/* Subroutine */ int dsytrd_(char *uplo, integer *n, doublereal *a, integer *
lda, doublereal *d__, doublereal *e, doublereal *tau, doublereal *
work, integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
/* Local variables */
integer i__, j, nb, kk, nx, iws;
extern logical lsame_(char *, char *);
integer nbmin, iinfo;
logical upper;
extern /* Subroutine */ int dsytd2_(char *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, integer *), dsyr2k_(char *, char *, integer *, integer *, doublereal
*, doublereal *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *), dlatrd_(char *,
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, doublereal *, integer *), xerbla_(char *,
integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
integer ldwork, lwkopt;
logical lquery;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSYTRD reduces a real symmetric matrix A to real symmetric */
/* tridiagonal form T by an orthogonal similarity transformation: */
/* Q**T * A * Q = T. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input/output) 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 diagonal and first superdiagonal */
/* of A are overwritten by the corresponding elements of the */
/* tridiagonal matrix T, and the elements above the first */
/* superdiagonal, with the array TAU, represent the orthogonal */
/* matrix Q as a product of elementary reflectors; if UPLO */
/* = 'L', the diagonal and first subdiagonal of A are over- */
/* written by the corresponding elements of the tridiagonal */
/* matrix T, and the elements below the first subdiagonal, with */
/* the array TAU, represent the orthogonal matrix Q as a product */
/* of elementary reflectors. See Further Details. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* D (output) DOUBLE PRECISION array, dimension (N) */
/* The diagonal elements of the tridiagonal matrix T: */
/* D(i) = A(i,i). */
/* E (output) DOUBLE PRECISION array, dimension (N-1) */
/* The off-diagonal elements of the tridiagonal matrix T: */
/* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. */
/* TAU (output) DOUBLE PRECISION array, dimension (N-1) */
/* The scalar factors of the elementary reflectors (see Further */
/* Details). */
/* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= 1. */
/* For optimum performance LWORK >= N*NB, where NB is the */
/* optimal blocksize. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* If UPLO = 'U', the matrix Q is represented as a product of elementary */
/* reflectors */
/* Q = H(n-1) . . . H(2) H(1). */
/* 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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in */
/* A(1:i-1,i+1), and tau in TAU(i). */
/* If UPLO = 'L', the matrix Q is represented as a product of elementary */
/* reflectors */
/* Q = H(1) H(2) . . . H(n-1). */
/* 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) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), */
/* and tau in TAU(i). */
/* The contents of A on exit are illustrated by the following examples */
/* with n = 5: */
/* if UPLO = 'U': if UPLO = 'L': */
/* ( d e v2 v3 v4 ) ( d ) */
/* ( d e v3 v4 ) ( e d ) */
/* ( d e v4 ) ( v1 e d ) */
/* ( d e ) ( v1 v2 e d ) */
/* ( d ) ( v1 v2 v3 e d ) */
/* where d and e denote diagonal and off-diagonal elements of T, and vi */
/* denotes an element of the vector defining H(i). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--d__;
--e;
--tau;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
lquery = *lwork == -1;
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*lwork < 1 && ! lquery) {
*info = -9;
}
if (*info == 0) {
/* Determine the block size. */
nb = ilaenv_(&c__1, "DSYTRD", uplo, n, &c_n1, &c_n1, &c_n1);
lwkopt = *n * nb;
work[1] = (doublereal) lwkopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSYTRD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
work[1] = 1.;
return 0;
}
nx = *n;
iws = 1;
if (nb > 1 && nb < *n) {
/* Determine when to cross over from blocked to unblocked code */
/* (last block is always handled by unblocked code). */
/* Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__3, "DSYTRD", uplo, n, &c_n1, &c_n1, &
c_n1);
nx = max(i__1,i__2);
if (nx < *n) {
/* Determine if workspace is large enough for blocked code. */
ldwork = *n;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: determine the */
/* minimum value of NB, and reduce NB or force use of */
/* unblocked code by setting NX = N. */
/* Computing MAX */
i__1 = *lwork / ldwork;
nb = max(i__1,1);
