/* Subroutine */ int zgerqf_(integer *m, integer *n, doublecomplex *a, integer *lda, doublecomplex *tau, doublecomplex *work, integer *lwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, i__1, i__2, i__3, i__4; /* Local variables */ integer i__, k, ib, nb, ki, kk, mu, nu, nx, iws, nbmin, iinfo; extern /* Subroutine */ int zgerq2_(integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), xerbla_( char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); extern /* Subroutine */ int zlarfb_(char *, char *, char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); integer ldwork; extern /* Subroutine */ int zlarft_(char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *); integer lwkopt; logical lquery; /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* ZGERQF computes an RQ factorization of a complex M-by-N matrix A: */ /* A = R * Q. */ /* Arguments */ /* ========= */ /* M (input) INTEGER */ /* The number of rows of the matrix A. M >= 0. */ /* N (input) INTEGER */ /* The number of columns of the matrix A. N >= 0. */ /* A (input/output) COMPLEX*16 array, dimension (LDA,N) */ /* On entry, the M-by-N matrix A. */ /* On exit, */ /* if m <= n, the upper triangle of the subarray */ /* A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R; */ /* if m >= n, the elements on and above the (m-n)-th subdiagonal */ /* contain the M-by-N upper trapezoidal matrix R; */ /* the remaining elements, with the array TAU, represent the */ /* unitary matrix Q as a product of min(m,n) elementary */ /* reflectors (see Further Details). */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,M). */ /* TAU (output) COMPLEX*16 array, dimension (min(M,N)) */ /* The scalar factors of the elementary reflectors (see Further */ /* Details). */ /* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) */ /* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ /* LWORK (input) INTEGER */ /* The dimension of the array WORK. LWORK >= max(1,M). */ /* For optimum performance LWORK >= M*NB, where NB is */ /* the optimal blocksize. */ /* If LWORK = -1, then a workspace query is assumed; the routine */ /* only calculates the optimal size of the WORK array, returns */ /* this value as the first entry of the WORK array, and no error */ /* message related to LWORK is issued by XERBLA. */ /* INFO (output) INTEGER */ /* = 0: successful exit */ /* < 0: if INFO = -i, the i-th argument had an illegal value */ /* Further Details */ /* =============== */ /* The matrix Q is represented as a product of elementary reflectors */ /* Q = H(1)' H(2)' . . . H(k)', where k = min(m,n). */ /* Each H(i) has the form */ /* H(i) = I - tau * v * v' */ /* where tau is a complex scalar, and v is a complex vector with */ /* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on */ /* exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i). */ /* ===================================================================== */ /* .. Local Scalars .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input arguments */ /* Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --tau; --work; /* Function Body */ *info = 0; lquery = *lwork == -1; if (*m < 0) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < max(1,*m)) { *info = -4; } if (*info == 0) { k = min(*m,*n); if (k == 0) { lwkopt = 1; } else { nb = ilaenv_(&c__1, "ZGERQF", " ", m, n, &c_n1, &c_n1); lwkopt = *m * nb; } work[1].r = (doublereal) lwkopt, work[1].i = 0.; if (*lwork < max(1,*m) && ! lquery) { *info = -7; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZGERQF", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (k == 0) { return 0; } nbmin = 2; nx = 1; iws = *m; if (nb > 1 && nb < k) { /* Determine when to cross over from blocked to unblocked code. */ /* Computing MAX */ i__1 = 0, i__2 = ilaenv_(&c__3, "ZGERQF", " ", m, n, &c_n1, &c_n1); nx = max(i__1,i__2); if (nx < k) { /* Determine if workspace is large enough for blocked code. */ ldwork = *m; iws = ldwork * nb; if (*lwork < iws) { /* Not enough workspace to use