#include "blaswrap.h"
#include "f2c.h"
/* Subroutine */ int dsbtrd_(char *vect, char *uplo, integer *n, integer *kd,
doublereal *ab, integer *ldab, doublereal *d__, doublereal *e,
doublereal *q, integer *ldq, doublereal *work, 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
=======
DSBTRD reduces a real symmetric band matrix A to symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.
Arguments
=========
VECT (input) CHARACTER*1
= 'N': do not form Q;
= 'V': form Q;
= 'U': update a matrix X, by forming X*Q.
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.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E (output) DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if VECT = 'U', then Q must contain an N-by-N
matrix X; if VECT = 'N' or 'V', then Q need not be set.
On exit:
if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
if VECT = 'U', Q contains the product X*Q;
if VECT = 'N', the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q.
LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
WORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
Modified by Linda Kaufman, Bell Labs.
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static doublereal c_b9 = 0.;
static doublereal c_b10 = 1.;
static integer c__1 = 1;
/* System generated locals */
integer ab_dim1, ab_offset, q_dim1, q_offset, i__1, i__2, i__3, i__4,
i__5;
/* Local variables */
static integer inca, jend, lend, jinc, incx, last;
static doublereal temp;
extern /* Subroutine */ int drot_(integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *);
static integer j1end, j1inc, i__, j, k, l, iqend;
extern logical lsame_(char *, char *);
static logical initq, wantq, upper;
static integer i2, j1, j2;
extern /* Subroutine */ int dlar2v_(integer *, doublereal *, doublereal *,
doublereal *, integer *, doublereal *, doublereal *, integer *);
static integer nq, nr, iqaend;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *),
dlartg_(doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *), xerbla_(char *, integer *), dlargv_(
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *), dlartv_(integer *, doublereal *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *);
static integer kd1, ibl, iqb, kdn, jin, nrt, kdm1;
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1 * 1;
ab -= ab_offset;
--d__;
--e;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--work;
/* Function Body */
initq = lsame_(vect, "V");
wantq = initq || lsame_(vect, "U");
upper = lsame_(uplo, "U");
kd1 = *kd + 1;
kdm1 = *kd - 1;
incx = *ldab - 1;
iqend = 1;
*info = 0;
if (! wantq && ! lsame_(vect, "N")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*kd < 0) {
*info = -4;
} else if (*ldab < kd1) {
*info = -6;
} else if (*ldq < max(1,*n) && wantq) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSBTRD", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Initialize Q to the unit matrix, if needed */
if (initq) {
dlaset_("Full", n, n, &c_b9, &c_b10, &q[q_offset], ldq);
}
/* Wherever possible, plane rotations are generated and applied in
vector operations of length NR over the index set J1:J2:KD1.
The cosines and sines of the plane rotations are stored in the
arrays D and WORK. */
inca = kd1 * *ldab;
/* Computing MIN */
i__1 = *n - 1;
kdn = min(i__1,*kd);
if (upper) {
if (*kd > 1) {
/* Reduce to tridiagonal form, working with upper triangle */
nr = 0;
j1 = kdn + 2;
j2 = 1;
i__1 = *n - 2;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th row of matrix to tridiagonal form */
for (k = kdn + 1; k >= 2; --k) {
j1 += kdn;
j2 += kdn;
if (nr > 0) {
/* generate plane rotations to annihilate nonzero
elements which have been created outside the band */
dlargv_(&nr, &ab_ref(1, j1 - 1), &inca, &work[j1], &
kd1, &d__[j1], &kd1);
/* apply rotations from the right
Dependent on the the number of diagonals either
DLARTV or DROT is used */
if (nr >= (*kd << 1) - 1) {
