/* -- translated by f2c (version 20191129). 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" /* Table of constant values */ static doublereal c_b4 = -1.; static doublereal c_b5 = 1.; static integer c__1 = 1; static doublereal c_b38 = 0.; /* > \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. =========== DOCUMENTATION =========== Online html documentation available at http://www.netlib.org/lapack/explore-html/ > \htmlonly > Download DLAHR2 + dependencies > > [TGZ] > > [ZIP] > > [TXT] > \endhtmlonly Definition: =========== SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) INTEGER K, LDA, LDT, LDY, N, NB DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), $ Y( LDY, NB ) > \par Purpose: ============= > > \verbatim > > DLAHR2 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**T * A * Q. The routine returns the matrices V and T which determine > Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T. > > This is an auxiliary routine called by DGEHRD. > \endverbatim Arguments: ========== > \param[in] N > \verbatim > N is INTEGER > The order of the matrix A. > \endverbatim > > \param[in] K > \verbatim > K is INTEGER > The offset for the reduction. Elements below the k-th > subdiagonal in the first NB columns are reduced to zero. > K < N. > \endverbatim > > \param[in] NB > \verbatim > NB is INTEGER > The number of columns to be reduced. > \endverbatim > > \param[in,out] A > \verbatim > A is 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. > \endverbatim > > \param[in] LDA > \verbatim > LDA is INTEGER > The leading dimension of the array A. LDA >= max(1,N). > \endverbatim > > \param[out] TAU > \verbatim > TAU is DOUBLE PRECISION array, dimension (NB) > The scalar factors of the elementary reflectors. See Further > Details. > \endverbatim > > \param[out] T > \verbatim > T is DOUBLE PRECISION array, dimension (LDT,NB) > The upper triangular matrix T. > \endverbatim > > \param[in] LDT > \verbatim > LDT is INTEGER > The leading dimension of the array T. LDT >= NB. > \endverbatim > > \param[out] Y > \verbatim > Y is DOUBLE PRECISION array, dimension (LDY,NB) > The n-by-nb matrix Y. > \endverbatim > > \param[in] LDY > \verbatim > LDY is INTEGER > The leading dimension of the array Y. LDY >= N. > \endverbatim Authors: ======== > \author Univ. of Tennessee > \author Univ. of California Berkeley > \author Univ. of Colorado Denver > \author NAG Ltd. > \date September 2012 > \ingroup doubleOTHERauxiliary > \par Further Details: ===================== > > \verbatim > > 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**T > > 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**T) * (A - Y*V**T). > > The contents of A on exit are illustrated by the following example > with n = 7, k = 3 and nb = 2: > > ( a a a a a ) > ( a a a a a ) > ( a a 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). > > This subroutine is a slight modification of LAPACK-3.0's DLAHRD > incorporating improvements proposed by Quintana-Orti and Van de > Gejin. Note that the entries of A(1:K,2:NB) differ from those > returned by the original LAPACK-3.0's DLAHRD routine. (This > subroutine is not backward compatible with LAPACK-3.0's DLAHRD.) > \endverbatim > \par References: ================ > > Gregorio Quintana-Orti and Robert van de Geijn, "Improving the > performance of reduction to Hessenberg form," ACM Transactions on > Mathematical Software, 32(2):180-194, June 2006. > ===================================================================== Subroutine */ int igraphdlahr2_(integer *n, integer *k, integer *nb, doublereal * a, integer *lda, doublereal *tau, doublereal *t, integer *ldt, doublereal *y, integer *ldy) { /* 