#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zlahrd_(integer *n, integer *k, integer *nb, doublecomplex *a, integer *lda, doublecomplex *tau, doublecomplex *t, integer *ldt, doublecomplex *y, integer *ldy) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by a unitary similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. This is an OBSOLETE auxiliary routine. This routine will be 'deprecated' in a future release. Please use the new routine ZLAHR2 instead. Arguments ========= N (input) INTEGER The order of the matrix A. K (input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. NB (input) INTEGER The number of columns to be reduced. A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). TAU (output) COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. T (output) COMPLEX*16 array, dimension (LDT,NB) The upper triangular matrix T. LDT (input) INTEGER The leading dimension of the array T. LDT >= NB. Y (output) COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y. LDY (input) INTEGER The leading dimension of the array Y. LDY >= max(1,N). Further Details =============== The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). 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(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i). The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V') * (A - Y*V'). The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a h a a a ) ( a h a a a ) ( a h a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(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, t_dim1, t_offset, y_dim1, y_offset, i__1, i__2, i__3; doublecomplex z__1; /* Local variables */ static integer i__; static doublecomplex ei; extern /* Subroutine */ int zscal_(integer *, doublecomplex *, doublecomplex *, integer *), zgemv_(char *, integer *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *), zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), zaxpy_(integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *), ztrmv_(char *, char *, char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *), zlarfg_(integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *), zlacgv_(integer *, doublecomplex *, integer *); --tau; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; t_dim1 = *ldt; t_offset = 1 + t_dim1; t -= t_offset; y_dim1 = *ldy; y_offset = 1 + y_dim1; y -= y_offset; /* Function Body */ if (*n <= 1) { return 0; } i__1 = *nb; for (i__ = 1; i__ <= i__1; ++i__) { if (i__ > 1) { /* Update A(1:n,i) Compute i-th column of A - Y * V' */ i__2 = i__ - 1; zlacgv_(&i__2, &a[*k + i__ - 1 + a_dim1], lda); i__2 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", n, &i__2, &z__1, &y[y_offset], ldy, &a[*k + i__ - 1 + a_dim1], lda, &c_b2, &a[i__ * a_dim1 + 1], & c__1); i__2 = i__ - 1; zlacgv_(&i__2, &a[*k + i__ - 1 + a_dim1], lda); /* Apply I - V * T' * V' to this column (call it b) from the left, using the last column of T as workspace Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) ( V2 ) ( b2 ) where V1 is unit lower triangular w := V1' * b1 */ i__2 = i__ - 1; zcopy_(&i__2, &a[*k + 1 + i__ * a_dim1], &c__1, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; ztrmv_("Lower", "Conjugate transpose", "Unit", &i__2, &a[*k + 1 + a_dim1], lda, &t[*nb * t_dim1 + 1], &c__1); /* w := w + V2'*b2 */ i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[*k + i__ + a_dim1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b2, & t[*nb * t_dim1 + 1], &c__1); /* w := T'*w */ i__2 = i__ - 1; ztrmv_("Upper", "Conjugate transpose", "Non-unit", &i__2, &t[ t_offset], ldt, &t[*nb * t_dim1 + 1], &c__1); /* b2 := b2 - V2*w */ i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", &i__2, &i__3, &z__1, &a[*k + i__ + a_dim1], lda, &t[*nb * t_dim1 + 1], &c__1, &c_b2, &a[*k + i__ + i__ * a_dim1], &c__1); /* b1 := b1 - V1*w */ i__2 = i__ - 1; ztrmv_("Lower", "No transpose", "Unit", &i__2, &a[*k + 1 + a_dim1] , lda, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; z__1.r = -1., z__1.i = -0.; zaxpy_(&i__2, &z__1, &t[*nb * t_dim1 + 1], &c__1, &a[*k + 1 + i__ * a_dim1], &c__1); i__2 = *k + i__ - 1 + (i__ - 1) * a_dim1; a[i__2].r = ei.r, a[i__2].i = ei.i; } /* Generate the elementary reflector H(i) to annihilate A(k+i+1:n,i) */ i__2 = *k + i__ + i__ * a_dim1; ei.r = a[i__2].r, ei.i = a[i__2].i; i__2 = *n - *k - i__ + 1; /* Computing MIN */ i__3 = *k + i__ + 1; zlarfg_(&i__2, &ei, &a[min(i__3,*n) + i__ * a_dim1], &c__1, &tau[i__]) ; i__2 = *k + i__ + i__ * a_dim1; a[i__2].r = 1., a[i__2].i = 0.; /* Compute Y(1:n,i) */ i__2 = *n - *k - i__ + 1; zgemv_("No transpose", n, &i__2, &c_b2, &a[(i__ + 1) * a_dim1 + 1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b1, &y[i__ * y_dim1 + 1], &c__1); i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; zgemv_("Conjugate transpose", &i__2, &i__3, &c_b2, &a[*k + i__ + a_dim1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b1, &t[ i__ * t_dim1 + 1], &c__1); i__2 = i__ - 1; z__1.r = -1., z__1.i = -0.; zgemv_("No transpose", n, &i__2, &z__1, &y[y_offset], ldy, &t[i__ * t_dim1 + 1], &c__1, &c_b2, &y[i__ * y_dim1 + 1], &c__1); zscal_(n, &tau[i__], &y[i__ * y_dim1 + 1], &c__1); /* Compute T(1:i,i) */ i__2 = i__ - 1; i__3 = i__; z__1.r = -tau[i__3].r, z__1.i = -tau[i__3].i; zscal_(&i__2, &z__1, &t[i__ * t_dim1 + 1], &c__1); i__2 = i__ - 1; ztrmv_("Upper", "No transpose", "Non-unit", &i__2, &t[t_offset], ldt, &t[i__ * t_dim1 + 1], &c__1) ; i__2 = i__ + i__ * t_dim1; i__3 = i__; t[i__2].r = tau[i__3].r, t[i__2].i = tau[i__3].i; /* L10: */ } i__1 = *k + *nb + *nb * a_dim1; a[i__1].r = ei.r, a[i__1].i = ei.i; return 0; /* End of ZLAHRD */ } /* zlahrd_ */