/* nag_dtgevc (f08ykc) Example Program. * * Copyright 2001 Numerical Algorithms Group. * * Mark 7, 2001. */ #include #include #include #include #include #include #include static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[], double alphai[], const char* title); int main(void) { /* Scalars */ Integer i, icols, ihi, ilo, irows, j, m, n, pda, pdb, pdq, pdz; Integer exit_status = 0; Nag_Boolean ileft, iright; NagError fail; Nag_OrderType order; /* Arrays */ double *a = 0, *alphai = 0, *alphar = 0, *b = 0, *beta = 0; double *lscale = 0, *q = 0, *rscale = 0, *tau = 0, *z = 0; #ifdef NAG_COLUMN_MAJOR #define A(I, J) a[(J-1)*pda + I - 1] #define B(I, J) b[(J-1)*pdb + I - 1] #define Q(I, J) q[(J-1)*pdq + I - 1] #define Z(I, J) z[(J-1)*pdz + I - 1] order = Nag_ColMajor; #else #define A(I, J) a[(I-1)*pda + J - 1] #define B(I, J) b[(I-1)*pdb + J - 1] #define Q(I, J) q[(I-1)*pdq + J - 1] #define Z(I, J) z[(I-1)*pdz + J - 1] order = Nag_RowMajor; #endif INIT_FAIL(fail); printf("nag_dtgevc (f08ykc) Example Program Results\n\n"); /* ileft is Nag_TRUE if left eigenvectors are required */ /* iright is Nag_TRUE if right eigenvectors are required */ ileft = Nag_TRUE; iright = Nag_TRUE; /* Skip heading in data file */ scanf("%*[^\n] "); scanf("%ld %*[^\n] ", &n); pda = n; pdb = n; pdq = n; pdz = n; /* Allocate memory */ if ( !(a = NAG_ALLOC(n * n, double)) || !(b = NAG_ALLOC(n * n, double)) || !(q = NAG_ALLOC(n * n, double)) || !(z = NAG_ALLOC(n * n, double)) || !(alphai = NAG_ALLOC(n, double)) || !(alphar = NAG_ALLOC(n, double)) || !(beta = NAG_ALLOC(n, double)) || !(lscale = NAG_ALLOC(n, double)) || !(rscale = NAG_ALLOC(n, double)) || !(tau = NAG_ALLOC(n, double))) { printf("Allocation failure\n"); exit_status = -1; goto END; } /* READ matrix A from data file */ for (i = 1; i <= n; ++i) for (j = 1; j <= n; ++j) scanf("%lf", &A(i, j)); scanf("%*[^\n] "); /* READ matrix B from data file */ for (i = 1; i <= n; ++i) for (j = 1; j <= n; ++j) scanf("%lf", &B(i, j)); scanf("%*[^\n] "); /* Balance the real general matrix pair (A,B) using nag_dggbal (f08whc). */ nag_dggbal(order, Nag_DoBoth, n, a, pda, b, pdb, &ilo, &ihi, lscale, rscale, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dggbal (f08whc).\n%s\n", fail.message); exit_status = 1; goto END; } /* Print matrices A and B after balancing using * nag_gen_real_mat_print (x04cac). */ fflush(stdout); nag_gen_real_mat_print(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "Matrix A after balancing", 0, &fail); if (fail.code == NE_NOERROR) { fflush(stdout); nag_gen_real_mat_print(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "Matrix B after balancing", 0, &fail); } if (fail.code != NE_NOERROR) { printf("Error from nag_gen_real_mat_print (x04cac).\n%s\n", fail.message); exit_status = 2; goto END; } printf("\n"); /* Reduce B to triangular form using QR and multipling both sides by Q^T */ irows = ihi + 1 - ilo; icols = n + 1 - ilo; /* nag_dgeqrf (f08aec). * QR factorization of real general rectangular matrix */ nag_dgeqrf(order, irows, icols, &B(ilo, ilo), pdb, tau, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dgeqrf (f08aec).\n%s\n", fail.message); exit_status = 3; goto END; } /* Apply the Q to matrix A - nag_dormqr (f08agc) * as determined by nag_dgeqrf (f08aec). */ nag_dormqr(order, Nag_LeftSide, Nag_Trans, irows, icols, irows, &B(ilo, ilo), pdb, tau, &A(ilo, ilo), pda, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dormqr (f08agc).\n%s\n", fail.message); exit_status = 4; goto END; } /* Initialize Q (if left eigenvectors are required) */ if (ileft) { /* Q = I. */ nag_dge_load(order, n, n, 0.0, 1.0, q, pdq, &fail); /* Copy B to Q using nag_dge_copy (f16qfc). */ nag_dge_copy(order, Nag_NoTrans, irows-1, irows-1, &B(ilo+1,ilo), pdb, &Q(ilo+1,ilo), pdq, &fail); /* nag_dorgqr (f08afc). * Form all or part of orthogonal Q from QR factorization * determined by nag_dgeqrf (f08aec) or nag_dgeqpf (f08bec) */ nag_dorgqr(order, irows, irows, irows, &Q(ilo, ilo), pdq, tau, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dorgqr (f08afc).