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 */ integer i__; doublereal ei; extern /* Subroutine */ int igraphdscal_(integer *, doublereal *, doublereal *, integer *), igraphdgemm_(char *, char *, integer *, integer *, integer * , doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), igraphdgemv_( char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), igraphdcopy_(integer *, doublereal *, integer *, doublereal *, integer *), igraphdtrmm_(char *, char *, char *, char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *), igraphdaxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *), igraphdtrmv_(char *, char *, char *, integer *, doublereal *, integer *, doublereal *, integer *), igraphdlarfg_( integer *, doublereal *, doublereal *, integer *, doublereal *), igraphdlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); /* -- LAPACK auxiliary routine (version 3.4.2) -- -- LAPACK is a software package provided by Univ. of Tennessee, -- -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- September 2012 ===================================================================== Quick return if possible Parameter adjustments */ --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(K+1:N,I) Update I-th column of A - Y * V**T */ i__2 = *n - *k; i__3 = i__ - 1; igraphdgemv_("NO TRANSPOSE", &i__2, &i__3, &c_b4, &y[*k + 1 + y_dim1], ldy, &a[*k + i__ - 1 + a_dim1], lda, &c_b5, &a[*k + 1 + i__ * a_dim1], &c__1); /* Apply I - V * T**T * V**T 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**T * b1 */ i__2 = i__ - 1; igraphdcopy_(&i__2, &a[*k + 1 + i__ * a_dim1], &c__1, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; igraphdtrmv_("Lower", "Transpose", "UNIT", &i__2, &a[*k + 1 + a_dim1], lda, &t[*nb * t_dim1 + 1], &c__1); /* w := w + V2**T * b2 */ i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; igraphdgemv_("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**T * w */ i__2 = i__ - 1; igraphdtrmv_("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; igraphdgemv_("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; igraphdtrmv_("Lower", "NO TRANSPOSE", "UNIT", &i__2, &a[*k + 1 + a_dim1] , lda, &t[*nb * t_dim1 + 1], &c__1); i__2 = i__ - 1; igraphdaxpy_(&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; igraphdlarfg_(&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(K+1:N,I) */ i__2 = *n - *k; i__3 = *n - *k - i__ + 1; igraphdgemv_("NO TRANSPOSE", &i__2, &i__3, &c_b5, &a[*k + 1 + (i__ + 1) * a_dim1], lda, &a[*k + i__ + i__ * a_dim1], &c__1, &c_b38, &y[* k + 1 + i__ * y_dim1], &c__1); i__2 = *n - *k - i__ + 1; i__3 = i__ - 1; igraphdgemv_("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 = *n - *k; i__3 = i__ - 1; igraphdgemv_("NO TRANSPOSE", &i__2, &i__3, &c_b4, &y[*k + 1 + y_dim1], ldy, &t[i__ * t_dim1 + 1], &c__1, &c_b5, &y[*k + 1 + i__ * y_dim1], &c__1); i__2 = *n - *k; igraphdscal_(&i__2, &tau[i__], &y[*k + 1 + i__ * y_dim1], &c__1); /* Compute T(1:I,I) */ i__2 = i__ - 1; d__1 = -tau[i__]; igraphdscal_(&i__2, &d__1, &t[i__ * t_dim1 + 1], &c__1); i__2 = i__ - 1; igraphdtrmv_("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; /* Compute Y(1:K,1:NB) */ igraphdlacpy_("ALL", k, nb, &a[(a_dim1 << 1) + 1], lda, &y[y_offset], ldy); igraphdtrmm_("RIGHT", "Lower", "NO TRANSPOSE", "UNIT", k, nb, &c_b5, &a[*k + 1 + a_dim1], lda, &y[y_offset], ldy); if (*n > *k + *nb) { i__1 = *n - *k - *nb; igraphdgemm_("NO TRANSPOSE", "NO TRANSPOSE", k, nb, &i__1, &c_b5, &a[(*nb + 2) * a_dim1 + 1], lda, &a[*k + 1 + *nb + a_dim1], lda, &c_b5, &y[y_offset], ldy); } igraphdtrmm_("RIGHT", "Upper", "NO TRANSPOSE", "NON-UNIT", k, nb, &c_b5, &t[ t_offset], ldt, &y[y_offset], ldy); return 0; /* End of DLAHR2 */ } /* igraphdlahr2_ */