i__2 = *kd - 1;
for (l = 1; l <= i__2; ++l) {
dlartv_(&nr, &ab_ref(l + 1, j1 - 1), &inca, &
ab_ref(l, j1), &inca, &d__[j1], &work[
j1], &kd1);
/* L10: */
}
} else {
jend = j1 + (nr - 1) * kd1;
i__2 = jend;
i__3 = kd1;
for (jinc = j1; i__3 < 0 ? jinc >= i__2 : jinc <=
i__2; jinc += i__3) {
drot_(&kdm1, &ab_ref(2, jinc - 1), &c__1, &
ab_ref(1, jinc), &c__1, &d__[jinc], &
work[jinc]);
/* L20: */
}
}
}
if (k > 2) {
if (k <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i,i+k-1)
within the band */
dlartg_(&ab_ref(*kd - k + 3, i__ + k - 2), &
ab_ref(*kd - k + 2, i__ + k - 1), &d__[
i__ + k - 1], &work[i__ + k - 1], &temp);
ab_ref(*kd - k + 3, i__ + k - 2) = temp;
/* apply rotation from the right */
i__3 = k - 3;
drot_(&i__3, &ab_ref(*kd - k + 4, i__ + k - 2), &
c__1, &ab_ref(*kd - k + 3, i__ + k - 1), &
c__1, &d__[i__ + k - 1], &work[i__ + k -
1]);
}
++nr;
j1 = j1 - kdn - 1;
}
/* apply plane rotations from both sides to diagonal
blocks */
if (nr > 0) {
dlar2v_(&nr, &ab_ref(kd1, j1 - 1), &ab_ref(kd1, j1), &
ab_ref(*kd, j1), &inca, &d__[j1], &work[j1], &
kd1);
}
/* apply plane rotations from the left */
if (nr > 0) {
if ((*kd << 1) - 1 < nr) {
/* Dependent on the the number of diagonals either
DLARTV or DROT is used */
i__3 = *kd - 1;
for (l = 1; l <= i__3; ++l) {
if (j2 + l > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
dlartv_(&nrt, &ab_ref(*kd - l, j1 + l), &
inca, &ab_ref(*kd - l + 1, j1 + l)
, &inca, &d__[j1], &work[j1], &
kd1);
}
/* L30: */
}
} else {
j1end = j1 + kd1 * (nr - 2);
if (j1end >= j1) {
i__3 = j1end;
i__2 = kd1;
for (jin = j1; i__2 < 0 ? jin >= i__3 : jin <=
i__3; jin += i__2) {
i__4 = *kd - 1;
drot_(&i__4, &ab_ref(*kd - 1, jin + 1), &
incx, &ab_ref(*kd, jin + 1), &
incx, &d__[jin], &work[jin]);
/* L40: */
}
}
/* Computing MIN */
i__2 = kdm1, i__3 = *n - j2;
lend = min(i__2,i__3);
last = j1end + kd1;
if (lend > 0) {
drot_(&lend, &ab_ref(*kd - 1, last + 1), &
incx, &ab_ref(*kd, last + 1), &incx, &
d__[last], &work[last]);
}
}
}
if (wantq) {
/* accumulate product of plane rotations in Q */
if (initq) {
/* take advantage of the fact that Q was
initially the Identity matrix */
iqend = max(iqend,j2);
/* Computing MAX */
i__2 = 0, i__3 = k - 3;
i2 = max(i__2,i__3);
iqaend = i__ * *kd + 1;
if (k == 2) {
iqaend += *kd;
}
iqaend = min(iqaend,iqend);
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j
+= i__3) {
ibl = i__ - i2 / kdm1;
++i2;
/* Computing MAX */
i__4 = 1, i__5 = j - ibl;
iqb = max(i__4,i__5);
nq = iqaend + 1 - iqb;
/* Computing MIN */
i__4 = iqaend + *kd;
iqaend = min(i__4,iqend);
drot_(&nq, &q_ref(iqb, j - 1), &c__1, &q_ref(
iqb, j), &c__1, &d__[j], &work[j]);
/* L50: */
}
} else {
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j
+= i__2) {
drot_(n, &q_ref(1, j - 1), &c__1, &q_ref(1, j)
, &c__1, &d__[j], &work[j]);
/* L60: */
}
}
}
if (j2 + kdn > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 = j2 - kdn - 1;
}
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3)
{
/* create nonzero element a(j-1,j+kd) outside the band
and store it in WORK */
work[j + *kd] = work[j] * ab_ref(1, j + *kd);
ab_ref(1, j + *kd) = d__[j] * ab_ref(1, j + *kd);
/* L70: */
}
/* L80: */
}
/* L90: */
}
}
if (*kd > 0) {
/* copy off-diagonal elements to E */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab_ref(*kd, i__ + 1);
/* L100: */
}
} else {
/* set E to zero if original matrix was diagonal */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.;
/* L110: */
}
}
/* copy diagonal elements to D */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab_ref(kd1, i__);
/* L120: */
}
} else {
if (*kd > 1) {
/* Reduce to tridiagonal form, working with lower triangle */
nr = 0;
j1 = kdn + 2;
j2 = 1;
i__1 = *n - 2;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th column of matrix to tridiagonal form */
for (k = kdn + 1; k >= 2; --k) {
j1 += kdn;
j2 += kdn;
if (nr > 0) {
/* generate plane rotations to annihilate nonzero