\n%s\n", fail.message); exit_status = 5; goto END; } } /* Initialize Z (if right eigenvectors are required) */ if (iright) { /* Z = I. */ nag_dge_load(order, n, n, 0.0, 1.0, z, pdz, &fail); } /* Compute the generalized Hessenberg form of (A,B) */ /* nag_dgghrd (f08wec). * Orthogonal reduction of a pair of real general matrices * to generalized upper Hessenberg form */ nag_dgghrd(order, Nag_UpdateSchur, Nag_UpdateZ, n, ilo, ihi, a, pda, b, pdb, q, pdq, z, pdz, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dgghrd (f08wec).\n%s\n", fail.message); exit_status = 6; goto END; } /* Matrix A in generalized Hessenberg form */ /* nag_gen_real_mat_print (x04cac), see above. */ fflush(stdout); nag_gen_real_mat_print(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, a, pda, "Matrix A in Hessenberg form", 0, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_gen_real_mat_print (x04cac).\n%s\n", fail.message); exit_status = 7; goto END; } printf("\n"); /* Matrix B in generalized Hessenberg form */ /* nag_gen_real_mat_print (x04cac), see above. */ fflush(stdout); nag_gen_real_mat_print(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, n, b, pdb, "Matrix B in Hessenberg form", 0, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_gen_real_mat_print (x04cac).\n%s\n", fail.message); exit_status = 1; goto END; } /* nag_dhgeqz (f08xec). * Eigenvalues and generalized Schur factorization of real * generalized upper Hessenberg form reduced from a pair of * real general matrices. */ nag_dhgeqz(order, Nag_Schur, Nag_AccumulateQ, Nag_AccumulateZ, n, ilo, ihi, a, pda, b, pdb, alphar, alphai, beta, q, pdq, z, pdz, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dhgeqz (f08xec).\n%s\n", fail.message); exit_status = 1; goto END; } /* Print the generalized eigenvalue parameters */ printf("\n Generalized eigenvalues\n"); for (i = 0; i < n; ++i) { if (beta[i] != 0.0) { printf(" %4ld (%7.3f,%7.3f)\n", i+1, alphar[i]/beta[i], alphai[i]/beta[i]); } else printf(" %4ldEigenvalue is infinite\n", i+1); } printf("\n"); /* Compute left and right generalized eigenvectors * of the balanced matrix - nag_dtgevc (f08ykc). */ nag_dtgevc(order, Nag_BothSides, Nag_BackTransform, NULL, n, a, pda, b, pdb, q, pdq, z, pdz, n, &m, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dtgevc (f08ykc).\n%s\n", fail.message); exit_status = 1; goto END; } if (iright) { /* Compute right eigenvectors of the original matrix pair * supplied tonag_dggbal (f08whc) using nag_dggbak (f08wjc). */ nag_dggbak(order, Nag_DoBoth, Nag_RightSide, n, ilo, ihi, lscale, rscale, n, z, pdz, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dggbak (f08wjc).\n%s\n", fail.message); exit_status = 1; goto END; } /* Normalize and print the right eigenvectors */ exit_status = normalize_vectors(order, n, z, alphai, "Right eigenvectors"); } printf("\n"); /* Compute left eigenvectors of the original matrix */ if (ileft) { /* nag_dggbak (f08wjc), see above. */ nag_dggbak(order, Nag_DoBoth, Nag_LeftSide, n, ilo, ihi, lscale, rscale, n, q, pdq, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_dggbak (f08wjc).\n%s\n", fail.message); exit_status = 1; goto END; } /* Normalize the left eigenvectors */ exit_status = normalize_vectors(order, n, q, alphai, "Left eigenvectors"); } END: NAG_FREE(a); NAG_FREE(b); NAG_FREE(q); NAG_FREE(z); NAG_FREE(alphai); NAG_FREE(alphar); NAG_FREE(beta); NAG_FREE(lscale); NAG_FREE(rscale); NAG_FREE(tau); return exit_status; } static Integer normalize_vectors(Nag_OrderType order, Integer n, double qz[], double alphai[], const char* title) { /* Real eigenvectors are scaled so that the maximum value of elements is 1.0; * each complex eigenvector z[] is normalized so that the element of largest * magnitude is scaled to be (1.0,0.0). */ double a, b, u, v, r, ri; Integer colinc, rowinc, i, j, k, indqz, errors=0; NagError fail; INIT_FAIL(fail); if (order==Nag_ColMajor) { rowinc = 1; colinc = n; } else { rowinc = n; colinc = 1; } indqz = 0; for (j=0; j=0.0) { if (alphai[j]==0.0) { /* Find element of eigenvector with largest absolute value using * nag_damax_val (f16jqc). */ nag_damax_val(n, &qz[indqz], rowinc, &k, &r, &fail); if (fail.code != NE_NOERROR) { printf("Error from nag_damax_val (f16jqc).\n%s\n", fail.message); errors = 1; goto END; } r = qz[indqz+k]; for (i=0; ir) { k = i; r = ri; } } a = qz[indqz+k]; b = qz[indqz+colinc+k]; r = a*a + b*b; for (i=0; i