#include "magneto1.4.h" //Magneto 1.3 4/24/2020 void CalculateCalibration(float *buf, int sampleCount, float BAinv[4][3]) { double* D; double J, hmb; int i, j, index; double maxval, norm, btqb, hm, norm1, norm2, norm3; double x, y, z, x2, nxsrej, xs, xave; double* raw; //raw obs // // calculate mean (norm) and standard deviation for possible outlier rejection // xs = 0; xave = 0; for (i = 0; i < sampleCount; i++) { x = buf[i * 3 + 0]; y = buf[i * 3 + 1]; z = buf[i * 3 + 2]; x2 = x * x + y * y + z * z; xs += x2; xave += sqrt(x2); } xave = xave / sampleCount; //mean vector length xs = sqrt(xs / sampleCount - (xave * xave)); //std. dev. // summarize statistics, give user opportunity to reject outlying measurements nxsrej = 0; // third time through! // allocate array space for accepted measurements D = (double*)malloc(10 * sampleCount * sizeof(double)); raw = (double*)malloc(3 * sampleCount * sizeof(double)); j = 0; //array index for good measurements // printf("\r\nAccepted measurements (file index, internal index, ...)\r\n"); for (i = 0; i < sampleCount; i++) { x = buf[i * 3 + 0]; y = buf[i * 3 + 1]; z = buf[i * 3 + 2]; x2 = sqrt(x * x + y * y + z * z); //vector length x2 = fabs(x2 - xave) / xs; //standard deviation from mean if ((nxsrej == 0) || (x2 <= nxsrej)) { // accepted measurement // printf("%d, %d: %6.1f %6.1f %6.1f\r\n",i,j,x,y,z); raw[3 * j] = x; raw[3 * j + 1] = y; raw[3 * j + 2] = z; D[j] = x * x; D[sampleCount + j] = y * y; D[sampleCount * 2 + j] = z * z; D[sampleCount * 3 + j] = 2.0 * y * z; D[sampleCount * 4 + j] = 2.0 * x * z; D[sampleCount * 5 + j] = 2.0 * x * y; D[sampleCount * 6 + j] = 2.0 * x; D[sampleCount * 7 + j] = 2.0 * y; D[sampleCount * 8 + j] = 2.0 * z; D[sampleCount * 9 + j] = 1.0; j++; //count good measurements } } free(raw); //printf("\r\nExpected norm of local field vector Hm? (Enter 0 for default %8.1f) ", xave); //scanf("%lf", &hm); //if (hm == 0.0) hm = xave; //printf("\r\nSet Hm = %8.1f\r\n", hm); hm = xave; // allocate memory for matrix S double S[10*10]; Multiply_Self_Transpose(S, D, 10, sampleCount); free(D); // Create pre-inverted constraint matrix C double C[6*6]; C[0] = 0.0; C[1] = 0.5; C[2] = 0.5; C[3] = 0.0; C[4] = 0.0; C[5] = 0.0; C[6] = 0.5; C[7] = 0.0; C[8] = 0.5; C[9] = 0.0; C[10] = 0.0; C[11] = 0.0; C[12] = 0.5; C[13] = 0.5; C[14] = 0.0; C[15] = 0.0; C[16] = 0.0; C[17] = 0.0; C[18] = 0.0; C[19] = 0.0; C[20] = 0.0; C[21] = -0.25; C[22] = 0.0; C[23] = 0.0; C[24] = 0.0; C[25] = 0.0; C[26] = 0.0; C[27] = 0.0; C[28] = -0.25; C[29] = 0.0; C[30] = 0.0; C[31] = 0.0; C[32] = 0.0; C[33] = 0.0; C[34] = 0.0; C[35] = -0.25; double S11[6 * 6]; Get_Submatrix(S11, 6, 6, S, 10, 0, 0); double S12[6 * 4]; Get_Submatrix(S12, 6, 4, S, 10, 0, 6); double S12t[4 * 6]; Get_Submatrix(S12t, 4, 6, S, 10, 6, 0); double S22[4 * 4]; Get_Submatrix(S22, 4, 4, S, 10, 6, 6); double S22_1[4 * 4]; for (i = 0; i < 16; i++) S22_1[i] = S22[i]; Choleski_LU_Decomposition(S22_1, 4); Choleski_LU_Inverse(S22_1, 4); // Calculate S22a = S22_1 * S12t 4*6 = 4x4 * 4x6 C = AB double S22a[4 * 6]; Multiply_Matrices(S22a, S22_1, 4, 4, S12t, 6); // Then calculate S22b = S12 * S22a ( 6x6 = 6x4 * 4x6) double S22b[6 * 6]; Multiply_Matrices(S22b, S12, 6, 4, S22a, 6); // Calculate SS = S11 - S22b double SS[6 * 6]; for (i = 0; i < 36; i++) SS[i] = S11[i] - S22b[i]; double E[6 * 6]; Multiply_Matrices(E, C, 6, 6, SS, 6); double SSS[6 * 6]; Hessenberg_Form_Elementary(E, SSS, 6); double eigen_real[6]; double eigen_imag[6]; QR_Hessenberg_Matrix(E, SSS, eigen_real, eigen_imag, 6, 100); index = 0; maxval = eigen_real[0]; for (i = 1; i < 6; i++) { if (eigen_real[i] > maxval) { maxval = eigen_real[i]; index = i; } } double v1[6]; v1[0] = SSS[index]; v1[1] = SSS[index + 6]; v1[2] = SSS[index + 12]; v1[3] = SSS[index + 18]; v1[4] = SSS[index + 24]; v1[5] = SSS[index + 30]; // normalize v1 norm = sqrt(v1[0] * v1[0] + v1[1] * v1[1] + v1[2] * v1[2] + v1[3] * v1[3] + v1[4] * v1[4] + v1[5] * v1[5]); v1[0] /= norm; v1[1] /= norm; v1[2] /= norm; v1[3] /= norm; v1[4] /= norm; v1[5] /= norm; if (v1[0] < 0.0) { v1[0] = -v1[0]; v1[1] = -v1[1]; v1[2] = -v1[2]; v1[3] = -v1[3]; v1[4] = -v1[4]; v1[5] = -v1[5]; } // Calculate v2 = S22a * v1 ( 4x1 = 4x6 * 6x1) double v2[4]; Multiply_Matrices(v2, S22a, 4, 6, v1, 1); double v[10]; v[0] = v1[0]; v[1] = v1[1]; v[2] = v1[2]; v[3] = v1[3]; v[4] = v1[4]; v[5] = v1[5]; v[6] = -v2[0]; v[7] = -v2[1]; v[8] = -v2[2]; v[9] = -v2[3]; double Q [3 * 3]; Q[0] = v[0]; Q[1] = v[5]; Q[2] = v[4]; Q[3] = v[5]; Q[4] = v[1]; Q[5] = v[3]; Q[6] = v[4]; Q[7] = v[3]; Q[8] = v[2]; double U[3]; U[0] = v[6]; U[1] = v[7]; U[2] = v[8]; double Q_1[3 * 3]; for (i = 0; i < 9; i++) Q_1[i] = Q[i]; Choleski_LU_Decomposition(Q_1, 3); Choleski_LU_Inverse(Q_1, 3); // Calculate B = Q-1 * U ( 3x1 = 3x3 * 3x1) double B[3]; Multiply_Matrices(B, Q_1, 3, 3, U, 1); B[0] = -B[0]; // x-axis combined bias B[1] = -B[1]; // y-axis combined bias B[2] = -B[2]; // z-axis combined bias // First calculate QB = Q * B ( 3x1 = 3x3 * 3x1) double QB[3]; Multiply_Matrices(QB, Q, 3, 3, B, 1); // Then calculate btqb = BT * QB ( 1x1 = 1x3 * 3x1) Multiply_Matrices(&btqb, B, 1, 3, QB, 1); // Calculate hmb = sqrt(btqb - J). J = v[9]; hmb = sqrt(btqb - J); // Calculate SQ, the square root of matrix Q double SSSS [3 * 3]; Hessenberg_Form_Elementary(Q, SSSS, 3); double eigen_real3[3]; double eigen_imag3[3]; QR_Hessenberg_Matrix(Q, SSSS, eigen_real3, eigen_imag3, 3, 100); // normalize eigenvectors norm1 = sqrt(SSSS[0] * SSSS[0] + SSSS[3] * SSSS[3] + SSSS[6] * SSSS[6]); SSSS[0] /= norm1; SSSS[3] /= norm1; SSSS[6] /= norm1; norm2 = sqrt(SSSS[1] * SSSS[1] + SSSS[4] * SSSS[4] + SSSS[7] * SSSS[7]); SSSS[1] /= norm2; SSSS[4] /= norm2; SSSS[7] /= norm2; norm3 = sqrt(SSSS[2] * SSSS[2] + SSSS[5] * SSSS[5] + SSSS[8] * SSSS[8]); SSSS[2] /= norm3; SSSS[5] /= norm3; SSSS[8] /= norm3; double Dz[3 * 3]; for (i = 0; i < 9; i++) Dz[i] = 0.0; Dz[0] = sqrt(eigen_real3[0]); Dz[4] = sqrt(eigen_real3[1]); Dz[8] = sqrt(eigen_real3[2]); double vdz[3 * 3]; Multiply_Matrices(vdz, SSSS, 3, 3, Dz, 3); Transpose_Square_Matrix(SSSS, 3); double SQ[3 * 3]; Multiply_Matrices(SQ, vdz, 3, 3, SSSS, 3); // hm = 0.569; double A_1[3 * 3]; for (i = 0; i < 9; i++) A_1[i] = SQ[i] * hm / hmb; for (i = 0; i < 3; i++) BAinv[0][i] = B[i]; for (i = 0; i < 3; i++) { BAinv[i + 1][0] = A_1[i * 3]; BAinv[i + 1][1] = A_1[i * 3 + 1]; BAinv[i + 1][2] = A_1[i * 3 + 2]; } } // Place here the source code of all the routines which have been forward-declared, available at // http://www.mymathlib.com/matrices/ //////////////////////////////////////////////////////////////////////////////// // File: copy_vector.c // // Routine(s): // // Copy_Vector // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Copy_Vector(double *d, double *s, int n) // // // // Description: // // Copy the n dimensional vector s(source) to the n dimensional // // vector d(destination). The memory locations of the source and // // destination vectors must not overlap, otherwise the results // // are installation dependent. // // // // Arguments: // // double *d Pointer to the first element of the destination vector d. // // double *s Pointer to the first element of the source vector s. // // int n The number of elements of the source / destination vectors.