/* -- translated by f2c (version 20100827).
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
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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*/
#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 ===========
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http://www.netlib.org/lapack/explore-html/
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> \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_ */