l];
                g = -SIGN(sqrt(s), f);
                h = f * g - s;
                a[i][l] = f - g;
                for (k = l; k < n; k++) {
                    rv1[k] = a[i][k] / h;
                }
                for (j = l; j < m; j++) {
                    for (s = 0.0, k = l; k < n; k++) {
                        s += a[j][k] * a[i][k];
                    }
                    for (k = l; k < n; k++) {
                        a[j][k] += s * rv1[k];
                    }
                }
                for (k = l; k < n; k++) {
                    a[i][k] *= scale;
                }
            }
        }
        anorm = MAX(anorm, (fabs(w[i]) + fabs(rv1[i])));
    }
    for (i = n - 1; i >= 0; i--) {
        if (i < n - 1) {
            if (g) {
                for (j = l; j < n; j++) {
                    v[j][i] = (a[i][j] / a[i][l]) / g;
                }
                for (j = l; j < n; j++) {
                    for (s = 0.0, k = l; k < n; k++) {
                        s += a[i][k] * v[k][j];
                    }
                    for (k = l; k < n; k++) {
                        v[k][j] += s * v[k][i];
                    }
                }
            }
            for (j = l; j < n; j++) {
                v[i][j] = v[j][i] = 0.0;
            }
        }
        v[i][i] = 1.0;
        g = rv1[i];
        l = i;
    }
    for (i = MIN(m, n) - 1; i >= 0; i--) {
        l = i + 1;
        g = w[i];
        for (j = l; j < n; j++) {
            a[i][j] = 0.0;
        }
        if (g) {
            g = 1.0 / g;
            for (j = l; j < n; j++) {
                for (s = 0.0, k = l; k < m; k++) {
                    s += a[k][i] * a[k][j];
                }
                f = (s / a[i][i]) * g;
                for (k = i; k < m; k++) {
                    a[k][j] += f * a[k][i];
                }
            }
            for (j = i; j < m; j++) {
                a[j][i] *= g;
            }
        } else {
            for (j = i; j < m; j++) {
                a[j][i] = 0.0;
            }
        }
        ++a[i][i];
    }
    for (k = n - 1; k >= 0; k--) {
        for (its = 1; its <= 30; its++) {
            flag = 1;
            for (l = k; l >= 0; l--) {
                nm = l - 1;
                if (fabs(rv1[l]) + anorm == anorm) {
                    flag = 0;
                    break;
                }
                if (fabs(w[nm]) + anorm == anorm) {
                    break;
                }
            }
            if (flag) {
                c = 0.0;
                s = 1.0;
                for (i = l; i <= k; i++) {
                    f = s * rv1[i];
                    if (fabs(f) + anorm != anorm) {
                        g = w[i];
                        h = sqrt(f * f + g * g);
                        w[i] = h;
                        h = 1.0 / h;
                        c = g * h;
                        s = (-f * h);
                        for (j = 0; j < m; j++) {
                            y = a[j][nm];
                            z = a[j][i];
                            a[j][nm] = y * c + z * s;
                            a[j][i] = z * c - y * s;
                        }
                    }
                }
            }
            z = w[k];
            if (l == k) {
                if (z < 0.0) {
                    w[k] = -z;
                    for (j = 0; j < n; j++) {
                        v[j][k] = (-v[j][k]);
                    }
                }
                break;
            }
            if (its == 30) {
                fprintf(stderr, "SVD failed: no convergence after %d iterations\n", its);
                return;
            }
            x = w[l];
            nm = k - 1;
            y = w[nm];
            g = rv1[nm];
            h = rv1[k];
            f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y);
            g = sqrt(f * f + 1.0);
            f = ((x - z) * (x + z) + h * ((y / (f + SIGN(g, f))) - h)) / x;
            c = s = 1.0;
            for (j = l; j <= nm; j++) {
                i = j + 1;
                g = rv1[i];
                y = w[i];
                h = s * g;
                g = c * g;
                z = sqrt(f * f + h * h);
                rv1[j] = z;
                c = f / z;
                s = h / z;
                f = (x * c) + (g * s);
                g = (g * c) - (x * s);
                h = y * s;
                y *= c;
                for (jj = 0; jj < n; jj++) {
                    x = v[jj][j];
                    z = v[jj][i];
                    v[jj][j] = x * c + z * s;
                    v[jj][i] = z * c - x * s;
                }
                z = sqrt(f * f + h * h);
                w[j] = z;
                if (z) {
                    z = 1.0 / z;
                    c = f * z;
                    s = h * z;
                }
                f = (c * g) + (s * y);
                x = (c * y) - (s * g);
                for (jj = 0; jj < m; jj++) {
                    y = a[jj][j];
                    z = a[jj][i];
                    a[jj][j] = y * c + z * s;
                    a[jj][i] = z * c - y * s;
                }
            }
            rv1[l] = 0.0;
            rv1[k] = f;
            w[k] = x;
        }
    }
    free(rv1);
}

其中，m和n分别是矩阵A的行数和列数，a是一个m行n列的矩阵，u是一个m行m列的矩阵，v是一个n行n列的矩阵，w是一个长度为n的一维数组，用于存储奇异值。

需要注意的是，这段代码是从Numerical Recipes in C中摘取的，但是有一些宏定义（如SIGN、MAX、MIN）需要自己定义或者修改。此外，代码中还需要使用一些基本的数学函数，如fabs（求绝对值）、sqrt（求平方根）等，需要添加头文件<math.h>。