// // // // Return Values: // // void // // // // Example: // // #define N // // double v[N], vd[N]; // // // // (your code to initialize the vector v) // // // // Copy_Vector(vd, v, N); // // printf(" Vector vd is \n"); // //////////////////////////////////////////////////////////////////////////////// void Copy_Vector(double* d, double* s, int n) { memcpy(d, s, sizeof(double) * n); } //////////////////////////////////////////////////////////////////////////////// // File: multiply_self_transpose.c // // Routine(s): // // Multiply_Self_Transpose // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Multiply_Self_Transpose(double *C, double *A, int nrows, int ncols ) // // // // Description: // // Post multiply an nrows x ncols matrix A by its transpose. The result // // is an nrows x nrows square symmetric matrix C, i.e. C = A A', where ' // // denotes the transpose. // // The matrix C should be declared as double C[nrows][nrows] in the // // calling routine. The memory allocated to C should not include any // // memory allocated to A. // // // // Arguments: // // double *C Pointer to the first element of the matrix C. // // double *A Pointer to the first element of the matrix A. // // int nrows The number of rows of matrix A. // // int ncols The number of columns of the matrices A. // // // // Return Values: // // void // // // // Example: // // #define N // // #define M // // double A[M][N], C[M][M]; // // // // (your code to initialize the matrix A) // // // // Multiply_Self_Transpose(&C[0][0], &A[0][0], M, N); // // printf("The matrix C = AA ' is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Multiply_Self_Transpose(double* C, double* A, int nrows, int ncols) { int i, j, k; double* pA = nullptr; double* p_A = A; double* pB; double* pCdiag = C; double* pC = C; double* pCt; for (i = 0; i < nrows; pCdiag += nrows + 1, p_A = pA, i++) { pC = pCdiag; pCt = pCdiag; pB = p_A; for (j = i; j < nrows; pC++, pCt += nrows, j++) { pA = p_A; *pC = 0.0; for (k = 0; k < ncols; k++) *pC += *(pA++) * *(pB++); *pCt = *pC; } } } //////////////////////////////////////////////////////////////////////////////// // File: get_submatrix.c // // Routine(s): // // Get_Submatrix // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Get_Submatrix(double *S, int mrows, int mcols, // // double *A, int ncols, int row, int col) // // // // Description: // // Copy the mrows and mcols of the nrows x ncols matrix A starting with // // A[row][col] to the submatrix S. // // Note that S should be declared double S[mrows][mcols] in the calling // // routine. // // // // Arguments: // // double *S Destination address of the submatrix. // // int mrows The number of rows of the matrix S. // // int mcols The number of columns of the matrix S. // // double *A Pointer to the first element of the matrix A[nrows][ncols]// // int ncols The number of columns of the matrix A. // // int row The row of A corresponding to the first row of S. // // int col The column of A corresponding to the first column of S. // // // // Return Values: // // void // // // // Example: // // #define N // // #define M // // #define NB // // #define MB // // double A[M][N], B[MB][NB]; // // int row, col; // // // // (your code to set the matrix A, the row number row and column number // // col) // // // // if ( (row >= 0) && (col >= 0) && ((row + MB) < M) && ((col + NB) < N) )// // Get_Submatrix(&B[0][0], MB, NB, &A[0][0], N, row, col); // // printf("The submatrix B is \n"); ... } // //////////////////////////////////////////////////////////////////////////////// void Get_Submatrix(double* S, int mrows, int mcols, double* A, int ncols, int row, int col) { int number_of_bytes = sizeof(double) * mcols; for (A += row * ncols + col; mrows > 0; A += ncols, S += mcols, mrows--) memcpy(S, A, number_of_bytes); } //////////////////////////////////////////////////////////////////////////////// // File: choleski.c // // Contents: // // Choleski_LU_Decomposition // // Choleski_LU_Solve // // Choleski_LU_Inverse // // // // Required Externally Defined Routines: // // Lower_Triangular_Solve // // Lower_Triangular_Inverse // // Upper_Triangular_Solve // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // int Choleski_LU_Decomposition(double *A, int n) // // // // Description: // // This routine uses Choleski's method to decompose the n x n positive // // definite symmetric matrix A into the product of a lower triangular // // matrix L and an upper triangular matrix U equal to the transpose of L. // // The original matrix A is replaced by L and U with L stored in the // // lower triangular part of A and the transpose U in the upper triangular // // part of A. The original matrix A is therefore destroyed. // // // // Choleski's decomposition is performed by evaluating, in order, the // // following pair of expressions for k = 0, ... ,n-1 : // // L[k][k] = sqrt( A[k][k] - ( L[k][0] ^ 2 + ... + L[k][k-1] ^ 2 ) ) // // L[i][k] = (A[i][k] - (L[i][0]*L[k][0] + ... + L[i][k-1]*L[k][k-1])) // // / L[k][k] // // and subsequently setting // // U[k][i] = L[i][k], for i = k+1, ... , n-1. // // // // After performing the LU decomposition for A, call Choleski_LU_Solve // // to solve the equation Ax = B or call Choleski_LU_Inverse to calculate // // the inverse of A. // // // // Arguments: // // double *A On input, the pointer to the first element of the matrix // // A[n][n]. On output, the matrix A is replaced by the lower // // and upper triangular Choleski factorizations of A. // // int n The number of rows and/or columns of the matrix A. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix A is not positive definite symmetric (within // // working accuracy). // // // // Example: // // #define N // // double A[N][N]; // // // // (your code to initialize the matrix A) // // err = Choleski_LU_Decomposition((double *) A, N); // // if (err < 0) printf(" Matrix A is singular\n"); // // else { printf(" The LLt decomposition of A is \n"); // // ... // //////////////////////////////////////////////////////////////////////////////// // // int Choleski_LU_Decomposition(double* A, int n) { int i, k, p; double* p_Lk0; // pointer to L[k][0] double* p_Lkp; // pointer to L[k][p] double* p_Lkk; // pointer to diagonal element on row k. double* p_Li0; // pointer to L[i][0] double reciprocal; for (k = 0, p_Lk0 = A; k < n; p_Lk0 += n, k++) { // Update pointer to row k diagonal element. p_Lkk = p_Lk0 + k; // Calculate the difference of the diagonal element in row k // from the sum of squares of elements row k from column 0 to // column k-1. for (p = 0, p_Lkp = p_Lk0; p < k; p_Lkp += 1, p++) *p_Lkk -= *p_Lkp * *p_Lkp; // If diagonal element is not positive, return the error code, // the matrix is not positive definite symmetric. if (*p_Lkk <= 0.0) return -1; // Otherwise take the square root of the diagonal element. *p_Lkk = sqrt(*p_Lkk); reciprocal = 1.0 / *p_Lkk; // For rows i = k+1 to n-1, column k, calculate the difference // between the i,k th element and the inner product of the first // k-1 columns of row i and row k, then divide the difference by // the diagonal element in row k. // Store the transposed element in the upper triangular matrix. p_Li0 = p_Lk0 + n; for (i = k + 1; i < n; p_Li0 += n, i++) { for (p = 0; p < k; p++) *(p_Li0 + k) -= *(p_Li0 + p) * *(p_Lk0 + p); *(p_Li0 + k) *= reciprocal; *(p_Lk0 + i) = *(p_Li0 + k); } } return 0; } //////////////////////////////////////////////////////////////////////////////// // int Choleski_LU_Solve(double *LU, double *B, double *x, int n) // // // // Description: // // This routine uses Choleski's method to solve the linear equation // // Ax = B. This routine is called after the matrix A has been decomposed // // into a product of a lower triangular matrix L and an upper triangular // // matrix U which is the transpose of L. The matrix A is the product LU. // // The solution proceeds by solving the linear equation Ly = B for y and // // subsequently solving the linear equation Ux = y for x. // // // // Arguments: // // double *LU Pointer to the first element of the matrix whose elements // // form the lower and upper triangular matrix factors of A. // // double *B Pointer to the column vector, (n x 1) matrix, B // // double *x Solution to the equation Ax = B. // // int n The number of rows and/or columns of the matrix LU. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix L is singular. // // // // Example: // // #define N // // double A[N][N], B[N], x[N]; // // // // (your code to create matrix A and column vector B) // // err = Choleski_LU_Decomposition(&A[0][0], N); // // if (err < 0) printf(" Matrix A is singular\n"); // // else { // // err = Choleski_LU_Solve(&A[0][0], B, x, n); // // if (err < 0) printf(" Matrix A is singular\n"); // // else printf(" The solution is \n"); // // ... // // } // //////////////////////////////////////////////////////////////////////////////// // // int Choleski_LU_Solve(double* LU, double B[], double x[], int n) { // Solve the linear equation Ly = B for y, where L is a lower // triangular matrix. if (Lower_Triangular_Solve(LU, B, x, n) < 0) return -1; // Solve the linear equation Ux = y, where y is the solution // obtained above of Ly = B and U is an upper triangular matrix. return Upper_Triangular_Solve(LU, x, x, n); } //////////////////////////////////////////////////////////////////////////////// // int Choleski_LU_Inverse(double *LU, int n) // // // // Description: // // This routine uses Choleski's method to find the inverse of the matrix // // A. This routine is called after the matrix A has been decomposed // // into a product of a lower triangular matrix L and an upper triangular // // matrix U which is the transpose of L. The matrix A is the product of // // the L and U. Upon completion, the inverse of A is stored in LU so // // that the matrix LU is destroyed. // // // // Arguments: // // double *LU On input, the pointer to the first element of the matrix // // whose elements form the lower and upper triangular matrix // // factors of A. On output, the matrix LU is replaced by the // // inverse of the matrix A equal to the product of L and U. // // int n The number of rows and/or columns of the matrix LU. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix L is singular. // // // // Example: // // #define N // // double A[N][N], B[N], x[N]; // // // // (your code to create matrix A and column vector B) // // err = Choleski_LU_Decomposition(&A[0][0], N); // // if (err < 0) printf(" Matrix A is singular\n"); // // else { // // err = Choleski_LU_Inverse(&A[0][0], n); // // if (err < 0) printf(" Matrix A is singular\n"); // // else printf(" The inverse is \n"); // // ... // // } // //////////////////////////////////////////////////////////////////////////////// // // int Choleski_LU_Inverse(double* LU, int n) { int i, j, k; double* p_i, * p_j, * p_k; double sum; if (Lower_Triangular_Inverse(LU, n) < 0) return -1; // Premultiply L inverse by the transpose of L inverse. for (i = 0, p_i = LU; i < n; i++, p_i += n) { for (j = 0, p_j = LU; j <= i; j++, p_j += n) { sum = 0.0; for (k = i, p_k = p_i; k < n; k++, p_k += n) sum += *(p_k + i) * *(p_k + j); *(p_i + j) = sum; *(p_j + i) = sum; } } return 0; } //////////////////////////////////////////////////////////////////////////////// // File: multiply_matrices.c // // Routine(s): // // Multiply_Matrices // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Multiply_Matrices(double *C, double *A, int nrows, int ncols, // // double *B, int mcols) // // // // Description: // // Post multiply the nrows x ncols matrix A by the ncols x mcols matrix // // B to form the nrows x mcols matrix C, i.e. C = A B. // // The matrix C should be declared as double C[nrows][mcols] in the // // calling routine. The memory allocated to C should not include any // // memory allocated to A or B. // // // // Arguments: // // double *C Pointer to the first element of the matrix C. // // double *A Pointer to the first element of the matrix A. // // int nrows The number of rows of the matrices A and C. // // int ncols The number of columns of the matrices A and the // // number of rows of the matrix B. // // double *B Pointer to the first element of the matrix B. // // int mcols The number of columns of the matrices B and C. // // // // Return Values: // // void // // // // Example: // // #define N // // #define M // // #define NB // // double A[M][N], B[N][NB], C[M][NB]; // // // // (your code to initialize the matrices A and B) // // // // Multiply_Matrices(&C[0][0], &A[0][0], M, N, &B[0][0], NB); // // printf("The matrix C is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Multiply_Matrices(double* C, double* A, int nrows, int ncols, double* B, int mcols) { double* pB; double* p_B; int i, j, k; for (i = 0; i < nrows; A += ncols, i++) for (p_B = B, j = 0; j < mcols; C++, p_B++, j++) { pB = p_B; *C = 0.0; for (k = 0; k < ncols; pB += mcols, k++) *C += *(A + k) * *pB; } } //////////////////////////////////////////////////////////////////////////////// // File: identity_matrix.c // // Routine(s): // // Identity_Matrix // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Identity_Matrix(double *A, int n) // // // // Description: // // Set the square n x n matrix A equal to the identity matrix, i.e. // // A[i][j] = 0 if i != j and A[i][i] = 1. // // // // Arguments: // // double *A Pointer to the first element of the matrix A. // // int n The number of rows and columns of the matrix A. // // // // Return Values: // // void // // // // Example: // // #define N // // double A[N][N]; // // // // Identity_Matrix(&A[0][0], N); // // printf("The matrix A is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Identity_Matrix(double* A, int n) { int i, j; for (i = 0; i < n - 1; i++) { *A++ = 1.0; for (j = 0; j < n; j++) *A++ = 0.0; } *A = 1.0; } //////////////////////////////////////////////////////////////////////////////// // File: hessenberg_elementary.c // // Routine(s): // // Hessenberg_Form_Elementary // // Hessenberg_Elementary_Transform // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // int Hessenberg_Form_Elementary(double *A, double *S, int n) // // // // Description: // // This program transforms the square matrix A to a similar matrix in // // Hessenberg form by a multiplying A on the right by a sequence of // // elementary transformations and on the left by the sequence of inverse // // transformations. // // Def: Two matrices A and B are said to be similar if there exists a // // nonsingular matrix S such that A S = S B. // // Def A Hessenberg matrix is the sum of an upper triangular matrix and // // a matrix all of whose components are 0 except possibly on its // // subdiagonal. A Hessenberg matrix is sometimes said to be almost // // upper triangular. // // The algorithm proceeds by successivly selecting columns j = 0,...,n-3 // // and then assuming that columns 0, ..., j-1 have been reduced to Hessen-// // berg form, for rows j+1 to n-1, select that row j' for which |a[j'][j]|// // is a maximum and interchange rows j+1 and j' and columns j+1 and j'. // // Then for each i = j+2 to n-1, let x = a[i][j] / a[j+1][j] subtract // // x * row j+1 from row i and add x * column i to column j+1. // // // // Arguments: // // double *A On input a pointer to the first element of the matrix // // A[n][n]. The matrix A is replaced with the matrix H, // // a matrix in Hessenberg form similar to A. // // double *S On output the transform such that A S = S H. // // The matrix S should be dimensioned at least n x n in the // // calling routine. // // int n The number of rows or columns of the matrix A. // // // // Return Values: // // 0 Success // // -1 Failure - Not enough memory // // // // Example: // // #define N // // double A[N][N], S[N][N]; // // // // (your code to create the matrix A) // // if (Hessenberg_Form_Elementary(&A[0][0], (double*)S, N ) < 0) { // // printf("Not enough memory\n"); exit(0); // // } // // // //////////////////////////////////////////////////////////////////////////////// // // int Hessenberg_Form_Elementary(double* A, double* S, int n) { int i, j, col, row; int* perm; double* p_row, * pS_row; double max; double s; double* pA, * pB, * pC, * pS; // n x n matrices for which n <= 2 are already in Hessenberg form if (n <= 1) { *S = 1.0; return 0; } if (n == 2) { *S++ = 1.0; *S++ = 0.0; *S++ = 1.0; *S = 0.0; return 0; } // Allocate working memory perm = (int*)malloc(n * sizeof(int)); if (perm == NULL) return -1; // not enough memory // For each column use Elementary transformations // to zero the entries below the subdiagonal. p_row = A + n; pS_row = S + n; for (col = 0; col < (n - 2); p_row += n, pS_row += n, col++) { // Find the row in column "col" with maximum magnitude where // row >= col + 1. row = col + 1; perm[row] = row; for (pA = p_row + col, max = 0.0, i = row; i < n; pA += n, i++) if (fabs(*pA) > max) { perm[row] = i; max = fabs(*pA); } // If perm[row] != row, then interchange row "row" and row // perm[row] and interchange column "row" and column perm[row]. if (perm[row] != row) { Interchange_Rows(A, row, perm[row], n); Interchange_Columns(A, row, perm[row], n, n); } // Zero out the components lying below the subdiagonal. pA = p_row + n; pS = pS_row + n; for (i = col + 2; i < n; pA += n, pS += n, i++) { s = *(pA + col) / *(p_row + col); for (j = 0; j < n; j++) *(pA + j) -= *(p_row + j) * s; *(pS + col) = s; for (j = 0, pB = A + col + 1, pC = A + i; j < n; pB += n, pC += n, j++) *pB += s * *pC; } } pA = A + n + n; pS = S + n + n; for (i = 2; i < n; pA += n, pS += n, i++) Copy_Vector(pA, pS, i - 1); Hessenberg_Elementary_Transform(A, S, perm, n); free(perm); return 0; } //////////////////////////////////////////////////////////////////////////////// // void Hessenberg_Elementary_Transform(double* H, double *S, // // int perm[], int n) // // // // Description: // // Given a n x n matrix A, let H be the matrix in Hessenberg form similar // // to A, i.e. A S = S H. If v is an eigenvector of H with eigenvalue z, // // i.e. Hv = zv, then ASv = SHv = z Sv, i.e. Sv is the eigenvector of A // // with corresponding eigenvalue z. // // This routine returns S where S is the similarity transformation such // // that A S = S H. // // // // Arguments: // // double* H On input a matrix in Hessenberg form with transformation // // elements stored below the subdiagonal part. // // On output the matrix in Hessenberg form with elements // // below the subdiagonal zeroed out. // // double* S On output, the transformations matrix such that // // A S = S H. // // int perm[] Array of row/column interchanges. // // int n The order of the matrices H and S. // // // // Return Values: // // void // // // //////////////////////////////////////////////////////////////////////////////// // // void Hessenberg_Elementary_Transform(double* H, double* S, int perm[], int n) { int i, j; double* pS, * pH; Identity_Matrix(S, n); for (i = n - 2; i >= 1; i--) { pH = H + n * (i + 1); pS = S + n * (i + 1); for (j = i + 1; j < n; pH += n, pS += n, j++) { *(pS + i) = *(pH + i - 1); *(pH + i - 1) = 0.0; } if (perm[i] != i) { pS = S + n * i; pH = S + n * perm[i]; for (j = i; j < n; j++) { *(pS + j) = *(pH + j); *(pH + j) = 0.0; } *(pH + i) = 1.0; } } } //////////////////////////////////////////////////////////////////////////////// // File: qr_hessenberg_matrix.c // // Routine(s): // // QR_Hessenberg_Matrix // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // int QR_Hessenberg_Matrix( double *H, double *S, double eigen_real[], // // double eigen_imag[], int n, int max_iteration_count) // // // // Description: // // This program calculates the eigenvalues and eigenvectors of a matrix // // in Hessenberg form. This routine is adapted from the routine 'hql2' // // appearing in 'Handbook for Automatic Computation, vol 2: Linear // // Algebra' published by Springer Verlag (1971) edited by Wilkinson and // // Reinsch, Contribution II/15 Eigenvectors of Real and Complex Matrices // // by LR and QR Triangularizations by Peters and Wilkinson. // // // // Arguments: // // double *H // // Pointer to the first element of the real n x n matrix H in upper// // Hessenberg form. // // double *S // // If H is the primary data matrix, the matrix S should be set // // to the identity n x n identity matrix on input. If H is // // derived from an n x n matrix A, then S should be the // // transformation matrix such that AS = SH. On output, the i-th // // column of S corresponds to the i-th eigenvalue if that eigen- // // value is real and the i-th and (i+1)-st columns of S correspond // // to the i-th eigenvector with the real part in the i-th column // // and positive imaginary part in the (i+1)-st column if that // // eigenvalue is complex with positive imaginary part. The // // eigenvector corresponding to the eigenvalue with negative // // imaginary part is the complex conjugate of the eigenvector // // corresponding to the complex conjugate of the eigenvalue. // // If on input, S was the identity matrix, then the columns are // // the eigenvectors of H as described, if S was a transformation // // matrix so that AS = SH, then the columns of S are the // // eigenvectors of A as described. // // double eigen_real[] // // Upon return, eigen_real[i] will contain the real part of the // // i-th eigenvalue. // // double eigen_imag[] // // Upon return, eigen_ima[i] will contain the imaginary part of // // the i-th eigenvalue. // // int n // // The number of rows or columns of the upper Hessenberg matrix A. // // int max_iteration_count // // The maximum number of iterations to try to find an eigenvalue // // before quitting. // // // // Return Values: // // 0 Success // // -1 Failure - Unable to find an eigenvalue within 'max_iteration_count' // // iterations. // // // // Example: // // #define N // // #define MAX_ITERATION_COUNT // // double H[N][N], S[N][N], eigen_real[N], eigen_imag[N]; // // int k; // // // // (code to initialize H[N][N] and S[N][N]) // // k = QR_Hessenberg_Matrix( (double*)H, (double*)S, eigen_real, // // eigen_imag, N, MAX_ITERATION_COUNT); // // if (k < 0) {printf("Failed"); exit(1);} // //////////////////////////////////////////////////////////////////////////////// // // int QR_Hessenberg_Matrix(double* H, double* S, double eigen_real[], double eigen_imag[], int n, int max_iteration_count) { int i; int row; int iteration; int found_eigenvalue; double shift = 0.0; double* pH; for (row = n - 1; row >= 0; row--) { found_eigenvalue = 0; for (iteration = 1; iteration <= max_iteration_count; iteration++) { // Search for small subdiagonal element for (i = row, pH = H + row * n; i > 0; i--, pH -= n) if (fabs(*(pH + i - 1)) <= DBL_EPSILON * (fabs(*(pH - n + i - 1)) + fabs(*(pH + i)))) break; // If the subdiagonal element on row "row" is small, then // that row element is an eigenvalue. If the subdiagonal // element on row "row-1" is small, then the eigenvalues // of the 2x2 diagonal block consisting rows "row-1" and // "row" are eigenvalues. Otherwise perform a double QR // iteration. switch (row - i) { case 0: // One real eigenvalue One_Real_Eigenvalue(pH, eigen_real, eigen_imag, i, shift); found_eigenvalue = 1; break; case 1: // Either two real eigenvalues or a complex pair row--; Two_Eigenvalues(H, S, eigen_real, eigen_imag, n, row, shift); found_eigenvalue = 1; break; default: Double_QR_Iteration(H, S, i, row, n, &shift, iteration); } if (found_eigenvalue) break; } if (iteration > max_iteration_count) return -1; } BackSubstitution(H, eigen_real, eigen_imag, n); Calculate_Eigenvectors(H, S, eigen_real, eigen_imag, n); return 0; } //////////////////////////////////////////////////////////////////////////////// // void One_Real_Eigenvalue( double Hrow[], double eigen_real[], // // double eigen_imag[], int row, double shift) // // // // Arguments: // // double Hrow[] // // Pointer to the row "row" of the matrix in Hessenberg form. // // double eigen_real[] // // Array of the real parts of the eigenvalues. // // double eigen_imag[] // // Array of the imaginary parts of the eigenvalues. // // int row // // The row to which the pointer Hrow[] points of the matrix H. // // double shift // // The cumulative exceptional shift of the diagonal elements of // // the matrix H. // //////////////////////////////////////////////////////////////////////////////// // // void One_Real_Eigenvalue(double Hrow[], double eigen_real[], double eigen_imag[], int row, double shift) { Hrow[row] += shift; eigen_real[row] = Hrow[row]; eigen_imag[row] = 0.0; } //////////////////////////////////////////////////////////////////////////////// // void Two_Eigenvalues( double *H, double *S, double eigen_real[], // // double eigen_imag[], int n, int row, double shift) // // // // Description: // // Given the 2x2 matrix A = (a[i][j]), the characteristic equation is: // // x^2 - Tr(A) x + Det(A) = 0, where Tr(A) = a[0][0] + a[1][1] and // // Det(A) = a[0][0] * a[1][1] - a[0][1] * a[1][0]. // // The solution for the eigenvalues x are: // // x1 = (Tr(A) + sqrt( (Tr(A))^2 + 4 * a[0][1] * a[1][0] ) / 2 and // // x2 = (Tr(A) - sqrt( (Tr(A))^2 + 4 * a[0][1] * a[1][0] ) / 2. // // Let p = (a[0][0] - a[1][1]) / 2 and q = p^2 - a[0][1] * a[1][0], then // // x1 = a[1][1] + p [+|-] sqrt(q) and x2 = a[0][0] + a[1][1] - x1. // // Choose the sign [+|-] to be the sign of p. // // If q > 0.0, then both roots are real and the transformation // // | cos sin | | a[0][0] a[0][1] | | cos -sin | // // |-sin cos | | a[1][0] a[1][1] | | sin cos | // // where sin = a[1][0] / r, cos = ( p + sqrt(q) ) / r, where r > 0 is // // determined sin^2 + cos^2 = 1 transforms the matrix A to an upper // // triangular matrix with x1 the upper diagonal element and x2 the lower.