optimal NB: reduce NB and */ /* determine the minimum value of NB. */ nb = *lwork / ldwork; /* Computing MAX */ i__1 = 2, i__2 = ilaenv_(&c__2, "ZGERQF", " ", m, n, &c_n1, & c_n1); nbmin = max(i__1,i__2); } } } if (nb >= nbmin && nb < k && nx < k) { /* Use blocked code initially. */ /* The last kk rows are handled by the block method. */ ki = (k - nx - 1) / nb * nb; /* Computing MIN */ i__1 = k, i__2 = ki + nb; kk = min(i__1,i__2); i__1 = k - kk + 1; i__2 = -nb; for (i__ = k - kk + ki + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) { /* Computing MIN */ i__3 = k - i__ + 1; ib = min(i__3,nb); /* Compute the RQ factorization of the current block */ /* A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) */ i__3 = *n - k + i__ + ib - 1; zgerq2_(&ib, &i__3, &a[*m - k + i__ + a_dim1], lda, &tau[i__], & work[1], &iinfo); if (*m - k + i__ > 1) { /* Form the triangular factor of the block reflector */ /* H = H(i+ib-1) . . . H(i+1) H(i) */ i__3 = *n - k + i__ + ib - 1; zlarft_("Backward", "Rowwise", &i__3, &ib, &a[*m - k + i__ + a_dim1], lda, &tau[i__], &work[1], &ldwork); /* Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right */ i__3 = *m - k + i__ - 1; i__4 = *n - k + i__ + ib - 1; zlarfb_("Right", "No transpose", "Backward", "Rowwise", &i__3, &i__4, &ib, &a[*m - k + i__ + a_dim1], lda, &work[1], &ldwork, &a[a_offset], lda, &work[ib + 1], &ldwork); } /* L10: */ } mu = *m - k + i__ + nb - 1; nu = *n - k + i__ + nb - 1; } else { mu = *m; nu = *n; } /* Use unblocked code to factor the last or only block */ if (mu > 0 && nu > 0) { zgerq2_(&mu, &nu, &a[a_offset], lda, &tau[1], &work[1], &iinfo); } work[1].r = (doublereal) iws, work[1].i = 0.; return 0; /* End of ZGERQF */ } /* zgerqf_ */
/* Subroutine */ int zggsvp_(char *jobu, char *jobv, char *jobq, integer *m, integer *p, integer *n, doublecomplex *a, integer *lda, doublecomplex *b, integer *ldb, doublereal *tola, doublereal *tolb, integer *k, integer *l, doublecomplex *u, integer *ldu, doublecomplex *v, integer *ldv, doublecomplex *q, integer *ldq, integer *iwork, doublereal * rwork, doublecomplex *tau, doublecomplex *work, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 Purpose ======= ZGGSVP computes unitary matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; L ( 0 0 A23 ) M-K-L ( 0 0 0 ) N-K-L K L = K ( 0 A12 A13 ) if M-K-L < 0; M-K ( 0 0 A23 ) N-K-L K L V'*B*Q = 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. K+L = the effective numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the conjugate transpose of Z. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine ZGGSVD. Arguments ========= JOBU (input) CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV (input) CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ (input) CHARACTER*1 = 'Q': Unitary matrix Q is computed; = '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. A (input/output) COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). B (input/output) COMPLEX*16 array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains the triangular matrix described in the Purpose section. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,P). TOLA (input) DOUBLE PRECISION TOLB (input) DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. K (output) INTEGER L (output) INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose section. K + L = effective numerical rank of (A',B')'. U (output) COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the unitary matrix 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 (output) COMPLEX*16 array, dimension (LDV,M) If JOBV = 'V', V contains the unitary matrix 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 (output) COMPLEX*16 array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the unitary matrix 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. IWORK (workspace) INTEGER array, dimension (N) RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) TAU (workspace) COMPLEX*16 array, dimension (N) WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. Further Details =============== The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization with column pivoting to detect the effective numerical rank of the a matrix. It may be replaced by a better rank determination strategy. ===================================================================== Test the input parameters Parameter adjustments */ /* Table of constant values */ static doublecomplex c_b1 = {0.,0.