/* ctgsja.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static complex c_b1 = {0.f,0.f}; static complex c_b2 = {1.f,0.f}; static integer c__1 = 1; static real c_b39 = -1.f; static real c_b42 = 1.f; /* Subroutine */ int ctgsja_(char *jobu, char *jobv, char *jobq, integer *m, integer *p, integer *n, integer *k, integer *l, complex *a, integer * lda, complex *b, integer *ldb, real *tola, real *tolb, real *alpha, real *beta, complex *u, integer *ldu, complex *v, integer *ldv, complex *q, integer *ldq, complex *work, integer *ncycle, integer * info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1, u_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4; real r__1; complex q__1; /* Builtin functions */ void r_cnjg(complex *, complex *); /* Local variables */ integer i__, j; real a1, b1, a3, b3; complex a2, b2; real csq, csu, csv; complex snq; real rwk; complex snu, snv; extern /* Subroutine */ int crot_(integer *, complex *, integer *, complex *, integer *, real *, complex *); real gamma; extern logical lsame_(char *, char *); extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *); logical initq, initu, initv, wantq, upper; real error, ssmin; logical wantu, wantv; extern /* Subroutine */ int clags2_(logical *, real *, complex *, real *, real *, complex *, real *, real *, complex *, real *, complex *, real *, complex *), clapll_(integer *, complex *, integer *, complex *, integer *, real *), csscal_(integer *, real *, complex *, integer *); integer kcycle; extern /* Subroutine */ int claset_(char *, integer *, integer *, complex *, complex *, complex *, integer *), xerbla_(char *, integer *), slartg_(real *, real *, real *, real *, real * ); /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* CTGSJA computes the generalized singular value decomposition (GSVD) */ /* of two complex upper triangular (or trapezoidal) matrices A and B. */ /* On entry, it is assumed that matrices A and B have the following */ /* forms, which may be obtained by the preprocessing subroutine CGGSVP */ /* from a general M-by-N matrix A and P-by-N matrix B: */ /* N-K-L K L */ /* A = K ( 0 A12 A13 ) if M-K-L >= 0; */ /* L ( 0 0 A23 ) */ /* M-K-L ( 0 0 0 ) */ /* N-K-L K L */ /* A = K ( 0 A12 A13 ) if M-K-L < 0; */ /* M-K ( 0 0 A23 ) */ /* N-K-L K L */ /* B = L ( 0 0 B13 ) */ /* P-L ( 0 0 0 ) */ /* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular */ /* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, */ /* otherwise A23 is (M-K)-by-L upper trapezoidal. */ /* On exit, */ /* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), */ /* where U, V and Q are unitary matrices, Z' denotes the conjugate */ /* transpose of Z, R is a nonsingular upper triangular matrix, and D1 */ /* and D2 are ``diagonal'' matrices, which are of the following */ /* structures: */ /* If M-K-L >= 0, */ /* K L */ /* D1 = K ( I 0 ) */ /* L ( 0 C ) */ /* M-K-L ( 0 0 ) */ /* K L */ /* D2 = L ( 0 S ) */ /* P-L ( 0 0 ) */ /* N-K-L K L */ /* ( 0 R ) = K ( 0 R11 R12 ) K */ /* L ( 0 0 R22 ) L */ /* where */ /* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */ /* S = diag( BETA(K+1), ... , BETA(K+L) ), */ /* C**2 + S**2 = I. */ /* R is stored in A(1:K+L,N-K-L+1:N) on exit. */ /* If M-K-L < 0, */ /* K M-K K+L-M */ /* D1 = K ( I 0 0 ) */ /* M-K ( 0 C 0 ) */ /* K M-K K+L-M */ /* D2 = M-K ( 0 S 0 ) */ /* K+L-M ( 0 0 I ) */ /* P-L ( 0 0 0 ) */ /* N-K-L K M-K K+L-M */ /* ( 0 R ) = K ( 0 R11 R12 R13 ) */ /* M-K ( 0 0 R22 R23 ) */ /* K+L-M ( 0 0 0 R33 ) */ /* where */ /* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */ /* S = diag( BETA(K+1), ... , BETA(M) ), */ /* C**2 + S**2 = I. */ /* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored */ /* ( 0 R22 R23 ) */ /* in B(M-K+1:L,N+M-K-L+1:N) on exit. */ /* The computation of the unitary transformation matrices U, V or Q */ /* is optional. These matrices may either be formed explicitly, or they */ /* may be postmultiplied into input matrices U1, V1, or Q1. */ /* Arguments */ /* ========= */ /* JOBU (input) CHARACTER*1 */ /* = 'U': U must contain a unitary matrix U1 on entry, and */ /* the product U1*U is returned; */ /* = 'I': U is initialized to the unit matrix, and the */ /* unitary matrix U is returned; */ /* = 'N': U is not computed. */ /* JOBV (input) CHARACTER*1 */ /* = 'V': V must contain a unitary matrix V1 on entry, and */ /* the product V1*V is returned; */ /* = 'I': V is initialized to the unit matrix, and the */ /* unitary matrix V is returned; */ /* = 'N': V is not computed. */ /* JOBQ (input) CHARACTER*1 */ /* = 'Q': Q must contain a unitary matrix Q1 on entry, and */ /* the product Q1*Q is returned; */ /* = 'I': Q is initialized to the unit matrix, and the */ /* unitary matrix Q is returned; */ /* = 'N': Q is not computed. */ /* M (input) INTEGER */ /* The number of rows of the matrix A. M >= 0. */ /* P (input) INTEGER */ /* The number of rows of the matrix B. P >= 0. */ /* N (input) INTEGER */ /* The number of columns of the matrices A and B. N >= 0. */ /* K (input) INTEGER */ /* L (input) INTEGER */ /* K and L specify the subblocks in the input matrices A and B: */ /* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) */ /* of A and B, whose GSVD is going to be computed by CTGSJA. */ /* See Further details. */ /* A (input/output) COMPLEX array, dimension (LDA,N) */ /* On entry, the M-by-N matrix A. */ /* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular */ /* matrix R or part of R. See Purpose for details. */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,M). */ /* B (input/output) COMPLEX array, dimension (LDB,N) */ /* On entry, the P-by-N matrix B. */ /* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains */ /* a part of R. See Purpose for details. */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,P). */ /* TOLA (input) REAL */ /* TOLB (input) REAL */ /* TOLA and TOLB are the convergence criteria for the Jacobi- */ /* Kogbetliantz iteration procedure. Generally, they are the */ /* same as used in the preprocessing step, say */ /* TOLA = MAX(M,N)*norm(A)*MACHEPS, */ /* TOLB = MAX(P,N)*norm(B)*MACHEPS. */ /* ALPHA (output) REAL array, dimension (N) */ /* BETA (output) REAL array, dimension (N) */ /* On exit, ALPHA and BETA contain the generalized singular */ /* value pairs of A and B; */ /* ALPHA(1:K) = 1, */ /* BETA(1:K) = 0, */ /* and if M-K-L >= 0, */ /* ALPHA(K+1:K+L) = diag(C), */ /* BETA(K+1:K+L) = diag(S), */ /* or if M-K-L < 0, */ /* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 */ /* BETA(K+1:M) = S, BETA(M+1:K+L) = 1. */ /* Furthermore, if K+L < N, */ /* ALPHA(K+L+1:N) = 0 */ /* BETA(K+L+1:N) = 0. */ /* U (input/output) COMPLEX array, dimension (LDU,M) */ /* On entry, if JOBU = 'U', U must contain a matrix U1 (usually */ /* the unitary matrix returned by CGGSVP). */ /* On exit, */ /* if JOBU = 'I', U contains the unitary matrix U; */ /* if JOBU = 'U', U contains the product U1*U. */ /* If JOBU = 'N', U is not referenced. */ /* LDU (input) INTEGER */ /* The leading dimension of the array U. LDU >= max(1,M) if */ /* JOBU = 'U'; LDU >= 1 otherwise. */ /* V (input/output) COMPLEX array, dimension (LDV,P) */ /* On entry, if JOBV = 'V', V must contain a matrix V1 (usually */ /* the unitary matrix returned by CGGSVP). */ /* On exit, */ /* if JOBV = 'I', V contains the unitary matrix V; */ /* if JOBV = 'V', V contains the product V1*V. */ /* If JOBV = 'N', V is not referenced. */ /* LDV (input) INTEGER */ /* The leading dimension of the array V. LDV >= max(1,P) if */ /* JOBV = 'V'; LDV >= 1 otherwise. */ /* Q (input/output) COMPLEX array, dimension (LDQ,N) */ /* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually */ /* the unitary matrix returned by CGGSVP). */ /* On exit, */ /* if JOBQ = 'I', Q contains the unitary matrix Q; */ /* if JOBQ = 'Q', Q contains the product Q1*Q. */ /* If JOBQ = 'N', Q is not referenced. */ /* LDQ (input) INTEGER */ /* The leading dimension of the array Q. LDQ >= max(1,N) if */ /* JOBQ = 'Q'; LDQ >= 1 otherwise. */ /* WORK (workspace) COMPLEX array, dimension (2*N) */ /* NCYCLE (output) INTEGER */ /* The number of cycles required for convergence. */ /* INFO (output) INTEGER */ /* = 0: successful exit */ /* < 0: if INFO = -i, the i-th argument had an illegal value. */ /* = 1: the procedure does not converge after MAXIT cycles. */ /* Internal Parameters */ /* =================== */ /* MAXIT INTEGER */ /* MAXIT specifies the total loops that the iterative procedure */ /* may take. If after MAXIT cycles, the routine fails to */ /* converge, we return INFO = 1. */ /* Further Details */ /* =============== */ /* CTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce */ /* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L */ /* matrix B13 to the form: */ /* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, */ /* where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate */ /* transpose of Z. C1 and S1 are diagonal matrices satisfying */ /* C1**2 + S1**2 = I, */ /* and R1 is an L-by-L nonsingular upper triangular matrix. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /* Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; --alpha; --beta; u_dim1 = *ldu; u_offset = 1 + u_dim1; u -= u_offset; v_dim1 = *ldv; v_offset = 1 + v_dim1; v -= v_offset; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; --work; /* Function Body */ initu = lsame_(jobu, "I"); wantu = initu || lsame_(jobu, "U"); initv = lsame_(jobv, "I"); wantv = initv || lsame_(jobv, "V"); initq = lsame_(jobq, "I"); wantq = initq || lsame_(jobq, "Q"); *info = 0; if (! (initu || wantu || lsame_(jobu, "N"))) { *info = -1; } else if (! (initv || wantv || lsame_(jobv, "N"))) { *info = -2; } else if (! (initq || wantq || lsame_(jobq, "N"))) { *info = -3; } else if (*m < 0) { *info = -4; } else if (*p < 0) { *info = -5; } else if (*n < 0) { *info = -6; } else if (*lda < max(1,*m)) { *info = -10; } else if (*ldb < max(1,*p)) { *info = -12; } else if (*ldu < 1 || wantu && *ldu < *m) { *info = -18; } else if (*ldv < 1 || wantv && *ldv < *p) { *info = -20; } else if (*ldq < 1 || wantq && *ldq < *n) { *info = -22; } if (*info != 0) { i__1 = -(*info); xerbla_("CTGSJA", &i__1); return 0; } /* Initialize U, V and Q, if necessary */ if (initu) { claset_("Full", m, m, &c_b1, &c_b2, &u[u_offset], ldu); } if (initv) { claset_("Full", p, p, &c_b1, &c_b2, &v[v_offset], ldv); } if (initq) { claset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq); } /* Loop until convergence */ upper = FALSE_; for (kcycle = 1; kcycle <= 40; ++kcycle) { upper = ! upper; i__1 = *l - 1; for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *l; for (j = i__ + 1; j <= i__2; ++j) { a1 = 0.f; a2.r = 0.f, a2.i = 0.f; a3 = 0.f; if (*k + i__ <= *m) { i__3 = *k + i__ + (*n - *l + i__) * a_dim1; a1 = a[i__3].r; } if (*k + j <= *m) { i__3 = *k + j + (*n - *l + j) * a_dim1; a3 = a[i__3].r; } i__3 = i__ + (*n - *l + i__) * b_dim1; b1 = b[i__3].r; i__3 = j + (*n - *l + j) * b_dim1; b3 = b[i__3].r; if (upper) { if (*k + i__ <= *m) { i__3 = *k + i__ + (*n - *l + j) * a_dim1; a2.r = a[i__3].r, a2.i = a[i__3].i; } i__3 = i__ + (*n - *l + j) * b_dim1; b2.r = b[i__3].r, b2.i = b[i__3].i; } else { if (*k + j <= *m) { i__3 = *k + j + (*n - *l + i__) * a_dim1; a2.r = a[i__3].r, a2.i = a[i__3].i; } i__3 = j + (*n - *l + i__) * b_dim1; b2.r = b[i__3].r, b2.i = b[i__3].i; } clags2_(&upper, &a1, &a2, &a3, &b1, &b2, &b3, &csu, &snu, & csv, &snv, &csq, &snq); /* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A */ if (*k + j <= *m) { r_cnjg(&q__1, &snu); crot_(l, &a[*k + j + (*n - *l + 1) * a_dim1], lda, &a[*k + i__ + (*n - *l + 1) * a_dim1], lda, &csu, &q__1) ; } /* Update I-th and J-th rows of matrix B: V'*B */ r_cnjg(&q__1, &snv); crot_(l, &b[j + (*n - *l + 1) * b_dim1], ldb, &b[i__ + (*n - * l + 1) * b_dim1], ldb, &csv, &q__1); /* Update (N-L+I)-th and (N-L+J)-th columns of matrices */ /* A and B: A*Q and B*Q */ /* Computing MIN */ i__4 = *k + *l; i__3 = min(i__4,*m); crot_(&i__3, &a[(*n - *l + j) * a_dim1 + 1], &c__1, &a[(*n - * l + i__) * a_dim1 + 1], &c__1, &csq, &snq); crot_(l, &b[(*n - *l + j) * b_dim1 + 1], &c__1, &b[(*n - *l + i__) * b_dim1 + 1], &c__1, &csq, &snq); if (upper) { if (*k + i__ <= *m) { i__3 = *k + i__ + (*n - *l + j) * a_dim1; a[i__3].r = 0.f, a[i__3].i = 0.f; } i__3 = i__ + (*n - *l + j) * b_dim1; b[i__3].r = 0.f, b[i__3].i = 0.f; } else { if (*k + j <= *m) { i__3 = *k + j + (*n - *l + i__) * a_dim1; a[i__3].r = 0.f, a[i__3].i = 0.f; } i__3 = j + (*n - *l + i__) * b_dim1; b[i__3].r = 0.f, b[i__3].i = 0.f; } /* Ensure that the diagonal elements of A and B are real. */ if (*k + i__ <= *m) { i__3 = *k + i__ + (*n - *l + i__) * a_dim1; i__4 = *k + i__ + (*n - *l + i__) * a_dim1; r__1 = a[i__4].r; a[i__3].r = r__1, a[i__3].i = 0.f; } if (*k + j <= *m) { i__3 = *k + j + (*n - *l + j) * a_dim1; i__4 = *k + j + (*n - *l + j) * a_dim1; r__1 = a[i__4].r; a[i__3].r = r__1, a[i__3].i = 0.f; } i__3 = i__ + (*n - *l + i__) * b_dim1; i__4 = i__ + (*n - *l + i__) * b_dim1; r__1 = b[i__4].r; b[i__3].r = r__1, b[i__3].i = 0.f; i__3 = j + (*n - *l + j) * b_dim1; i__4 = j + (*n - *l + j) * b_dim1; r__1 = b[i__4].r; b[i__3].r = r__1, b[i__3].i = 0.f; /* Update unitary matrices U, V, Q, if desired. */ if (wantu && *k + j <= *m) { crot_(m, &u[(*k + j) * u_dim1 + 1], &c__1, &u[(*k + i__) * u_dim1 + 1], &c__1, &csu, &snu); } if (wantv) { crot_(p, &v[j * v_dim1 + 1], &c__1, &v[i__ * v_dim1 + 1], &c__1, &csv, &snv); } if (wantq) { crot_(n, &q[(*n - *l + j) * q_dim1 + 1], &c__1, &q[(*n - * l + i__) * q_dim1 + 1], &c__1, &csq, &snq); } /* L10: */ } /* L20: */ } if (! upper) { /* The matrices A13 and B13 were lower triangular at the start */ /* of the cycle, and are now upper triangular. */ /* Convergence test: test the parallelism of the corresponding */ /* rows of A and B. */ error = 0.f; /* Computing MIN */ i__2 = *l, i__3 = *m - *k; i__1 = min(i__2,i__3); for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *l - i__ + 1; ccopy_(&i__2, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda, & work[1], &c__1); i__2 = *l - i__ + 1; ccopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &work[* l + 1], &c__1); i__2 = *l - i__ + 1; clapll_(&i__2, &work[1], &c__1, &work[*l + 1], &c__1, &ssmin); error = dmax(error,ssmin); /* L30: */ } if (dabs(error) <= dmin(*tola,*tolb)) { goto L50; } } /* End of cycle loop */ /* L40: */ } /* The algorithm has not converged after MAXIT cycles. */ *info = 1; goto L100; L50: /* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged. */ /* Compute the generalized singular value pairs (ALPHA, BETA), and */ /* set the triangular matrix R to array A. */ i__1 = *k; for (i__ = 1; i__ <= i__1; ++i__) { alpha[i__] = 1.f; beta[i__] = 0.f; /* L60: */ } /* Computing MIN */ i__2 = *l, i__3 = *m - *k; i__1 = min(i__2,i__3); for (i__ = 1; i__ <= i__1; ++i__) { i__2 = *k + i__ + (*n - *l + i__) * a_dim1; a1 = a[i__2].r; i__2 = i__ + (*n - *l + i__) * b_dim1; b1 = b[i__2].r; if (a1 != 0.f) { gamma = b1 / a1; if (gamma < 0.f) { i__2 = *l - i__ + 1; csscal_(&i__2, &c_b39, &b[i__ + (*n - *l + i__) * b_dim1], ldb); if (wantv) { csscal_(p, &c_b39, &v[i__ * v_dim1 + 1], &c__1); } } r__1 = dabs(gamma); slartg_(&r__1, &c_b42, &beta[*k + i__], &alpha[*k + i__], &rwk); if (alpha[*k + i__] >= beta[*k + i__]) { i__2 = *l - i__ + 1; r__1 = 1.f / alpha[*k + i__]; csscal_(&i__2, &r__1, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda); } else { i__2 = *l - i__ + 1; r__1 = 1.f / beta[*k + i__]; csscal_(&i__2, &r__1, &b[i__ + (*n - *l + i__) * b_dim1], ldb) ; i__2 = *l - i__ + 1; ccopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda); } } else { alpha[*k + i__] = 0.f; beta[*k + i__] = 1.f; i__2 = *l - i__ + 1; ccopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda); } /* L70: */ } /* Post-assignment */ i__1 = *k + *l; for (i__ = *m + 1; i__ <= i__1; ++i__) { alpha[i__] = 0.f; beta[i__] = 1.f; /* L80: */ } if (*k + *l < *n) { i__1 = *n; for (i__ = *k + *l + 1; i__ <= i__1; ++i__) { alpha[i__] = 0.f; beta[i__] = 0.f; /* L90: */ } } L100: *ncycle = kcycle; return 0; /* End of CTGSJA */ } /* ctgsja_ */