// // If q < 0.0, then both roots form a complex conjugate pair. // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // double *S // // Pointer to the first element of the transformation matrix. // // double eigen_real[] // // Array of the real parts of the eigenvalues. // // double eigen_imag[] // // Array of the imaginary parts of the eigenvalues. // // int n // // The dimensions of the matrix H and S. // // int row // // The upper most row of the block diagonal 2 x 2 submatrix of H. // // double shift // // The cumulative exceptional shift of the diagonal elements of // // the matrix H. // //////////////////////////////////////////////////////////////////////////////// // // void Two_Eigenvalues(double* H, double* S, double eigen_real[], double eigen_imag[], int n, int row, double shift) { double p, q, x, discriminant, r; double cos, sin; double* Hrow = H + n * row; double* Hnextrow = Hrow + n; int nextrow = row + 1; p = 0.5 * (Hrow[row] - Hnextrow[nextrow]); x = Hrow[nextrow] * Hnextrow[row]; discriminant = p * p + x; Hrow[row] += shift; Hnextrow[nextrow] += shift; if (discriminant > 0.0) { // pair of real roots q = sqrt(discriminant); if (p < 0.0) q = p - q; else q += p; eigen_real[row] = Hnextrow[nextrow] + q; eigen_real[nextrow] = Hnextrow[nextrow] - x / q; eigen_imag[row] = 0.0; eigen_imag[nextrow] = 0.0; r = sqrt(Hnextrow[row] * Hnextrow[row] + q * q); sin = Hnextrow[row] / r; cos = q / r; Update_Row(Hrow, cos, sin, n, row); Update_Column(H, cos, sin, n, row); Update_Transformation(S, cos, sin, n, row); } else { // pair of complex roots eigen_real[nextrow] = eigen_real[row] = Hnextrow[nextrow] + p; eigen_imag[row] = sqrt(fabs(discriminant)); eigen_imag[nextrow] = -eigen_imag[row]; } } //////////////////////////////////////////////////////////////////////////////// // void Update_Row(double *Hrow, double cos, double sin, int n, // // int row) // // // // Description: // // Update rows 'row' and 'row + 1' using the rotation matrix: // // | cos sin | // // |-sin cos |. // // I.e. multiply the matrix H on the left by the identity matrix with // // the 2x2 diagonal block starting at row 'row' replaced by the above // // 2x2 rotation matrix. // // // // Arguments: // // double Hrow[] // // Pointer to the row "row" of the matrix in Hessenberg form. // // double cos // // Cosine of the rotation angle. // // double sin // // Sine of the rotation angle. // // int n // // The dimension of the matrix H. // // int row // // The row to which the pointer Hrow[] points of the matrix H // // in Hessenberg form. // //////////////////////////////////////////////////////////////////////////////// // // void Update_Row(double* Hrow, double cos, double sin, int n, int row) { double x; double* Hnextrow = Hrow + n; int i; for (i = row; i < n; i++) { x = Hrow[i]; Hrow[i] = cos * x + sin * Hnextrow[i]; Hnextrow[i] = cos * Hnextrow[i] - sin * x; } } //////////////////////////////////////////////////////////////////////////////// // void Update_Column(double* H, double cos, double sin, int n, // // int col) // // // // Description: // // Update columns 'col' and 'col + 1' using the rotation matrix: // // | cos -sin | // // | sin cos |. // // I.e. multiply the matrix H on the right by the identity matrix with // // the 2x2 diagonal block starting at row 'col' replaced by the above // // 2x2 rotation matrix. // // // // Arguments: // // double *H // // Pointer to the matrix in Hessenberg form. // // double cos // // Cosine of the rotation angle. // // double sin // // Sine of the rotation angle. // // int n // // The dimension of the matrix H. // // int col // // The left-most column of the matrix H to update. // //////////////////////////////////////////////////////////////////////////////// // // void Update_Column(double* H, double cos, double sin, int n, int col) { double x; int i; int next_col = col + 1; for (i = 0; i <= next_col; i++, H += n) { x = H[col]; H[col] = cos * x + sin * H[next_col]; H[next_col] = cos * H[next_col] - sin * x; } } //////////////////////////////////////////////////////////////////////////////// // void Update_Transformation(double *S, double cos, double sin, // // int n, int k) // // // // Description: // // Update columns 'k' and 'k + 1' using the rotation matrix: // // | cos -sin | // // | sin cos |. // // I.e. multiply the matrix S on the right by the identity matrix with // // the 2x2 diagonal block starting at row 'k' replaced by the above // // 2x2 rotation matrix. // // // // Arguments: // // double *S // // Pointer to the row "row" of the matrix in Hessenberg form. // // double cos // // Pointer to the first element of the matrix in Hessenberg form. // // double sin // // Pointer to the first element of the transformation matrix. // // int n // // The dimensions of the matrix H and S. // // int k // // The row to which the pointer Hrow[] points of the matrix H. // //////////////////////////////////////////////////////////////////////////////// // // void Update_Transformation(double* S, double cos, double sin, int n, int k) { double x; int i; int k1 = k + 1; for (i = 0; i < n; i++, S += n) { x = S[k]; S[k] = cos * x + sin * S[k1]; S[k1] = cos * S[k1] - sin * x; } } //////////////////////////////////////////////////////////////////////////////// // void Double_QR_Iteration(double *H, double *S, int min_row, // // int max_row, int n, double* shift, int iteration) // // // // Description: // // Calculate the trace and determinant of the 2x2 matrix: // // | H[k-1][k-1] H[k-1][k] | // // | H[k][k-1] H[k][k] | // // unless iteration = 0 (mod 10) in which case increment the shift and // // decrement the first k elements of the matrix H, then fudge the trace // // and determinant by trace = 3/2( |H[k][k-1]| + |H[k-1][k-2]| and // // det = 4/9 trace^2. // // // // Arguments: // // double *H // // Pointer to the matrix H in Hessenberg form. // // double *S // // Pointer to the transformation matrix S. // // int min_row // // The top-most row in which the off-diagonal element of H is // // negligible. If no such row exists, then min_row = 0. // // int max_row // // The maximum row of the block 2 x 2 diagonal matrix used to // // estimate the two eigenvalues for the two implicit shifts. // // int n // // The dimensions of the matrix H and S. // // double *shift // // The cumulative exceptional shift of the diagonal elements of // // the matrix H. // // int iteration // // Current iteration count. // //////////////////////////////////////////////////////////////////////////////// // // void Double_QR_Iteration(double* H, double* S, int min_row, int max_row, int n, double* shift, int iteration) { int k; double trace, det; Product_and_Sum_of_Shifts(H, n, max_row, shift, &trace, &det, iteration); k = Two_Consecutive_Small_Subdiagonal(H, min_row, max_row, n, trace, det); Double_QR_Step(H, min_row, max_row, k, trace, det, S, n); } //////////////////////////////////////////////////////////////////////////////// // void Product_and_Sum_of_Shifts(double *H, int n, int max_row, // // double* shift, double *trace, double *det, int iteration) // // // // Description: // // Calculate the trace and determinant of the 2x2 matrix: // // | H[k-1][k-1] H[k-1][k] | // // | H[k][k-1] H[k][k] | // // unless iteration = 0 (mod 10) in which case increment the shift and // // decrement the first k elements of the matrix H, then fudge the trace // // and determinant by trace = 3/2( |H[k][k-1]| + |H[k-1][k-2]| and // // det = 4/9 trace^2. // // // // Arguments: // // double *H // // Pointer to the matrix H in Hessenberg form. // // int n // // The dimension of the matrix H. // // int max_row // // The maximum row of the block 2 x 2 diagonal matrix used to // // estimate the two eigenvalues for the two implicit shifts. // // double *shift // // The cumulative exceptional shift of the diagonal elements of // // the matrix H. Modified if an exceptional shift occurs. // // double *trace // // Returns the trace of the 2 x 2 block diagonal matrix starting // // at the row/column max_row-1. For an exceptional shift, the // // trace is set as described above. // // double *det // // Returns the determinant of the 2 x 2 block diagonal matrix // // starting at the row/column max_row-1. For an exceptional shift,// // the determinant is set as described above. // // int iteration // // Current iteration count. // //////////////////////////////////////////////////////////////////////////////// // // void Product_and_Sum_of_Shifts(double* H, int n, int max_row, double* shift, double* trace, double* det, int iteration) { double* pH = H + max_row * n; double* p_aux; int i; int min_col = max_row - 1; if ((iteration % 10) == 0) { *shift += pH[max_row]; for (i = 0, p_aux = H; i <= max_row; p_aux += n, i++) p_aux[i] -= pH[max_row]; p_aux = pH - n; *trace = fabs(pH[min_col]) + fabs(p_aux[min_col - 1]); *det = *trace * *trace; *trace *= 1.5; } else { p_aux = pH - n; *trace = p_aux[min_col] + pH[max_row]; *det = p_aux[min_col] * pH[max_row] - p_aux[max_row] * pH[min_col]; } }; //////////////////////////////////////////////////////////////////////////////// // int Two_Consecutive_Small_Subdiagonal(double* H, int min_row, // // int max_row, int n, double trace, double det) // // // // Description: // // To reduce the amount of computation in Francis' double QR step search // // for two consecutive small subdiagonal elements from row nn to row m, // // where m < nn. // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // int min_row // // The row in which to end the search (search is from upwards). // // int max_row // // The row in which to begin the search. // // int n // // The dimension of H. // // double trace // // The trace of the lower 2 x 2 block diagonal matrix. // // double det // // The determinant of the lower 2 x 2 block diagonal matrix. // // // // Return Value: // // Row with negligible subdiagonal element or min_row if none found. // //////////////////////////////////////////////////////////////////////////////// // // int Two_Consecutive_Small_Subdiagonal(double* H, int min_row, int max_row, int n, double trace, double det) { double x, y, z, s; double* pH; int i, k; for (k = max_row - 2, pH = H + k * n; k >= min_row; pH -= n, k--) { x = (pH[k] * (pH[k] - trace) + det) / pH[n + k] + pH[k + 1]; y = pH[k] + pH[n + k + 1] - trace; z = pH[n + n + k + 1]; s = fabs(x) + fabs(y) + fabs(z); x /= s; y /= s; z /= s; if (k == min_row) break; if ((fabs(pH[k - 1]) * (fabs(y) + fabs(z))) <= DBL_EPSILON * fabs(x) * (fabs(pH[k - 1 - n]) + fabs(pH[k]) + fabs(pH[n + k + 1]))) break; } for (i = k + 2, pH = H + i * n; i <= max_row; pH += n, i++) pH[i - 2] = 0.0; for (i = k + 3, pH = H + i * n; i <= max_row; pH += n, i++) pH[i - 3] = 0.0; return k; }; //////////////////////////////////////////////////////////////////////////////// // void Double_QR_Step(double *H, int min_row, int max_row, // // int min_col, double *S, int n) // // // // Description: // // Perform Francis' double QR step from rows 'min_row' to 'max_row' // // and columns 'min_col' to 'max_row'. // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // int min_row // // The row in which to begin. // // int max_row // // The row in which to end. // // int min_col // // The column in which to begin. // // double trace // // The trace of the lower 2 x 2 block diagonal matrix. // // double det // // The determinant of the lower 2 x 2 block diagonal matrix. // // double *S // // Pointer to the first element of the transformation matrix. // // int n // // The dimensions of H and S. // //////////////////////////////////////////////////////////////////////////////// // // void Double_QR_Step(double* H, int min_row, int max_row, int min_col, double trace, double det, double* S, int n) { double s, x, y, z; double a, b, c; double* pH; double* tH; double* pS; int i, j, k; int last_test_row_col = max_row - 1; k = min_col; pH = H + min_col * n; a = (pH[k] * (pH[k] - trace) + det) / pH[n + k] + pH[k + 1]; b = pH[k] + pH[n + k + 1] - trace; c = pH[n + n + k + 1]; s = fabs(a) + fabs(b) + fabs(c); a /= s; b /= s; c /= s; for (; k <= last_test_row_col; k++, pH += n) { if (k > min_col) { c = (k == last_test_row_col) ? 0.0 : pH[n + n + k - 1]; x = fabs(pH[k - 1]) + fabs(pH[n + k - 1]) + fabs(c); if (x == 0.0) continue; a = pH[k - 1] / x; b = pH[n + k - 1] / x; c /= x; } s = sqrt(a * a + b * b + c * c); if (a < 0.0) s = -s; if (k > min_col) pH[k - 1] = -s * x; else if (min_row != min_col) pH[k - 1] = -pH[k - 1]; a += s; x = a / s; y = b / s; z = c / s; b /= a; c /= a; // Update rows k, k+1, k+2 for (j = k; j < n; j++) { a = pH[j] + b * pH[n + j]; if (k != last_test_row_col) { a += c * pH[n + n + j]; pH[n + n + j] -= a * z; } pH[n + j] -= a * y; pH[j] -= a * x; } // Update column k+1 j = k + 3; if (j > max_row) j = max_row; for (i = 0, tH = H; i <= j; i++, tH += n) { a = x * tH[k] + y * tH[k + 1]; if (k != last_test_row_col) { a += z * tH[k + 2]; tH[k + 2] -= a * c; } tH[k + 1] -= a * b; tH[k] -= a; } // Update transformation matrix for (i = 0, pS = S; i < n; pS += n, i++) { a = x * pS[k] + y * pS[k + 1]; if (k != last_test_row_col) { a += z * pS[k + 2]; pS[k + 2] -= a * c; } pS[k + 1] -= a * b; pS[k] -= a; } }; } //////////////////////////////////////////////////////////////////////////////// // void BackSubstitution(double *H, double eigen_real[], // // double eigen_imag[], int n) // // // // Description: // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // double eigen_real[] // // The real part of an eigenvalue. // // double eigen_imag[] // // The imaginary part of an eigenvalue. // // int n // // The dimension of H, eigen_real, and eigen_imag. // //////////////////////////////////////////////////////////////////////////////// // // void BackSubstitution(double* H, double eigen_real[], double eigen_imag[], int n) { double zero_tolerance; double* pH; int i, j, row; // Calculate the zero tolerance pH = H; zero_tolerance = fabs(pH[0]); for (pH += n, i = 1; i < n; pH += n, i++) for (j = i - 1; j < n; j++) zero_tolerance += fabs(pH[j]); zero_tolerance *= DBL_EPSILON; // Start Backsubstitution for (row = n - 1; row >= 0; row--) { if (eigen_imag[row] == 0.0) BackSubstitute_Real_Vector(H, eigen_real, eigen_imag, row, zero_tolerance, n); else if (eigen_imag[row] < 0.0) BackSubstitute_Complex_Vector(H, eigen_real, eigen_imag, row, zero_tolerance, n); } } //////////////////////////////////////////////////////////////////////////////// // void BackSubstitute_Real_Vector(double *H, double eigen_real[], // // double eigen_imag[], int row, double zero_tolerance, int n) // // // // Description: // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // double eigen_real[] // // The real part of an eigenvalue. // // double eigen_imag[] // // The imaginary part of an eigenvalue. // // int row // // double zero_tolerance // // Zero substitute. To avoid dividing by zero. // // int n // // The dimension of H, eigen_real, and eigen_imag. // //////////////////////////////////////////////////////////////////////////////// // // void BackSubstitute_Real_Vector(double* H, double eigen_real[], double eigen_imag[], int row, double zero_tolerance, int n) { double* pH; double* pV; double x; double u[4]; double v[2]; int i, j, k; k = row; pH = H + row * n; pH[row] = 1.0; for (i = row - 1, pH -= n; i >= 0; i--, pH -= n) { u[0] = pH[i] - eigen_real[row]; v[0] = pH[row]; pV = H + n * k; for (j = k; j < row; j++, pV += n) v[0] += pH[j] * pV[row]; if (eigen_imag[i] < 0.0) { u[3] = u[0]; v[1] = v[0]; } else { k = i; if (eigen_imag[i] == 0.0) { if (u[0] != 0.0) pH[row] = -v[0] / u[0]; else