/* Subroutine */ int slaed3_(integer *k, integer *n, integer *n1, real *d__,
real *q, integer *ldq, real *rho, real *dlamda, real *q2, integer *
indx, integer *ctot, real *w, real *s, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
integer i__, j, n2, n12, ii, n23, iq2;
real temp;
extern doublereal snrm2_(integer *, real *, integer *);
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *), scopy_(integer *, real *,
integer *, real *, integer *), slaed4_(integer *, integer *, real
*, real *, real *, real *, real *, integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int xerbla_(char *, integer *), slacpy_(
char *, integer *, integer *, real *, integer *, real *, integer *
), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAED3 finds the roots of the secular equation, as defined by the */
/* values in D, W, and RHO, between 1 and K. It makes the */
/* appropriate calls to SLAED4 and then updates the eigenvectors by */
/* multiplying the matrix of eigenvectors of the pair of eigensystems */
/* being combined by the matrix of eigenvectors of the K-by-K system */
/* which is solved here. */
/* This code makes very mild assumptions about floating point */
/* arithmetic. It will work on machines with a guard digit in */
/* add/subtract, or on those binary machines without guard digits */
/* which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. */
/* It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. */
/* Arguments */
/* ========= */
/* K (input) INTEGER */
/* The number of terms in the rational function to be solved by */
/* SLAED4. K >= 0. */
/* N (input) INTEGER */
/* The number of rows and columns in the Q matrix. */
/* N >= K (deflation may result in N>K). */
/* N1 (input) INTEGER */
/* The location of the last eigenvalue in the leading submatrix. */
/* min(1,N) <= N1 <= N/2. */
/* D (output) REAL array, dimension (N) */
/* D(I) contains the updated eigenvalues for */
/* 1 <= I <= K. */
/* Q (output) REAL array, dimension (LDQ,N) */
/* Initially the first K columns are used as workspace. */
/* On output the columns 1 to K contain */
/* the updated eigenvectors. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= max(1,N). */
/* RHO (input) REAL */
/* The value of the parameter in the rank one update equation. */
/* RHO >= 0 required. */
/* DLAMDA (input/output) REAL array, dimension (K) */
/* The first K elements of this array contain the old roots */
/* of the deflated updating problem. These are the poles */
/* of the secular equation. May be changed on output by */
/* having lowest order bit set to zero on Cray X-MP, Cray Y-MP, */
/* Cray-2, or Cray C-90, as described above. */
/* Q2 (input) REAL array, dimension (LDQ2, N) */
/* The first K columns of this matrix contain the non-deflated */
/* eigenvectors for the split problem. */
/* INDX (input) INTEGER array, dimension (N) */
/* The permutation used to arrange the columns of the deflated */
/* Q matrix into three groups (see SLAED2). */
/* The rows of the eigenvectors found by SLAED4 must be likewise */
/* permuted before the matrix multiply can take place. */
/* CTOT (input) INTEGER array, dimension (4) */
/* A count of the total number of the various types of columns */
/* in Q, as described in INDX. The fourth column type is any */
/* column which has been deflated. */
/* W (input/output) REAL array, dimension (K) */
/* The first K elements of this array contain the components */
/* of the deflation-adjusted updating vector. Destroyed on */
/* output. */
/* S (workspace) REAL array, dimension (N1 + 1)*K */
/* Will contain the eigenvectors of the repaired matrix which */
/* will be multiplied by the previously accumulated eigenvectors */
/* to update the system. */
/* LDS (input) INTEGER */
/* The leading dimension of S. LDS >= max(1,K). */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an eigenvalue did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Jeff Rutter, Computer Science Division, University of California */
/* at Berkeley, USA */
/* Modified by Francoise Tisseur, University of Tennessee. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--dlamda;
--q2;
--indx;
--ctot;
--w;
--s;
/* Function Body */
*info = 0;
if (*k < 0) {
*info = -1;
} else if (*n < *k) {
*info = -2;
} else if (*ldq < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAED3", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 0) {
return 0;
}
/* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can */
/* be computed with high relative accuracy (barring over/underflow). */
/* This is a problem on machines without a guard digit in */
/* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). */
/* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I), */
/* which on any of these machines zeros out the bottommost */
/* bit of DLAMDA(I) if it is 1; this makes the subsequent */
/* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation */
/* occurs. On binary machines with a guard digit (almost all */
/* machines) it does not change DLAMDA(I) at all. On hexadecimal */
/* and decimal machines with a guard digit, it slightly */
/* changes the bottommost bits of DLAMDA(I). It does not account */
/* for hexadecimal or decimal machines without guard digits */
/* (we know of none). We use a subroutine call to compute */
/* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating */
/* this code. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
dlamda[i__] = slamc3_(&dlamda[i__], &dlamda[i__]) - dlamda[i__];
/* L10: */
}
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
slaed4_(k, &j, &dlamda[1], &w[1], &q[j * q_dim1 + 1], rho, &d__[j],
info);
/* If the zero finder fails, the computation is terminated. */
if (*info != 0) {
goto L120;
}
/* L20: */
}
if (*k == 1) {
goto L110;
}
if (*k == 2) {
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
w[1] = q[j * q_dim1 + 1];
w[2] = q[j * q_dim1 + 2];
ii = indx[1];
q[j * q_dim1 + 1] = w[ii];
ii = indx[2];
q[j * q_dim1 + 2] = w[ii];
/* L30: */
}
goto L110;
}
/* Compute updated W. */
scopy_(k, &w[1], &c__1, &s[1], &c__1);
/* Initialize W(I) = Q(I,I) */
i__1 = *ldq + 1;
scopy_(k, &q[q_offset], &i__1, &w[1], &c__1);
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
w[i__] *= q[i__ + j * q_dim1] / (dlamda[i__] - dlamda[j]);
/* L40: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
w[i__] *= q[i__ + j * q_dim1] / (dlamda[i__] - dlamda[j]);
/* L50: */
}
/* L60: */
}
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__1 = sqrt(-w[i__]);
w[i__] = r_sign(&r__1, &s[i__]);
/* L70: */
}
/* Compute eigenvectors of the modified rank-1 modification. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
s[i__] = w[i__] / q[i__ + j * q_dim1];
/* L80: */
}
temp = snrm2_(k, &s[1], &c__1);
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
ii = indx[i__];
q[i__ + j * q_dim1] = s[ii] / temp;
/* L90: */
}
/* L100: */
}
/* Compute the updated eigenvectors. */
L110:
n2 = *n - *n1;
n12 = ctot[1] + ctot[2];
n23 = ctot[2] + ctot[3];
slacpy_("A", &n23, k, &q[ctot[1] + 1 + q_dim1], ldq, &s[1], &n23);
iq2 = *n1 * n12 + 1;
if (n23 != 0) {
sgemm_("N", "N", &n2, k, &n23, &c_b22, &q2[iq2], &n2, &s[1], &n23, &
c_b23, &q[*n1 + 1 + q_dim1], ldq);
} else {
slaset_("A", &n2, k, &c_b23, &c_b23, &q[*n1 + 1 + q_dim1], ldq);
}
slacpy_("A", &n12, k, &q[q_offset], ldq, &s[1], &n12);
if (n12 != 0) {
sgemm_("N", "N", n1, k, &n12, &c_b22, &q2[1], n1, &s[1], &n12, &c_b23,
&q[q_offset], ldq);
} else {
slaset_("A", n1, k, &c_b23, &c_b23, &q[q_dim1 + 1], ldq);
}
L120:
return 0;
/* End of SLAED3 */
} /* slaed3_ */
/* Subroutine */ int slaed9_(integer *k, integer *kstart, integer *kstop,
integer *n, real *d__, real *q, integer *ldq, real *rho, real *dlamda,
real *w, real *s, integer *lds, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, s_dim1, s_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
integer i__, j;
real temp;
extern doublereal snrm2_(integer *, real *, integer *);
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slaed4_(integer *, integer *, real *, real *, real *,
real *, real *, integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int xerbla_(char *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAED9 finds the roots of the secular equation, as defined by the */
/* values in D, Z, and RHO, between KSTART and KSTOP. It makes the */
/* appropriate calls to SLAED4 and then stores the new matrix of */
/* eigenvectors for use in calculating the next level of Z vectors. */
/* Arguments */
/* ========= */
/* K (input) INTEGER */
/* The number of terms in the rational function to be solved by */
/* SLAED4. K >= 0. */
/* KSTART (input) INTEGER */
/* KSTOP (input) INTEGER */
/* The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP */
/* are to be computed. 1 <= KSTART <= KSTOP <= K. */
/* N (input) INTEGER */
/* The number of rows and columns in the Q matrix. */
/* N >= K (delation may result in N > K). */
/* D (output) REAL array, dimension (N) */
/* D(I) contains the updated eigenvalues */
/* for KSTART <= I <= KSTOP. */
/* Q (workspace) REAL array, dimension (LDQ,N) */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= max( 1, N ). */
/* RHO (input) REAL */
/* The value of the parameter in the rank one update equation. */
/* RHO >= 0 required. */
/* DLAMDA (input) REAL array, dimension (K) */
/* The first K elements of this array contain the old roots */
/* of the deflated updating problem. These are the poles */
/* of the secular equation. */
/* W (input) REAL array, dimension (K) */
/* The first K elements of this array contain the components */
/* of the deflation-adjusted updating vector. */
/* S (output) REAL array, dimension (LDS, K) */
/* Will contain the eigenvectors of the repaired matrix which */
/* will be stored for subsequent Z vector calculation and */
/* multiplied by the previously accumulated eigenvectors */