}; static doublecomplex c_b2 = {1.,0.}; /* 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; doublereal d__1, d__2; /* Builtin functions */ double d_imag(doublecomplex *); /* Local variables */ static integer i__, j; extern logical lsame_(char *, char *); static logical wantq, wantu, wantv; extern /* Subroutine */ int zgeqr2_(integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zgerq2_( integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zung2r_(integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zunm2r_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *), zunmr2_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *), xerbla_( char *, integer *), zgeqpf_(integer *, integer *, doublecomplex *, integer *, integer *, doublecomplex *, doublecomplex *, doublereal *, integer *), zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static logical forwrd; extern /* Subroutine */ int zlaset_(char *, integer *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *), zlapmt_(logical *, integer *, integer *, doublecomplex *, integer *, integer *); #define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1 #define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)] #define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1 #define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)] #define u_subscr(a_1,a_2) (a_2)*u_dim1 + a_1 #define u_ref(a_1,a_2) u[u_subscr(a_1,a_2)] #define v_subscr(a_1,a_2) (a_2)*v_dim1 + a_1 #define v_ref(a_1,a_2) v[v_subscr(a_1,a_2)] a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; u_dim1 = *ldu; u_offset = 1 + u_dim1 * 1; u -= u_offset; v_dim1 = *ldv; v_offset = 1 + v_dim1 * 1; v -= v_offset; q_dim1 = *ldq; q_offset = 1 + q_dim1 * 1; q -= q_offset; --iwork; --rwork; --tau; --work; /* Function Body */ wantu = lsame_(jobu, "U"); wantv = lsame_(jobv, "V"); wantq = lsame_(jobq, "Q"); forwrd = TRUE_; *info = 0; if (! (wantu || lsame_(jobu, "N"))) { *info = -1; } else if (! (wantv || lsame_(jobv, "N"))) { *info = -2; } else if (! (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 = -8; } else if (*ldb < max(1,*p)) { *info = -10; } else if (*ldu < 1 || wantu && *ldu < *m) { *info = -16; } else if (*ldv < 1 || wantv && *ldv < *p) { *info = -18; } else if (*ldq < 1 || wantq && *ldq < *n) { *info = -20; } if (*info != 0) { i__1 = -(*info); xerbla_("ZGGSVP", &i__1); return 0; } /* QR with column pivoting of B: B*P = V*( S11 S12 ) ( 0 0 ) */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { iwork[i__] = 0; /* L10: */ } zgeqpf_(p, n, &b[b_offset], ldb, &iwork[1], &tau[1], &work[1], &rwork[1], info); /* Update A := A*P */ zlapmt_(&forwrd, m, n, &a[a_offset], lda, &iwork[1]); /* Determine the effective rank of matrix B. */ *l = 0; i__1 = min(*p,*n); for (i__ = 1; i__ <= i__1; ++i__) { i__2 = b_subscr(i__, i__); if ((d__1 = b[i__2].r, abs(d__1)) + (d__2 = d_imag(&b_ref(i__, i__)), abs(d__2)) > *tolb) { ++(*l); } /* L20: */ } if (wantv) { /* Copy the details of V, and form V. */ zlaset_("Full", p, p, &c_b1, &c_b1, &v[v_offset], ldv); if (*p > 1) { i__1 = *p - 1; zlacpy_("Lower", &i__1, n, &b_ref(2, 1), ldb, &v_ref(2, 1), ldv); } i__1 = min(*p,*n); zung2r_(p, p, &i__1, &v[v_offset], ldv, &tau[1], &work[1], info); } /* Clean up B */ i__1 = *l - 1; for (j = 1; j <= i__1; ++j) { i__2 = *l; for (i__ = j + 1; i__ <= i__2; ++i__) { i__3 = b_subscr(i__, j); b[i__3].r = 0., b[i__3].i = 0.; /* L30: */ } /* L40: */ } if (*p > *l) { i__1 = *p - *l; zlaset_("Full", &i__1, n, &c_b1, &c_b1, &b_ref(*l + 1, 1), ldb); } if (wantq) { /* Set Q = I and Update Q := Q*P */ zlaset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq); zlapmt_(&forwrd, n, n, &q[q_offset], ldq, &iwork[1]); } if (*p >= *l && *n != *l) { /* RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z */ zgerq2_(l, n, &b[b_offset], ldb, &tau[1], &work[1], info); /* Update A := A*Z' */ zunmr2_("Right", "Conjugate transpose", m, n, l, &b[b_offset], ldb, & tau[1], &a[a_offset], lda, &work[1], info); if (wantq) { /* Update Q := Q*Z' */ zunmr2_("Right", "Conjugate transpose", n, n, l, &b[b_offset], ldb, &tau[1], &q[q_offset], ldq, &work[1], info); } /* Clean up B */ i__1 = *n - *l; zlaset_("Full", l, &i__1, &c_b1, &c_b1, &b[b_offset], ldb); i__1 = *n; for (j = *n - *l + 1; j <= i__1; ++j) { i__2 = *l; for (i__ = j - *n + *l + 1; i__ <= i__2; ++i__) { i__3 = b_subscr(i__, j); b[i__3].r = 0., b[i__3].i = 0.; /* L50: */ } /* L60: */ } } /* Let N-L L A = ( A11 A12 ) M, then the following does the complete QR decomposition of A11: A11 = U*( 0 T12 )*P1' ( 0 0 ) */ i__1 = *n - *l; for (i__ = 1; i__ <= i__1; ++i__) { iwork[i__] = 0; /* L70: */ } i__1 = *n - *l; zgeqpf_(m, &i__1, &a[a_offset], lda, &iwork[1], &tau[1], &work[1], &rwork[ 1], info); /* Determine the effective rank of A11 */ *k = 0; /* Computing MIN */ i__2 = *m, i__3 = *n - *l; i__1 = min(i__2,i__3); for (i__ = 1; i__ <= i__1; ++i__) { i__2 = a_subscr(i__, i__); if ((d__1 = a[i__2].r, abs(d__1)) + (d__2 = d_imag(&a_ref(i__, i__)), abs(d__2)) > *tola) { ++(*k); } /* L80: */ } /* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) Computing MIN */ i__2 = *m, i__3 = *n - *l; i__1 = min(i__2,i__3); zunm2r_("Left", "Conjugate transpose", m, l, &i__1, &a[a_offset], lda, & tau[1], &a_ref(1, *n - *l + 1), lda, &work[1], info); if (wantu) { /* Copy the details of U, and form U */ zlaset_("Full", m, m, &c_b1, &c_b1, &u[u_offset], ldu); if (*m > 1) { i__1 = *m - 1; i__2 = *n - *l; zlacpy_("Lower", &i__1, &i__2, &a_ref(2, 1), lda, &u_ref(2, 1), ldu); } /* Computing MIN */ i__2 = *m, i__3 = *n - *l; i__1 = min(i__2,i__3); zung2r_(m, m, &i__1, &u[u_offset], ldu, &tau[1], &work[1], info); } if (wantq) { /* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 */ i__1 = *n - *l; zlapmt_(&forwrd, n, &i__1, &q[q_offset], ldq, &iwork[1]); } /* Clean up A: set the strictly lower triangular part of A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. */ i__1 = *k - 1; for (j = 1; j <= i__1; ++j) { i__2 = *k; for (i__ = j + 1; i__ <= i__2; ++i__) { i__3 = a_subscr(i__, j); a[i__3].r = 0., a[i__3].i = 0.; /* L90: */ } /* L100: */ } if (*m > *k) { i__1 = *m - *k; i__2 = *n - *l; zlaset_("Full", &i__1, &i__2, &c_b1, &c_b1, &a_ref(*k + 1, 1), lda); } if (*n - *l > *k) { /* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 */ i__1 = *n - *l; zgerq2_(k, &i__1, &a[a_offset], lda, &tau[1], &work[1], info); if (wantq) { /* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' */ i__1 = *n - *l; zunmr2_("Right", "Conjugate transpose", n, &i__1, k, &a[a_offset], lda, &tau[1], &q[q_offset], ldq, &work[1], info); } /* Clean up A */ i__1 = *n - *l - *k; zlaset_("Full", k, &i__1, &c_b1, &c_b1, &a[a_offset], lda); i__1 = *n - *l; for (j = *n - *l - *k + 1; j <= i__1; ++j) { i__2 = *k; for (i__ = j - *n + *l + *k + 1; i__ <= i__2; ++i__) { i__3 = a_subscr(i__, j); a[i__3].r = 0., a[i__3].i = 0.; /* L110: */ } /* L120: */ } } if (*m > *k) { /* QR factorization of A( K+1:M,N-L+1:N ) */ i__1 = *m - *k; zgeqr2_(&i__1, l, &a_ref(*k + 1, *n - *l + 1), lda, &tau[1], &work[1], info); if (wantu) { /* Update U(:,K+1:M) := U(:,K+1:M)*U1 */ i__1 = *m - *k; /* Computing MIN */ i__3 = *m - *k; i__2 = min(i__3,*l); zunm2r_("Right", "No transpose", m, &i__1, &i__2, &a_ref(*k + 1, * n - *l + 1), lda, &tau[1], &u_ref(1, *k + 1), ldu, &work[ 1], info); } /* Clean up */ i__1 = *n; for (j = *n - *l + 1; j <= i__1; ++j) { i__2 = *m; for (i__ = j - *n + *k + *l + 1; i__ <= i__2; ++i__) { i__3 = a_subscr(i__, j); a[i__3].r = 0., a[i__3].i = 0.; /* L130: */ } /* L140: */ } } return 0; /* End of ZGGSVP */ } /* zggsvp_ */