#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zlabrd_(integer *m, integer *n, integer *nb, doublecomplex *a, integer *lda, doublereal *d__, doublereal *e, doublecomplex *tauq, doublecomplex *taup, doublecomplex *x, integer * ldx, doublecomplex *y, integer *ldy) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. This is an auxiliary routine called by ZGEBRD Arguments ========= M (input) INTEGER The number of rows in the matrix A. N (input) INTEGER The number of columns in the matrix A. NB (input) INTEGER The number of leading rows and columns of A to be reduced. A (input/output) COMPLEX*16 array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). D (output) DOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). E (output) DOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. TAUQ (output) COMPLEX*16 array dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. TAUP (output) COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. X (output) COMPLEX*16 array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,M). Y (output) COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. LDY (input) INTEGER The leading dimension of the array Y. LDY >= max(1,N). Further Details =============== The matrices Q and P are represented as products of elementary reflectors: Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are complex scalars, and v and u are complex vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'. The contents of A on exit are illustrated by the following examples with nb = 2: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). ===================================================================== Quick return if possible Parameter adjustments */ /* Table of constant values */ static doublecomplex c_b1 = {0.,0.}; static doublecomplex c_b2 = {1.,0.}; static integer c__1 = 1; /* System generated locals */ integer a_dim1, a_offset, x_dim1, x_offset, y_dim1, y_offset, i__1, i__2, i__3; doublecomplex z__1; /* Local variables */ static integer i__; static doublecomplex alpha; extern /* Subroutine */ int zscal_(integer *, doublecomplex *, doublecomplex *, integer *), zgemv_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zlarfg_(integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *), zlacgv_(integer *, doublecomplex *, integer *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --d__; --e; --tauq; --taup; x_dim1 = *ldx; x_offset = 1 + x_dim1; x -= x_offset; y_dim1 = *ldy; y_offset = 1 + y_dim1; y -= y_offset; /* Function Body */ if (*m <= 0 || *n <= 0) { return 0; } if (*m >= *n) { /* Reduce to upper bidiagonal form */ i__1 = *nb; for (i__ = 1; i__ <= i__1; ++i__) { /* Update A(i:m,i) */ i__2 = i__ - 1; zlacgv_(&i__2, &y[i__ + y_dim1], ldy); i__2 = *m - i__ + 1; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &a[i__ + a_dim1], lda, &y[i__ + y_dim1], ldy, &c_b2, &a[i__ + i__ * a_dim1], & c__1); i__2 = i__ - 1; zlacgv_(&i__2, &y[i__ + y_dim1], ldy); i__2 = *m - i__ + 1; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &x[i__ + x_dim1], ldx, &a[i__ * a_dim1 + 1], &c__1, &c_b2, &a[i__ + i__ * a_dim1], &c__1); /* Generate reflection Q(i) to annihilate A(i+1:m,i) */ i__2 = i__ + i__ * a_dim1; alpha.r = a[i__2].r, alpha.i = a[i__2].i; i__2 = *m - i__ + 1; /* Computing MIN */ i__3 = i__ + 1; zlarfg_(&i__2, &alpha, &a[min(i__3,*m) + i__ * a_dim1], &c__1, & tauq[i__]); i__2 = i__; d__[i__2] = alpha.r; if (i__ < *n) { i__2 = i__ + i__ * a_dim1; a[i__2].r = 1., a[i__2].i = 0.; /* Compute Y(i+1:n,i) */ i__2 = *m - i__ + 1; i__3 = *n - i__; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[i__ + ( i__ + 1) * a_dim1], lda, &a[i__ + i__ * a_dim1], & c__1, &c_b1, &y[i__ + 1 + i__ * y_dim1], &c__1); i__2 = *m - i__ + 1; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[i__ + a_dim1], lda, &a[i__ + i__ * a_dim1], &c__1, &c_b1, & y[i__ * y_dim1 + 1], &c__1); i__2 = *n - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &y[i__ + 1 + y_dim1], ldy, &y[i__ * y_dim1 + 1], &c__1, &c_b2, &y[ i__ + 1 + i__ * y_dim1], &c__1); i__2 = *m - i__ + 1; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &x[i__ + x_dim1], ldx, &a[i__ + i__ * a_dim1], &c__1, &c_b1, & y[i__ * y_dim1 + 1], &c__1); i__2 = i__ - 1; i__3 = *n - i__; z__1.r = -1., z__1.i = -0.; zgemv_("Conjugate transpose", &i__2, &i__3, &z__1, &a[(i__ + 1) * a_dim1 + 1], lda, &y[i__ * y_dim1 + 1], &c__1, & c_b2, &y[i__ + 1 + i__ * y_dim1], &c__1); i__2 = *n - i__; zscal_(&i__2, &tauq[i__], &y[i__ + 1 + i__ * y_dim1], &c__1); /* Update A(i,i+1:n) */ i__2 = *n - i__; zlacgv_(&i__2, &a[i__ + (i__ + 1) * a_dim1], lda); zlacgv_(&i__, &a[i__ + a_dim1], lda); i__2 = *n - i__; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__, &z__1, &y[i__ + 1 + y_dim1], ldy, &a[i__ + a_dim1], lda, &c_b2, &a[i__ + ( i__ + 1) * a_dim1], lda); zlacgv_(&i__, &a[i__ + a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &x[i__ + x_dim1], ldx); i__2 = i__ - 1; i__3 = *n - i__; z__1.r = -1., z__1.i = -0.; zgemv_("Conjugate transpose", &i__2, &i__3, &z__1, &a[(i__ + 1) * a_dim1 + 1], lda, &x[i__ + x_dim1], ldx, &c_b2, & a[i__ + (i__ + 1) * a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &x[i__ + x_dim1], ldx); /* Generate reflection P(i) to annihilate A(i,i+2:n) */ i__2 = i__ + (i__ + 1) * a_dim1; alpha.r = a[i__2].r, alpha.i = a[i__2].i; i__2 = *n - i__; /* Computing MIN */ i__3 = i__ + 2; zlarfg_(&i__2, &alpha, &a[i__ + min(i__3,*n) * a_dim1], lda, & taup[i__]); i__2 = i__; e[i__2] = alpha.r; i__2 = i__ + (i__ + 1) * a_dim1; a[i__2].r = 1., a[i__2].i = 0.; /* Compute X(i+1:m,i) */ i__2 = *m - i__; i__3 = *n - i__; zgemv_("No transpose", &i__2, &i__3, &c_b2, &a[i__ + 1 + (i__ + 1) * a_dim1], lda, &a[i__ + (i__ + 1) * a_dim1], lda, &c_b1, &x[i__ + 1 + i__ * x_dim1], &c__1); i__2 = *n - i__; zgemv_("Conjugate transpose", &i__2, &i__, &c_b2, &y[i__ + 1 + y_dim1], ldy, &a[i__ + (i__ + 1) * a_dim1], lda, & c_b1, &x[i__ * x_dim1 + 1], &c__1); i__2 = *m - i__; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__, &z__1, &a[i__ + 1 + a_dim1], lda, &x[i__ * x_dim1 + 1], &c__1, &c_b2, &x[ i__ + 1 + i__ * x_dim1], &c__1); i__2 = i__ - 1; i__3 = *n - i__; zgemv_("No transpose", &i__2, &i__3, &c_b2, &a[(i__ + 1) * a_dim1 + 1], lda, &a[i__ + (i__ + 1) * a_dim1], lda, & c_b1, &x[i__ * x_dim1 + 1], &c__1); i__2 = *m - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &x[i__ + 1 + x_dim1], ldx, &x[i__ * x_dim1 + 1], &c__1, &c_b2, &x[ i__ + 1 + i__ * x_dim1], &c__1); i__2 = *m - i__; zscal_(&i__2, &taup[i__], &x[i__ + 1 + i__ * x_dim1], &c__1); i__2 = *n - i__; zlacgv_(&i__2, &a[i__ + (i__ + 1) * a_dim1], lda); } /* L10: */ } } else { /* Reduce to lower bidiagonal form */ i__1 = *nb; for (i__ = 1; i__ <= i__1; ++i__) { /* Update A(i,i:n) */ i__2 = *n - i__ + 1; zlacgv_(&i__2, &a[i__ + i__ * a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &a[i__ + a_dim1], lda); i__2 = *n - i__ + 1; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &y[i__ + y_dim1], ldy, &a[i__ + a_dim1], lda, &c_b2, &a[i__ + i__ * a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &a[i__ + a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &x[i__ + x_dim1], ldx); i__2 = i__ - 1; i__3 = *n - i__ + 