/* to update the system. */
/* LDS (input) INTEGER */
/* The leading dimension of S. LDS >= max( 1, K ). */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an eigenvalue did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Jeff Rutter, Computer Science Division, University of California */
/* at Berkeley, USA */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--dlamda;
--w;
s_dim1 = *lds;
s_offset = 1 + s_dim1;
s -= s_offset;
/* Function Body */
*info = 0;
if (*k < 0) {
*info = -1;
} else if (*kstart < 1 || *kstart > max(1,*k)) {
*info = -2;
} else if (max(1,*kstop) < *kstart || *kstop > max(1,*k)) {
*info = -3;
} else if (*n < *k) {
*info = -4;
} else if (*ldq < max(1,*k)) {
*info = -7;
} else if (*lds < max(1,*k)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAED9", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 0) {
return 0;
}
/* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can */
/* be computed with high relative accuracy (barring over/underflow). */
/* This is a problem on machines without a guard digit in */
/* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). */
/* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I), */
/* which on any of these machines zeros out the bottommost */
/* bit of DLAMDA(I) if it is 1; this makes the subsequent */
/* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation */
/* occurs. On binary machines with a guard digit (almost all */
/* machines) it does not change DLAMDA(I) at all. On hexadecimal */
/* and decimal machines with a guard digit, it slightly */
/* changes the bottommost bits of DLAMDA(I). It does not account */
/* for hexadecimal or decimal machines without guard digits */
/* (we know of none). We use a subroutine call to compute */
/* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating */
/* this code. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
dlamda[i__] = slamc3_(&dlamda[i__], &dlamda[i__]) - dlamda[i__];
/* L10: */
}
i__1 = *kstop;
for (j = *kstart; j <= i__1; ++j) {
slaed4_(k, &j, &dlamda[1], &w[1], &q[j * q_dim1 + 1], rho, &d__[j],
info);
/* If the zero finder fails, the computation is terminated. */
if (*info != 0) {
goto L120;
}
/* L20: */
}
if (*k == 1 || *k == 2) {
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *k;
for (j = 1; j <= i__2; ++j) {
s[j + i__ * s_dim1] = q[j + i__ * q_dim1];
/* L30: */
}
/* L40: */
}
goto L120;
}
/* Compute updated W. */
scopy_(k, &w[1], &c__1, &s[s_offset], &c__1);
/* Initialize W(I) = Q(I,I) */
i__1 = *ldq + 1;
scopy_(k, &q[q_offset], &i__1, &w[1], &c__1);
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
w[i__] *= q[i__ + j * q_dim1] / (dlamda[i__] - dlamda[j]);
/* L50: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
w[i__] *= q[i__ + j * q_dim1] / (dlamda[i__] - dlamda[j]);
/* L60: */
}
/* L70: */
}
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__1 = sqrt(-w[i__]);
w[i__] = r_sign(&r__1, &s[i__ + s_dim1]);
/* L80: */
}
/* Compute eigenvectors of the modified rank-1 modification. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
q[i__ + j * q_dim1] = w[i__] / q[i__ + j * q_dim1];
/* L90: */
}
temp = snrm2_(k, &q[j * q_dim1 + 1], &c__1);
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
s[i__ + j * s_dim1] = q[i__ + j * q_dim1] / temp;
/* L100: */
}
/* L110: */
}
L120:
return 0;
/* End of SLAED9 */
} /* slaed9_ */
/* Subroutine */ int slaed3_(integer *k, integer *n, integer *n1, real *d__,
real *q, integer *ldq, real *rho, real *dlamda, real *q2, integer *
indx, integer *ctot, real *w, real *s, integer *info)
{
/* System generated locals */
integer q_dim1, q_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
static real temp;
extern doublereal snrm2_(integer *, real *, integer *);
static integer i__, j;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *), scopy_(integer *, real *,
integer *, real *, integer *);
static integer n2;
extern /* Subroutine */ int slaed4_(integer *, integer *, real *, real *,
real *, real *, real *, integer *);
extern doublereal slamc3_(real *, real *);
static integer n12, ii, n23;
extern /* Subroutine */ int xerbla_(char *, integer *), slacpy_(
char *, integer *, integer *, real *, integer *, real *, integer *
), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
static integer iq2;
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
/* -- LAPACK routine (instrumented to count operations, version 3.0) --
Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
Courant Institute, NAG Ltd., and Rice University
June 30, 1999
Common block to return operation count and iteration count
ITCNT is unchanged, OPS is only incremented
Purpose
=======
SLAED3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K. It makes the
appropriate calls to SLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
=========
K (input) INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
N (input) INTEGER
The number of rows and columns in the Q matrix.
