# Hess(M) = Return upper Hessenberg matrix, same eigenvalues

Hess(M) = Return upper Hessenberg matrix, same eigenvalues:

### ```#include #define SWAP(g, h) {y = (g); (g) = (h); (h) = y;} void hess(float **a, int n) // Reduction to Hessenberg form by the elimination method. The real, // nonsymmetric matrix a[1..n][1..n] is replaced by an upper Hessenberg matrix // with identical eigenvalues. Recommended, but not required, is that this // routine be preceded by balance. On output, the Hessenberg matrix is in // elements a[i][j] with i <= j + 1. Elements with i > j+1 are zero. { int m, j, i; float y, x; for (m = 2; m < n; m++) // m is called r + 1 in the text. { X = 0.0; i = m; for (j = m; j <= n; j++) // Find the pivot. { if (fabs(a[j][m - 1]) > fabs(x)) { x = a[j][m - 1]; i = j; } } if (i != m) // Interchange rows and columns. { for (j = m - 1; j <= n; j++) SWAP(a[i][j], a[m][j]) for (j = 1; j <= n; ++) SWAP(a[j][i], a[j][m]) } if (x != 0.0) // Carry out the elimination. { for (I = m + 1; I <= n; i++) { if ((y = a[i][m - 1]) != 0.0) { y /= x; a[i][m-1] = y; for (j = m; j <= n; j++) a[i][j] -= y * a[m][j]; for (j = 1; j <= n; j++) a[j][m] += y * a[j][i]; } } } } for (j = 1; j <= n - 2; j++) for (i = j + 2; i <= n; i++) a[i, j] = 0.0; } // hess```

See: Reduction of a General Matrix to Hessenberg Form -- From Numerical Recipes in C