pH[row] = -v[0] / zero_tolerance; } else { u[1] = pH[i + 1]; u[2] = pH[n + i]; x = (eigen_real[i] - eigen_real[row]); x *= x; x += eigen_imag[i] * eigen_imag[i]; pH[row] = (u[1] * v[1] - u[3] * v[0]) / x; if (fabs(u[1]) > fabs(u[3])) pH[n + row] = -(v[0] + u[0] * pH[row]) / u[1]; else pH[n + row] = -(v[1] + u[2] * pH[row]) / u[3]; } } } } //////////////////////////////////////////////////////////////////////////////// // void BackSubstitute_Complex_Vector(double *H, double eigen_real[], // // double eigen_imag[], int row, double zero_tolerance, int n) // // // // Description: // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // double eigen_real[] // // The real part of an eigenvalue. // // double eigen_imag[] // // The imaginary part of an eigenvalue. // // int row // // double zero_tolerance // // Zero substitute. To avoid dividing by zero. // // int n // // The dimension of H, eigen_real, and eigen_imag. // //////////////////////////////////////////////////////////////////////////////// // // void BackSubstitute_Complex_Vector(double* H, double eigen_real[], double eigen_imag[], int row, double zero_tolerance, int n) { double* pH; double* pV; double x, y; double u[4]; double v[2]; double w[2]; int i, j, k; k = row - 1; pH = H + n * row; if (fabs(pH[k]) > fabs(pH[row - n])) { pH[k - n] = -(pH[row] - eigen_real[row]) / pH[k]; pH[row - n] = -eigen_imag[row] / pH[k]; } else Complex_Division(-pH[row - n], 0.0, pH[k - n] - eigen_real[row], eigen_imag[row], &pH[k - n], &pH[row - n]); pH[k] = 1.0; pH[row] = 0.0; for (i = row - 2, pH = H + n * i; i >= 0; pH -= n, i--) { u[0] = pH[i] - eigen_real[row]; w[0] = pH[row]; w[1] = 0.0; pV = H + k * n; for (j = k; j < row; j++, pV += n) { w[0] += pH[j] * pV[row - 1]; w[1] += pH[j] * pV[row]; } if (eigen_imag[i] < 0.0) { u[3] = u[0]; v[0] = w[0]; v[1] = w[1]; } else { k = i; if (eigen_imag[i] == 0.0) { Complex_Division(-w[0], -w[1], u[0], eigen_imag[row], &pH[row - 1], &pH[row]); } else { u[1] = pH[i + 1]; u[2] = pH[n + i]; x = eigen_real[i] - eigen_real[row]; y = 2.0 * x * eigen_imag[row]; x = x * x + eigen_imag[i] * eigen_imag[i] - eigen_imag[row] * eigen_imag[row]; if (x == 0.0 && y == 0.0) x = zero_tolerance * (fabs(u[0]) + fabs(u[1]) + fabs(u[2]) + fabs(u[3]) + fabs(eigen_imag[row])); Complex_Division(u[1] * v[0] - u[3] * w[0] + w[1] * eigen_imag[row], u[1] * v[1] - u[3] * w[1] - w[0] * eigen_imag[row], x, y, &pH[row - 1], &pH[row]); if (fabs(u[1]) > (fabs(u[3]) + fabs(eigen_imag[row]))) { pH[n + row - 1] = -w[0] - u[0] * pH[row - 1] + eigen_imag[row] * pH[row] / u[1]; pH[n + row] = -w[1] - u[0] * pH[row] - eigen_imag[row] * pH[row - 1] / u[1]; } else { Complex_Division(-v[0] - u[2] * pH[row - 1], -v[1] - u[2] * pH[row], u[3], eigen_imag[row], &pH[n + row - 1], &pH[n + row]); } } } } } //////////////////////////////////////////////////////////////////////////////// // void Calculate_Eigenvectors(double *H, double *S, // // double eigen_real[], double eigen_imag[], int n) // // // // Description: // // Multiply by transformation matrix. // // // // Arguments: // // double *H // // Pointer to the first element of the matrix in Hessenberg form. // // double *S // // Pointer to the first element of the transformation matrix. // // double eigen_real[] // // The real part of an eigenvalue. // // double eigen_imag[] // // The imaginary part of an eigenvalue. // // int n // // The dimension of H, S, eigen_real, and eigen_imag. // //////////////////////////////////////////////////////////////////////////////// // // void Calculate_Eigenvectors(double* H, double* S, double eigen_real[], double eigen_imag[], int n) { double* pH; double* pS; double x, y; int i, j, k; for (k = n - 1; k >= 0; k--) { if (eigen_imag[k] < 0.0) { for (i = 0, pS = S; i < n; pS += n, i++) { x = 0.0; y = 0.0; for (j = 0, pH = H; j <= k; pH += n, j++) { x += pS[j] * pH[k - 1]; y += pS[j] * pH[k]; } pS[k - 1] = x; pS[k] = y; } } else if (eigen_imag[k] == 0.0) { for (i = 0, pS = S; i < n; i++, pS += n) { x = 0.0; for (j = 0, pH = H; j <= k; j++, pH += n) x += pS[j] * pH[k]; pS[k] = x; } } } } //////////////////////////////////////////////////////////////////////////////// // void Complex_Division(double x, double y, double u, double v, // // double* a, double* b) // // // // Description: // // a + i b = (x + iy) / (u + iv) // // = (x * u + y * v) / r^2 + i (y * u - x * v) / r^2, // // where r^2 = u^2 + v^2. // // // // Arguments: // // double x // // Real part of the numerator. // // double y // // Imaginary part of the numerator. // // double u // // Real part of the denominator. // // double v // // Imaginary part of the denominator. // // double *a // // Real part of the quotient. // // double *b // // Imaginary part of the quotient. // //////////////////////////////////////////////////////////////////////////////// // // void Complex_Division(double x, double y, double u, double v, double* a, double* b) { double q = u * u + v * v; *a = (x * u + y * v) / q; *b = (y * u - x * v) / q; } //////////////////////////////////////////////////////////////////////////////// // File: transpose_square_matrix.c // // Routine(s): // // Transpose_Square_Matrix // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Transpose_Square_Matrix( double *A, int n ) // // // // Description: // // Take the transpose of A and store in place. // // // // Arguments: // // double *A Pointer to the first element of the matrix A. // // int n The number of rows and columns of the matrix A. // // // // Return Values: // // void // // // // Example: // // #define N // // double A[N][N]; // // // // (your code to initialize the matrix A) // // // // Transpose_Square_Matrix( &A[0][0], N); // // printf("The transpose of A is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Transpose_Square_Matrix(double* A, int n) { double* pA, * pAt; double temp; int i, j; for (i = 0; i < n; A += n + 1, i++) { pA = A + 1; pAt = A + n; for (j = i + 1; j < n; pA++, pAt += n, j++) { temp = *pAt; *pAt = *pA; *pA = temp; } } } /////////////////////////////////////////////////////////////////////////////// // File: lower_triangular.c // // Routines: // // Lower_Triangular_Solve // // Lower_Triangular_Inverse // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // int Lower_Triangular_Solve(double *L, double *B, double x[], int n) // // // // Description: // // This routine solves the linear equation Lx = B, where L is an n x n // // lower triangular matrix. (The superdiagonal part of the matrix is // // not addressed.) // // The algorithm follows: // // x[0] = B[0]/L[0][0], and // // x[i] = [B[i] - (L[i][0] * x[0] + ... + L[i][i-1] * x[i-1])] / L[i][i],// // for i = 1, ..., n-1. // // // // Arguments: // // double *L Pointer to the first element of the lower triangular // // matrix. // // double *B Pointer to the column vector, (n x 1) matrix, B. // // double *x Pointer to the column vector, (n x 1) matrix, x. // // int n The number of rows or columns of the matrix L. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix L is singular. // // // // Example: // // #define N // // double A[N][N], B[N], x[N]; // // // // (your code to create matrix A and column vector B) // // err = Lower_Triangular_Solve(&A[0][0], B, x, n); // // if (err < 0) printf(" Matrix A is singular\n"); // // else printf(" The solution is \n"); // // ... // //////////////////////////////////////////////////////////////////////////////// // // int Lower_Triangular_Solve(double* L, double B[], double x[], int n) { int i, k; // Solve the linear equation Lx = B for x, where L is a lower // triangular matrix. for (k = 0; k < n; L += n, k++) { if (*(L + k) == 0.0) return -1; // The matrix L is singular x[k] = B[k]; for (i = 0; i < k; i++) x[k] -= x[i] * *(L + i); x[k] /= *(L + k); } return 0; } //////////////////////////////////////////////////////////////////////////////// // int Lower_Triangular_Inverse(double *L, int n) // // // // Description: // // This routine calculates the inverse of the lower triangular matrix L. // // The superdiagonal part of the matrix is not addressed. // // The algorithm follows: // // Let M be the inverse of L, then L M = I, // // M[i][i] = 1.0 / L[i][i] for i = 0, ..., n-1, and // // M[i][j] = -[(L[i][j] M[j][j] + ... + L[i][i-1] M[i-1][j])] / L[i][i], // // for i = 1, ..., n-1, j = 0, ..., i - 1. // // // // // // Arguments: // // double *L On input, the pointer to the first element of the matrix // // whose lower triangular elements form the matrix which is // // to be inverted. On output, the lower triangular part is // // replaced by the inverse. The superdiagonal elements are // // not modified. // // its inverse. // // int n The number of rows and/or columns of the matrix L. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix L is singular. // // // // Example: // // #define N // // double L[N][N]; // // // // (your code to create the matrix L) // // err = Lower_Triangular_Inverse(&L[0][0], N); // // if (err < 0) printf(" Matrix L is singular\n"); // // else { // // printf(" The inverse is \n"); // // ... // // } // //////////////////////////////////////////////////////////////////////////////// // // int Lower_Triangular_Inverse(double* L, int n) { int i, j, k; double* p_i, * p_j, * p_k; double sum; // Invert the diagonal elements of the lower triangular matrix L. for (k = 0, p_k = L; k < n; p_k += (n + 1), k++) { if (*p_k == 0.0) return -1; else *p_k = 1.0 / *p_k; } // Invert the remaining lower triangular matrix L row by row. for (i = 1, p_i = L + n; i < n; i++, p_i += n) { for (j = 0, p_j = L; j < i; p_j += n, j++) { sum = 0.0; for (k = j, p_k = p_j; k < i; k++, p_k += n) sum += *(p_i + k) * *(p_k + j); *(p_i + j) = -*(p_i + i) * sum; } } return 0; } //////////////////////////////////////////////////////////////////////////////// // File: upper_triangular.c // // Routines: // // Upper_Triangular_Solve // // Upper_Triangular_Inverse // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // int Upper_Triangular_Solve(double *U, double *B, double x[], int n) // // // // Description: // // This routine solves the linear equation Ux = B, where U is an n x n // // upper triangular matrix. (The subdiagonal part of the matrix is // // not addressed.) // // The algorithm follows: // // x[n-1] = B[n-1]/U[n-1][n-1], and // // x[i] = [B[i] - (U[i][i+1] * x[i+1] + ... + U[i][n-1] * x[n-1])] // // / U[i][i], // // for i = n-2, ..., 0. // // // // Arguments: // // double *U Pointer to the first element of the upper triangular // // matrix. // // double *B Pointer to the column vector, (n x 1) matrix, B. // // double *x Pointer to the column vector, (n x 1) matrix, x. // // int n The number of rows or columns of the matrix U. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix U is singular. // // // // Example: // // #define N // // double A[N][N], B[N], x[N]; // // // // (your code to create matrix A and column vector B) // // err = Upper_Triangular_Solve(&A[0][0], B, x, n); // // if (err < 0) printf(" Matrix A is singular\n"); // // else printf(" The solution is \n"); // // ... // //////////////////////////////////////////////////////////////////////////////// // // int Upper_Triangular_Solve(double* U, double B[], double x[], int n) { int i, k; // Solve the linear equation Ux = B for x, where U is an upper // triangular matrix. for (k = n - 1, U += n * (n - 1); k >= 0; U -= n, k--) { if (*(U + k) == 0.0) return -1; // The matrix U is singular x[k] = B[k]; for (i = k + 1; i < n; i++) x[k] -= x[i] * *(U + i); x[k] /= *(U + k); } return 0; } //////////////////////////////////////////////////////////////////////////////// // int Upper_Triangular_Inverse(double *U, int n) // // // // Description: // // This routine calculates the inverse of the upper triangular matrix U. // // The subdiagonal part of the matrix is not addressed. // // The algorithm follows: // // Let M be the inverse of U, then U M = I, // // M[n-1][n-1] = 1.0 / U[n-1][n-1] and // // M[i][j] = -( U[i][i+1] M[i+1][j] + ... + U[i][j] M[j][j] ) / U[i][i], // // for i = n-2, ... , 0, j = n-1, ..., i+1. // // // // // // Arguments: // // double *U On input, the pointer to the first element of the matrix // // whose upper triangular elements form the matrix which is // // to be inverted. On output, the upper triangular part is // // replaced by the inverse. The subdiagonal elements are // // not modified. // // int n The number of rows and/or columns of the matrix U. // // // // Return Values: // // 0 Success // // -1 Failure - The matrix U is singular. // // // // Example: // // #define N // // double U[N][N]; // // // // (your code to create the matrix U) // // err = Upper_Triangular_Inverse(&U[0][0], N); // // if (err < 0) printf(" Matrix U is singular\n"); // // else { // // printf(" The inverse is \n"); // // ... // // } // //////////////////////////////////////////////////////////////////////////////// // // int Upper_Triangular_Inverse(double* U, int n) { int i, j, k; double* p_i, * p_j, * p_k; double sum; // Invert the diagonal elements of the upper triangular matrix U. for (k = 0, p_k = U; k < n; p_k += (n + 1), k++) { if (*p_k == 0.0) return -1; else *p_k = 1.0 / *p_k; } // Invert the remaining upper triangular matrix U. for (i = n - 2, p_i = U + n * (n - 2); i >= 0; p_i -= n, i--) { for (j = n - 1; j > i; j--) { sum = 0.0; for (k = i + 1, p_k = p_i + n; k <= j; p_k += n, k++) { sum += *(p_i + k) * *(p_k + j); } *(p_i + j) = -*(p_i + i) * sum; } } return 0; } //////////////////////////////////////////////////////////////////////////////// // File: interchange_cols.c // // Routine(s): // // Interchange_Columns // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Interchange_Columns(double *A, int col1, int col2, int nrows, // // int ncols) // // // // Description: // // Interchange the columns 'col1' and 'col2' of the nrows x ncols // // matrix A. // // // // Arguments: // // double *A Pointer to the first element of the matrix A. // // int col1 The column of A which is to be interchanged with col2. // // int col2 The column of A which is to be interchanged with col1. // // int nrows The number of rows matrix A. // // int ncols The number of columns of the matrix A. // // // // Return Values: // // void // // // // Example: // // #define N // // #define M // // double A[M][N]; // // int i,j; // // // // (your code to initialize the matrix A, the column number i and column // // number j) // // // // if ( (i >= 0) && ( i < N ) && ( j >= 0 ) && (j < N) ) // // Interchange_Columns(&A[0][0], i, j, M, N); // // printf("The matrix A is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Interchange_Columns(double* A, int col1, int col2, int nrows, int ncols) { int i; double* pA1, * pA2; double temp; pA1 = A + col1; pA2 = A + col2; for (i = 0; i < nrows; pA1 += ncols, pA2 += ncols, i++) { temp = *pA1; *pA1 = *pA2; *pA2 = temp; } } //////////////////////////////////////////////////////////////////////////////// // File: interchange_rows.c // // Routine(s): // // Interchange_Rows // //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// // void Interchange_Rows(double *A, int row1, int row2, int ncols) // // // // Description: // // Interchange the rows 'row1' and 'row2' of the nrows x ncols matrix A. // // // // Arguments: // // double *A Pointer to the first element of the matrix A. // // int row1 The row of A which is to be interchanged with row row2. // // int row2 The row of A which is to be interchanged with row row1. // // int ncols The number of columns of the matrix A. // // // // Return Values: // // void // // // // Example: // // #define N // // #define M // // double A[M][N]; // // int i, j; // // // // (your code to initialize the matrix A, the row number i and row number j) // // // // if ( (i >= 0) && ( i < M ) && (j > 0) && ( j < M ) ) // // Interchange_Rows(&A[0][0], i, j, N); // // printf("The matrix A is \n"); ... // //////////////////////////////////////////////////////////////////////////////// void Interchange_Rows(double* A, int row1, int row2, int ncols) { int i; double* pA1, * pA2; double temp; pA1 = A + row1 * ncols; pA2 = A + row2 * ncols; for (i = 0; i < ncols; i++) { temp = *pA1; *pA1++ = *pA2; *pA2++ = temp; } }