elements which have been created outside the band */
dlargv_(&nr, &ab_ref(kd1, j1 - kd1), &inca, &work[j1],
&kd1, &d__[j1], &kd1);
/* apply plane rotations from one side
Dependent on the the number of diagonals either
DLARTV or DROT is used */
if (nr > (*kd << 1) - 1) {
i__3 = *kd - 1;
for (l = 1; l <= i__3; ++l) {
dlartv_(&nr, &ab_ref(kd1 - l, j1 - kd1 + l), &
inca, &ab_ref(kd1 - l + 1, j1 - kd1 +
l), &inca, &d__[j1], &work[j1], &kd1);
/* L130: */
}
} else {
jend = j1 + kd1 * (nr - 1);
i__3 = jend;
i__2 = kd1;
for (jinc = j1; i__2 < 0 ? jinc >= i__3 : jinc <=
i__3; jinc += i__2) {
drot_(&kdm1, &ab_ref(*kd, jinc - *kd), &incx,
&ab_ref(kd1, jinc - *kd), &incx, &d__[
jinc], &work[jinc]);
/* L140: */
}
}
}
if (k > 2) {
if (k <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i+k-1,i)
within the band */
dlartg_(&ab_ref(k - 1, i__), &ab_ref(k, i__), &
d__[i__ + k - 1], &work[i__ + k - 1], &
temp);
ab_ref(k - 1, i__) = temp;
/* apply rotation from the left */
i__2 = k - 3;
i__3 = *ldab - 1;
i__4 = *ldab - 1;
drot_(&i__2, &ab_ref(k - 2, i__ + 1), &i__3, &
ab_ref(k - 1, i__ + 1), &i__4, &d__[i__ +
k - 1], &work[i__ + k - 1]);
}
++nr;
j1 = j1 - kdn - 1;
}
/* apply plane rotations from both sides to diagonal
blocks */
if (nr > 0) {
dlar2v_(&nr, &ab_ref(1, j1 - 1), &ab_ref(1, j1), &
ab_ref(2, j1 - 1), &inca, &d__[j1], &work[j1],
&kd1);
}
/* apply plane rotations from the right
Dependent on the the number of diagonals either
DLARTV or DROT is used */
if (nr > 0) {
if (nr > (*kd << 1) - 1) {
i__2 = *kd - 1;
for (l = 1; l <= i__2; ++l) {
if (j2 + l > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
dlartv_(&nrt, &ab_ref(l + 2, j1 - 1), &
inca, &ab_ref(l + 1, j1), &inca, &
d__[j1], &work[j1], &kd1);
}
/* L150: */
}
} else {
j1end = j1 + kd1 * (nr - 2);
if (j1end >= j1) {
i__2 = j1end;
i__3 = kd1;
for (j1inc = j1; i__3 < 0 ? j1inc >= i__2 :
j1inc <= i__2; j1inc += i__3) {
drot_(&kdm1, &ab_ref(3, j1inc - 1), &c__1,
&ab_ref(2, j1inc), &c__1, &d__[
j1inc], &work[j1inc]);
/* L160: */
}
}
/* Computing MIN */
i__3 = kdm1, i__2 = *n - j2;
lend = min(i__3,i__2);
last = j1end + kd1;
if (lend > 0) {
drot_(&lend, &ab_ref(3, last - 1), &c__1, &
ab_ref(2, last), &c__1, &d__[last], &
work[last]);
}
}
}
if (wantq) {
/* accumulate product of plane rotations in Q */
if (initq) {
/* take advantage of the fact that Q was
initially the Identity matrix */
iqend = max(iqend,j2);
/* Computing MAX */
i__3 = 0, i__2 = k - 3;
i2 = max(i__3,i__2);
iqaend = i__ * *kd + 1;
if (k == 2) {
iqaend += *kd;
}
iqaend = min(iqaend,iqend);
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j
+= i__2) {
ibl = i__ - i2 / kdm1;
++i2;
/* Computing MAX */
i__4 = 1, i__5 = j - ibl;
iqb = max(i__4,i__5);
nq = iqaend + 1 - iqb;
/* Computing MIN */
i__4 = iqaend + *kd;
iqaend = min(i__4,iqend);
drot_(&nq, &q_ref(iqb, j - 1), &c__1, &q_ref(
iqb, j), &c__1, &d__[j], &work[j]);
/* L170: */
}
} else {
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j
+= i__3) {
drot_(n, &q_ref(1, j - 1), &c__1, &q_ref(1, j)
, &c__1, &d__[j], &work[j]);
/* L180: */
}
}
}
if (j2 + kdn > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 = j2 - kdn - 1;
}
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2)
{
/* create nonzero element a(j+kd,j-1) outside the
band and store it in WORK */
work[j + *kd] = work[j] * ab_ref(kd1, j);
ab_ref(kd1, j) = d__[j] * ab_ref(kd1, j);
/* L190: */
}
/* L200: */
}
/* L210: */
}
}
if (*kd > 0) {
/* copy off-diagonal elements to E */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab_ref(2, i__);
/* L220: */
}
} else {
/* set E to zero if original matrix was diagonal */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.;
/* L230: */
}
}
/* copy diagonal elements to D */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab_ref(1, i__);
/* L240: */
}
}
return 0;
/* End of DSBTRD */
} /* dsbtrd_ */
#undef ab_ref
#undef q_ref