/* -- 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
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_b5 = -1.;
static doublereal c_b6 = 1.;
static integer c__1 = 1;
static doublereal c_b16 = 0.;
/* > \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiago
nal form by an orthogonal similarity transformation.
=========== DOCUMENTATION ===========
Online html documentation available at
http://www.netlib.org/lapack/explore-html/
> \htmlonly
> Download DLATRD + dependencies
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.
f">
> [TGZ]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.
f">
> [ZIP]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.
f">
> [TXT]</a>
> \endhtmlonly
Definition:
===========
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
> \par Purpose:
=============
>
> \verbatim
>
> DLATRD reduces NB rows and columns of a real symmetric matrix A to
> symmetric tridiagonal form by an orthogonal similarity
> transformation Q**T * A * Q, and returns the matrices V and W which are
> needed to apply the transformation to the unreduced part of A.
>
> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
> matrix, of which the upper triangle is supplied;
> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
> matrix, of which the lower triangle is supplied.
>
> This is an auxiliary routine called by DSYTRD.
> \endverbatim
Arguments:
==========
> \param[in] UPLO
> \verbatim
> UPLO is CHARACTER*1
> Specifies whether the upper or lower triangular part of the
> symmetric matrix A is stored:
> = 'U': Upper triangular
> = 'L': Lower triangular
> \endverbatim
>
> \param[in] N
> \verbatim
> N is INTEGER
> The order of the matrix A.
> \endverbatim
>
> \param[in] NB
> \verbatim
> NB is INTEGER
> The number of rows and columns to be reduced.
> \endverbatim
>
> \param[in,out] A
> \verbatim
> A is DOUBLE PRECISION array, dimension (LDA,N)
> On entry, the symmetric matrix A. If UPLO = 'U', the leading
> n-by-n upper triangular part of A contains the upper
> triangular part of the matrix A, and the strictly lower
> triangular part of A is not referenced. If UPLO = 'L', the
> leading n-by-n lower triangular part of A contains the lower
> triangular part of the matrix A, and the strictly upper
> triangular part of A is not referenced.
> On exit:
> if UPLO = 'U', the last NB columns have been reduced to
> tridiagonal form, with the diagonal elements overwriting
> the diagonal elements of A; the elements above the diagonal
> with the array TAU, represent the orthogonal matrix Q as a
> product of elementary reflectors;
> if UPLO = 'L', the first NB columns have been reduced to
> tridiagonal form, with the diagonal elements overwriting
> the diagonal elements of A; the elements below the diagonal
> with the array TAU, represent the orthogonal matrix Q as a
> product of elementary reflectors.
> See Further Details.
> \endverbatim
>
> \param[in] LDA
> \verbatim
> LDA is INTEGER
> The leading dimension of the array A. LDA >= (1,N).
> \endverbatim
>
> \param[out] E
> \verbatim
> E is DOUBLE PRECISION array, dimension (N-1)
> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
> elements of the last NB columns of the reduced matrix;
> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
> the first NB columns of the reduced matrix.
> \endverbatim
>
> \param[out] TAU
> \verbatim
> TAU is DOUBLE PRECISION array, dimension (N-1)
> The scalar factors of the elementary reflectors, stored in
> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
> See Further Details.
> \endverbatim
>
> \param[out] W
> \verbatim
> W is DOUBLE PRECISION array, dimension (LDW,NB)
> The n-by-nb matrix W required to update the unreduced part
> of A.
> \endverbatim
>
> \param[in] LDW
> \verbatim
> LDW is INTEGER
> The leading dimension of the array W. LDW >= max(1,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
>
> If UPLO = 'U', the matrix Q is represented as a product of elementary
> reflectors
>
> Q = H(n) H(n-1) . . . H(n-nb+1).
>
> 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
> and tau in TAU(i-1).
>
> If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
> and tau in TAU(i).
>
> The elements of the vectors v together form the n-by-nb matrix V
> which is needed, with W, to apply the transformation to the unreduced
> part of the matrix, using a symmetric rank-2k update of the form:
> A := A - V*W**T - W*V**T.