1; z__1.r = -1., z__1.i = -0.; zgemv_("Conjugate transpose", &i__2, &i__3, &z__1, &a[i__ * a_dim1 + 1], lda, &x[i__ + x_dim1], ldx, &c_b2, &a[i__ + i__ * a_dim1], lda); i__2 = i__ - 1; zlacgv_(&i__2, &x[i__ + x_dim1], ldx); /* Generate reflection P(i) to annihilate A(i,i+1:n) */ i__2 = i__ + i__ * a_dim1; alpha.r = a[i__2].r, alpha.i = a[i__2].i; i__2 = *n - i__ + 1; /* Computing MIN */ i__3 = i__ + 1; zlarfg_(&i__2, &alpha, &a[i__ + min(i__3,*n) * a_dim1], lda, & taup[i__]); i__2 = i__; d__[i__2] = alpha.r; if (i__ < *m) { i__2 = i__ + i__ * a_dim1; a[i__2].r = 1., a[i__2].i = 0.; /* Compute X(i+1:m,i) */ i__2 = *m - i__; i__3 = *n - i__ + 1; zgemv_("No transpose", &i__2, &i__3, &c_b2, &a[i__ + 1 + i__ * a_dim1], lda, &a[i__ + i__ * a_dim1], lda, &c_b1, &x[ i__ + 1 + i__ * x_dim1], &c__1); i__2 = *n - i__ + 1; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &y[i__ + y_dim1], ldy, &a[i__ + i__ * a_dim1], lda, &c_b1, &x[ i__ * x_dim1 + 1], &c__1); i__2 = *m - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &a[i__ + 1 + a_dim1], lda, &x[i__ * x_dim1 + 1], &c__1, &c_b2, &x[ i__ + 1 + i__ * x_dim1], &c__1); i__2 = i__ - 1; i__3 = *n - i__ + 1; zgemv_("No transpose", &i__2, &i__3, &c_b2, &a[i__ * a_dim1 + 1], lda, &a[i__ + i__ * a_dim1], lda, &c_b1, &x[i__ * x_dim1 + 1], &c__1); i__2 = *m - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &x[i__ + 1 + x_dim1], ldx, &x[i__ * x_dim1 + 1], &c__1, &c_b2, &x[ i__ + 1 + i__ * x_dim1], &c__1); i__2 = *m - i__; zscal_(&i__2, &taup[i__], &x[i__ + 1 + i__ * x_dim1], &c__1); i__2 = *n - i__ + 1; zlacgv_(&i__2, &a[i__ + i__ * a_dim1], lda); /* Update A(i+1:m,i) */ i__2 = i__ - 1; zlacgv_(&i__2, &y[i__ + y_dim1], ldy); i__2 = *m - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &a[i__ + 1 + a_dim1], lda, &y[i__ + y_dim1], ldy, &c_b2, &a[i__ + 1 + i__ * a_dim1], &c__1); i__2 = i__ - 1; zlacgv_(&i__2, &y[i__ + y_dim1], ldy); i__2 = *m - i__; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__, &z__1, &x[i__ + 1 + x_dim1], ldx, &a[i__ * a_dim1 + 1], &c__1, &c_b2, &a[ i__ + 1 + i__ * a_dim1], &c__1); /* Generate reflection Q(i) to annihilate A(i+2:m,i) */ i__2 = i__ + 1 + i__ * a_dim1; alpha.r = a[i__2].r, alpha.i = a[i__2].i; i__2 = *m - i__; /* Computing MIN */ i__3 = i__ + 2; zlarfg_(&i__2, &alpha, &a[min(i__3,*m) + i__ * a_dim1], &c__1, &tauq[i__]); i__2 = i__; e[i__2] = alpha.r; i__2 = i__ + 1 + i__ * a_dim1; a[i__2].r = 1., a[i__2].i = 0.; /* Compute Y(i+1:n,i) */ i__2 = *m - i__; i__3 = *n - i__; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[i__ + 1 + (i__ + 1) * a_dim1], lda, &a[i__ + 1 + i__ * a_dim1] , &c__1, &c_b1, &y[i__ + 1 + i__ * y_dim1], &c__1); i__2 = *m - i__; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[i__ + 1 + a_dim1], lda, &a[i__ + 1 + i__ * a_dim1], &c__1, & c_b1, &y[i__ * y_dim1 + 1], &c__1); i__2 = *n - i__; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &y[i__ + 1 + y_dim1], ldy, &y[i__ * y_dim1 + 1], &c__1, &c_b2, &y[ i__ + 1 + i__ * y_dim1], &c__1); i__2 = *m - i__; zgemv_("Conjugate transpose", &i__2, &i__, &c_b2, &x[i__ + 1 + x_dim1], ldx, &a[i__ + 1 + i__ * a_dim1], &c__1, & c_b1, &y[i__ * y_dim1 + 1], &c__1); i__2 = *n - i__; z__1.r = -1., z__1.i = -0.; zgemv_("Conjugate transpose", &i__, &i__2, &z__1, &a[(i__ + 1) * a_dim1 + 1], lda, &y[i__ * y_dim1 + 1], &c__1, & c_b2, &y[i__ + 1 + i__ * y_dim1], &c__1); i__2 = *n - i__; zscal_(&i__2, &tauq[i__], &y[i__ + 1 + i__ * y_dim1], &c__1); } else { i__2 = *n - i__ + 1; zlacgv_(&i__2, &a[i__ + i__ * a_dim1], lda); } /* L20: */ } } return 0; /* End of ZLABRD */ } /* zlabrd_ */