nbmin = ilaenv_(&c__2, "DSYTRD", uplo, n, &c_n1, &c_n1, &c_n1);
if (nb < nbmin) {
nx = *n;
}
}
} else {
nx = *n;
}
} else {
nb = 1;
}
if (upper) {
/* Reduce the upper triangle of A. */
/* Columns 1:kk are handled by the unblocked method. */
kk = *n - (*n - nx + nb - 1) / nb * nb;
i__1 = kk + 1;
i__2 = -nb;
for (i__ = *n - nb + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ +=
i__2) {
/* Reduce columns i:i+nb-1 to tridiagonal form and form the */
/* matrix W which is needed to update the unreduced part of */
/* the matrix */
i__3 = i__ + nb - 1;
dlatrd_(uplo, &i__3, &nb, &a[a_offset], lda, &e[1], &tau[1], &
work[1], &ldwork);
/* Update the unreduced submatrix A(1:i-1,1:i-1), using an */
/* update of the form: A := A - V*W' - W*V' */
i__3 = i__ - 1;
dsyr2k_(uplo, "No transpose", &i__3, &nb, &c_b22, &a[i__ * a_dim1
+ 1], lda, &work[1], &ldwork, &c_b23, &a[a_offset], lda);
/* Copy superdiagonal elements back into A, and diagonal */
/* elements into D */
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
a[j - 1 + j * a_dim1] = e[j - 1];
d__[j] = a[j + j * a_dim1];
/* L10: */
}
/* L20: */
}
/* Use unblocked code to reduce the last or only block */
dsytd2_(uplo, &kk, &a[a_offset], lda, &d__[1], &e[1], &tau[1], &iinfo);
} else {
/* Reduce the lower triangle of A */
i__2 = *n - nx;
i__1 = nb;
for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ += i__1) {
/* Reduce columns i:i+nb-1 to tridiagonal form and form the */
/* matrix W which is needed to update the unreduced part of */
/* the matrix */
i__3 = *n - i__ + 1;
dlatrd_(uplo, &i__3, &nb, &a[i__ + i__ * a_dim1], lda, &e[i__], &
tau[i__], &work[1], &ldwork);
/* Update the unreduced submatrix A(i+ib:n,i+ib:n), using */
/* an update of the form: A := A - V*W' - W*V' */
i__3 = *n - i__ - nb + 1;
dsyr2k_(uplo, "No transpose", &i__3, &nb, &c_b22, &a[i__ + nb +
i__ * a_dim1], lda, &work[nb + 1], &ldwork, &c_b23, &a[
i__ + nb + (i__ + nb) * a_dim1], lda);
/* Copy subdiagonal elements back into A, and diagonal */
/* elements into D */
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
a[j + 1 + j * a_dim1] = e[j];
d__[j] = a[j + j * a_dim1];
/* L30: */
}
/* L40: */
}
/* Use unblocked code to reduce the last or only block */
i__1 = *n - i__ + 1;
dsytd2_(uplo, &i__1, &a[i__ + i__ * a_dim1], lda, &d__[i__], &e[i__],
&tau[i__], &iinfo);
}
work[1] = (doublereal) lwkopt;
return 0;
/* End of DSYTRD */
} /* dsytrd_ */
/* Subroutine */ int dsytrd_(char *uplo, integer *n, doublereal *a, integer *
lda, doublereal *d__, doublereal *e, doublereal *tau, doublereal *
work, integer *lwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
DSYTRD reduces a real symmetric matrix A to real symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) 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 diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E (output) DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1.
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
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) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static integer c__3 = 3;
static integer c__2 = 2;
static doublereal c_b22 = -1.;
static doublereal c_b23 = 1.;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
/* Local variables */
static integer i__, j;
extern logical lsame_(char *, char *);
static integer nbmin, iinfo;
static logical upper;
extern /* Subroutine */ int dsytd2_(char *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, integer *), dsyr2k_(char *, char *, integer *, integer *, doublereal
*, doublereal *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *);
static integer nb, kk, nx;
extern /* Subroutine */ int dlatrd_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ldwork, lwkopt;
static logical lquery;
static integer iws;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--d__;
--e;
--tau;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
lquery = *lwork == -1;
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*lwork < 1 && ! lquery) {
*info = -9;
}
if (*info == 0) {
/* Determine the block size. */
nb = ilaenv_(&c__1, "DSYTRD", uplo, n, &c_n1, &c_n1, &c_n1, (ftnlen)6,
(ftnlen)1);
lwkopt = *n * nb;
work[1] = (doublereal) lwkopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSYTRD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
work[1] = 1.;
return 0;
}
nx = *n;
iws = 1;
if (nb > 1 && nb < *n) {
/* Determine when to cross over from blocked to unblocked code
(last block is always handled by unblocked code).
Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__3, "DSYTRD", uplo, n, &c_n1, &c_n1, &
c_n1, (ftnlen)6, (ftnlen)1);
nx = max(i__1,i__2);
if (nx < *n) {
/* Determine if workspace is large enough for blocked code. */
ldwork = *n;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: determine the
minimum value of NB, and reduce NB or force use of
unblocked code by setting NX = N.
Computing MAX */
i__1 = *lwork / ldwork;
nb = max(i__1,1);
nbmin = ilaenv_(&c__2, "DSYTRD", uplo, n, &c_n1, &c_n1, &c_n1,
(ftnlen)6, (ftnlen)1);
if (nb < nbmin) {
nx = *n;
}
}
} else {
nx = *n;
}
} else {
nb = 1;
}
if (upper) {
/* Reduce the upper triangle of A.
Columns 1:kk are handled by the unblocked method. */
kk = *n - (*n - nx + nb - 1) / nb * nb;
i__1 = kk + 1;
i__2 = -nb;
for (i__ = *n - nb + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ +=
i__2) {
/* Reduce columns i:i+nb-1 to tridiagonal form and form the
matrix W which is needed to update the unreduced part of
the matrix */
i__3 = i__ + nb - 1;
dlatrd_(uplo, &i__3, &nb, &a[a_offset], lda, &e[1], &tau[1], &
work[1], &ldwork);
/* Update the unreduced submatrix A(1:i-1,1:i-1), using an
update of the form: A := A - V*W' - W*V' */
i__3 = i__ - 1;
dsyr2k_(uplo, "No transpose", &i__3, &nb, &c_b22, &a_ref(1, i__),
lda, &work[1], &ldwork, &c_b23, &a[a_offset], lda);
/* Copy superdiagonal elements back into A, and diagonal
elements into D */
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
a_ref(j - 1, j) = e[j - 1];
d__[j] = a_ref(j, j);
/* L10: */
}
/* L20: */
}
/* Use unblocked code to reduce the last or only block */
dsytd2_(uplo, &kk, &a[a_offset], lda, &d__[1], &e[1], &tau[1], &iinfo);
} else {
/* Reduce the lower triangle of A */
i__2 = *n - nx;
i__1 = nb;
for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ += i__1) {
/* Reduce columns i:i+nb-1 to tridiagonal form and form the
matrix W which is needed to update the unreduced part of
the matrix */
i__3 = *n - i__ + 1;
dlatrd_(uplo, &i__3, &nb, &a_ref(i__, i__), lda, &e[i__], &tau[
i__], &work[1], &ldwork);
/* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
an update of the form: A := A - V*W' - W*V' */
i__3 = *n - i__ - nb + 1;
dsyr2k_(uplo, "No transpose", &i__3, &nb, &c_b22, &a_ref(i__ + nb,
i__), lda, &work[nb + 1], &ldwork, &c_b23, &a_ref(i__ +
nb, i__ + nb), lda);
/* Copy subdiagonal elements back into A, and diagonal
elements into D */
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
a_ref(j + 1, j) = e[j];
d__[j] = a_ref(j, j);
/* L30: */
}
/* L40: */
}
/* Use unblocked code to reduce the last or only block */
i__1 = *n - i__ + 1;
dsytd2_(uplo, &i__1, &a_ref(i__, i__), lda, &d__[i__], &e[i__], &tau[
i__], &iinfo);
}
work[1] = (doublereal) lwkopt;
return 0;
/* End of DSYTRD */
} /* dsytrd_ */