>
> The contents of A on exit are illustrated by the following examples
> with n = 5 and nb = 2:
>
> if UPLO = 'U': if UPLO = 'L':
>
> ( a a a v4 v5 ) ( d )
> ( a a v4 v5 ) ( 1 d )
> ( a 1 v5 ) ( v1 1 a )
> ( d 1 ) ( v1 v2 a a )
> ( d ) ( v1 v2 a a a )
>
> where d denotes a diagonal element of the reduced matrix, a denotes
> an element of the original matrix that is unchanged, and vi denotes
> an element of the vector defining H(i).
> \endverbatim
>
=====================================================================
Subroutine */ int igraphdlatrd_(char *uplo, integer *n, integer *nb, doublereal *
a, integer *lda, doublereal *e, doublereal *tau, doublereal *w,
integer *ldw)
{
/* System generated locals */
integer a_dim1, a_offset, w_dim1, w_offset, i__1, i__2, i__3;
/* Local variables */
integer i__, iw;
extern doublereal igraphddot_(integer *, doublereal *, integer *, doublereal *,
integer *);
doublereal alpha;
extern /* Subroutine */ int igraphdscal_(integer *, doublereal *, doublereal *,
integer *);
extern logical igraphlsame_(char *, char *);
extern /* Subroutine */ int igraphdgemv_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *), igraphdaxpy_(integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *),
igraphdsymv_(char *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *), igraphdlarfg_(integer *, doublereal *, doublereal *, integer *,
doublereal *);
/* -- 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 */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--e;
--tau;
w_dim1 = *ldw;
w_offset = 1 + w_dim1;
w -= w_offset;
/* Function Body */
if (*n <= 0) {
return 0;
}
if (igraphlsame_(uplo, "U")) {
/* Reduce last NB columns of upper triangle */
i__1 = *n - *nb + 1;
for (i__ = *n; i__ >= i__1; --i__) {
iw = i__ - *n + *nb;
if (i__ < *n) {
/* Update A(1:i,i) */
i__2 = *n - i__;
igraphdgemv_("No transpose", &i__, &i__2, &c_b5, &a[(i__ + 1) *
a_dim1 + 1], lda, &w[i__ + (iw + 1) * w_dim1], ldw, &
c_b6, &a[i__ * a_dim1 + 1], &c__1);
i__2 = *n - i__;
igraphdgemv_("No transpose", &i__, &i__2, &c_b5, &w[(iw + 1) *
w_dim1 + 1], ldw, &a[i__ + (i__ + 1) * a_dim1], lda, &
c_b6, &a[i__ * a_dim1 + 1], &c__1);
}
if (i__ > 1) {
/* Generate elementary reflector H(i) to annihilate
A(1:i-2,i) */
i__2 = i__ - 1;
igraphdlarfg_(&i__2, &a[i__ - 1 + i__ * a_dim1], &a[i__ * a_dim1 +
1], &c__1, &tau[i__ - 1]);
e[i__ - 1] = a[i__ - 1 + i__ * a_dim1];
a[i__ - 1 + i__ * a_dim1] = 1.;
/* Compute W(1:i-1,i) */
i__2 = i__ - 1;
igraphdsymv_("Upper", &i__2, &c_b6, &a[a_offset], lda, &a[i__ *
a_dim1 + 1], &c__1, &c_b16, &w[iw * w_dim1 + 1], &
c__1);
if (i__ < *n) {
i__2 = i__ - 1;
i__3 = *n - i__;
igraphdgemv_("Transpose", &i__2, &i__3, &c_b6, &w[(iw + 1) *
w_dim1 + 1], ldw, &a[i__ * a_dim1 + 1], &c__1, &
c_b16, &w[i__ + 1 + iw * w_dim1], &c__1);
i__2 = i__ - 1;
i__3 = *n - i__;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &a[(i__ + 1) *
