#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dlahrd_(integer *n, integer *k, integer *nb, doublereal * a, integer *lda, doublereal *tau, doublereal *t, integer *ldt, doublereal *y, integer *ldy) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal 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 DLAHR2 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. T (output) DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T. LDT (input) INTEGER The leading dimension of the array T. LDT >= NB. Y (output) DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y. LDY (input) INTEGER The leading dimension of the array Y. LDY >= 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 real scalar, and v is a real 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 doublereal c_b4 = -1.; static doublereal c_b5 = 1.; static integer c__1 = 1; static doublereal c_b38 = 0.; /* System generated locals */ integer a_dim1, a_offset, t_dim1, t_offset, y_dim1, y_offset, i__1, i__2, i__3; doublereal d__1; /* Local variables */ static integer i__; static doublereal ei; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *), dgemv_(char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *), dtrmv_(char *, char *, char *, integer *, doublereal *, integer *, doublereal *, integer *), dlarfg_(integer *, doublereal *, doublereal *, integer *, doublereal *); --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; dgemv_("No transpose", n, &i__2, &c_b4, &y[y_offset], ldy, &a[*k + i__ - 1 + a_dim1], lda, &c_b5, &a[i__ * a_dim1 + 1], & c__1); /* 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; dcopy_(&i__2, &a[*k + 1 + i__ * a_dim1], &c__1, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; dtrmv_("Lower", "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; dgemv_("Transpose", &i__2, &i__3, &c_b5, &a[*k + i__ + a_dim1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b5, &t[*nb * t_dim1 + 1], &c__1); /* w := T'*w */ i__2 = i__ - 1; dtrmv_("Upper", "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; dgemv_("No transpose", &i__2, &i__3, &c_b4, &a[*k + i__ + a_dim1], lda, &t[*nb * t_dim1 + 1], &c__1, &c_b5, &a[*k + i__ + i__ * a_dim1], &c__1); /* b1 := b1 - V1*w */ i__2 = i__ - 1; dtrmv_("Lower", "No transpose", "Unit", &i__2, &a[*k + 1 + a_dim1] , lda, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; daxpy_(&i__2, &c_b4, &t[*nb * t_dim1 + 1], &c__1, &a[*k + 1 + i__ * a_dim1], &c__1); a[*k + i__ - 1 + (i__ - 1) * a_dim1] = ei; } /* Generate the elementary reflector H(i) to annihilate A(k+i+1:n,i) */ i__2 = *n - *k - i__ + 1; /* Computing MIN */ i__3 = *k + i__ + 1; dlarfg_(&i__2, &a[*k + i__ + i__ * a_dim1], &a[min(i__3,*n) + i__ * a_dim1], &c__1, &tau[i__]); ei = a[*k + i__ + i__ * a_dim1]; a[*k + i__ + i__ * a_dim1] = 1.; /* Compute Y(1:n,i) */ i__2 = *n - *k - i__ + 1; dgemv_("No transpose", n, &i__2, &c_b5, &a[(i__ + 1) * a_dim1 + 1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b38, &y[i__ * y_dim1 + 1], &c__1); i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; dgemv_("Transpose", &i__2, &i__3, &c_b5, &a[*k + i__ + a_dim1], lda, & a[*k + i__ + i__ * a_dim1], &c__1, &c_b38, &t[i__ * t_dim1 + 1], &c__1); i__2 = i__ - 1; dgemv_("No transpose", n, &i__2, &c_b4, &y[y_offset], ldy, &t[i__ * t_dim1 + 1], &c__1, &c_b5, &y[i__ * y_dim1 + 1], &c__1); dscal_(n, &tau[i__], &y[i__ * y_dim1 + 1], &c__1); /* Compute T(1:i,i) */ i__2 = i__ - 1; d__1 = -tau[i__]; dscal_(&i__2, &d__1, &t[i__ * t_dim1 + 1], &c__1); i__2 = i__ - 1; dtrmv_("Upper", "No transpose", "Non-unit", &i__2, &t[t_offset], ldt, &t[i__ * t_dim1 + 1], &c__1) ; t[i__ + i__ * t_dim1] = tau[i__]; /* L10: */ } a[*k + *nb + *nb * a_dim1] = ei; return 0; /* End of DLAHRD */ } /* dlahrd_ */