a_dim1 + 1], lda, &w[i__ + 1 + iw * w_dim1], &
c__1, &c_b6, &w[iw * w_dim1 + 1], &c__1);
i__2 = i__ - 1;
i__3 = *n - i__;
igraphdgemv_("Transpose", &i__2, &i__3, &c_b6, &a[(i__ + 1) *
a_dim1 + 1], lda, &a[i__ * a_dim1 + 1], &c__1, &
c_b16, &w[i__ + 1 + iw * w_dim1], &c__1);
i__2 = i__ - 1;
i__3 = *n - i__;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &w[(iw + 1) *
w_dim1 + 1], ldw, &w[i__ + 1 + iw * w_dim1], &
c__1, &c_b6, &w[iw * w_dim1 + 1], &c__1);
}
i__2 = i__ - 1;
igraphdscal_(&i__2, &tau[i__ - 1], &w[iw * w_dim1 + 1], &c__1);
i__2 = i__ - 1;
alpha = tau[i__ - 1] * -.5 * igraphddot_(&i__2, &w[iw * w_dim1 + 1],
&c__1, &a[i__ * a_dim1 + 1], &c__1);
i__2 = i__ - 1;
igraphdaxpy_(&i__2, &alpha, &a[i__ * a_dim1 + 1], &c__1, &w[iw *
w_dim1 + 1], &c__1);
}
/* L10: */
}
} else {
/* Reduce first NB columns of lower triangle */
i__1 = *nb;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Update A(i:n,i) */
i__2 = *n - i__ + 1;
i__3 = i__ - 1;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &a[i__ + a_dim1], lda,
&w[i__ + w_dim1], ldw, &c_b6, &a[i__ + i__ * a_dim1], &
c__1);
i__2 = *n - i__ + 1;
i__3 = i__ - 1;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &w[i__ + w_dim1], ldw,
&a[i__ + a_dim1], lda, &c_b6, &a[i__ + i__ * a_dim1], &
c__1);
if (i__ < *n) {
/* Generate elementary reflector H(i) to annihilate
A(i+2:n,i) */
i__2 = *n - i__;
/* Computing MIN */
i__3 = i__ + 2;
igraphdlarfg_(&i__2, &a[i__ + 1 + i__ * a_dim1], &a[min(i__3,*n) +
i__ * a_dim1], &c__1, &tau[i__]);
e[i__] = a[i__ + 1 + i__ * a_dim1];
a[i__ + 1 + i__ * a_dim1] = 1.;
/* Compute W(i+1:n,i) */
i__2 = *n - i__;
igraphdsymv_("Lower", &i__2, &c_b6, &a[i__ + 1 + (i__ + 1) * a_dim1]
, lda, &a[i__ + 1 + i__ * a_dim1], &c__1, &c_b16, &w[
i__ + 1 + i__ * w_dim1], &c__1);
i__2 = *n - i__;
i__3 = i__ - 1;
igraphdgemv_("Transpose", &i__2, &i__3, &c_b6, &w[i__ + 1 + w_dim1],
ldw, &a[i__ + 1 + i__ * a_dim1], &c__1, &c_b16, &w[
i__ * w_dim1 + 1], &c__1);
i__2 = *n - i__;
i__3 = i__ - 1;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &a[i__ + 1 +
a_dim1], lda, &w[i__ * w_dim1 + 1], &c__1, &c_b6, &w[
i__ + 1 + i__ * w_dim1], &c__1);
i__2 = *n - i__;
i__3 = i__ - 1;
igraphdgemv_("Transpose", &i__2, &i__3, &c_b6, &a[i__ + 1 + a_dim1],
lda, &a[i__ + 1 + i__ * a_dim1], &c__1, &c_b16, &w[
i__ * w_dim1 + 1], &c__1);
i__2 = *n - i__;
i__3 = i__ - 1;
igraphdgemv_("No transpose", &i__2, &i__3, &c_b5, &w[i__ + 1 +
w_dim1], ldw, &w[i__ * w_dim1 + 1], &c__1, &c_b6, &w[
i__ + 1 + i__ * w_dim1], &c__1);
i__2 = *n - i__;
igraphdscal_(&i__2, &tau[i__], &w[i__ + 1 + i__ * w_dim1], &c__1);
i__2 = *n - i__;
alpha = tau[i__] * -.5 * igraphddot_(&i__2, &w[i__ + 1 + i__ *
w_dim1], &c__1, &a[i__ + 1 + i__ * a_dim1], &c__1);
i__2 = *n - i__;
igraphdaxpy_(&i__2, &alpha, &a[i__ + 1 + i__ * a_dim1], &c__1, &w[
i__ + 1 + i__ * w_dim1], &c__1);
}
/* L20: */
}
}
return 0;
/* End of DLATRD */
} /* igraphdlatrd_ */