N >= K (deflation may result in N>K).
N1 (input) INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= N1 <= N/2.
D (output) REAL array, dimension (N)
D(I) contains the updated eigenvalues for
1 <= I <= K.
Q (output) REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace.
On output the columns 1 to K contain
the updated eigenvectors.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO (input) REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA (input/output) REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation. May be changed on output by
having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
Cray-2, or Cray C-90, as described above.
Q2 (input) REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the non-deflated
eigenvectors for the split problem.
INDX (input) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups (see SLAED2).
The rows of the eigenvectors found by SLAED4 must be likewise
permuted before the matrix multiply can take place.
CTOT (input) INTEGER array, dimension (4)
A count of the total number of the various types of columns
in Q, as described in INDX. The fourth column type is any
column which has been deflated.
W (input/output) REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector. Destroyed on
output.
S (workspace) REAL array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which
will be multiplied by the previously accumulated eigenvectors
to update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max(1,K).
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Further Details
===============
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee.
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--dlamda;
--q2;
--indx;
--ctot;
--w;
--s;
/* Function Body */
*info = 0;
if (*k < 0) {
*info = -1;
} else if (*n < *k) {
*info = -2;
} else if (*ldq < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAED3", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 0) {
return 0;
}
/* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
be computed with high relative accuracy (barring over/underflow).
This is a problem on machines without a guard digit in
add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
which on any of these machines zeros out the bottommost
bit of DLAMDA(I) if it is 1; this makes the subsequent
subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
occurs. On binary machines with a guard digit (almost all
machines) it does not change DLAMDA(I) at all. On hexadecimal
and decimal machines with a guard digit, it slightly
changes the bottommost bits of DLAMDA(I). It does not account
for hexadecimal or decimal machines without guard digits
(we know of none). We use a subroutine call to compute
2*DLAMBDA(I) to prevent optimizing compilers from eliminating
this code. */
latime_1.ops += *n << 1;
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
dlamda[i__] = slamc3_(&dlamda[i__], &dlamda[i__]) - dlamda[i__];
/* L10: */
}
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
slaed4_(k, &j, &dlamda[1], &w[1], &q_ref(1, j), rho, &d__[j], info);
/* If the zero finder fails, the computation is terminated. */
if (*info != 0) {
goto L120;
}
/* L20: */
}
if (*k == 1) {
goto L110;
}
if (*k == 2) {
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
w[1] = q_ref(1, j);
w[2] = q_ref(2, j);
ii = indx[1];
q_ref(1, j) = w[ii];
ii = indx[2];
q_ref(2, j) = w[ii];
/* L30: */
}
goto L110;
}
/* Compute updated W. */
scopy_(k, &w[1], &c__1, &s[1], &c__1);
/* Initialize W(I) = Q(I,I) */
i__1 = *ldq + 1;
scopy_(k, &q[q_offset], &i__1, &w[1], &c__1);
latime_1.ops += *k * 3 * (*k - 1);
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
w[i__] *= q_ref(i__, j) / (dlamda[i__] - dlamda[j]);
/* L40: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
w[i__] *= q_ref(i__, j) / (dlamda[i__] - dlamda[j]);
/* L50: */
}
/* L60: */
}
latime_1.ops += *k;
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__1 = sqrt(-w[i__]);
w[i__] = r_sign(&r__1, &s[i__]);
/* L70: */
}
/* Compute eigenvectors of the modified rank-1 modification. */
latime_1.ops += (*k << 2) * *k;
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
s[i__] = w[i__] / q_ref(i__, j);
/* L80: */
}
temp = snrm2_(k, &s[1], &c__1);
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
ii = indx[i__];
q_ref(i__, j) = s[ii] / temp;
/* L90: */
}
/* L100: */
}
/* Compute the updated eigenvectors. */
L110:
n2 = *n - *n1;
n12 = ctot[1] + ctot[2];
n23 = ctot[2] + ctot[3];
slacpy_("A", &n23, k, &q_ref(ctot[1] + 1, 1), ldq, &s[1], &n23)
;
iq2 = *n1 * n12 + 1;
if (n23 != 0) {
latime_1.ops += (real) n2 * 2 * *k * n23;
sgemm_("N", "N", &n2, k, &n23, &c_b22, &q2[iq2], &n2, &s[1], &n23, &
c_b23, &q_ref(*n1 + 1, 1), ldq);
} else {
slaset_("A", &n2, k, &c_b23, &c_b23, &q_ref(*n1 + 1, 1), ldq);
}
slacpy_("A", &n12, k, &q[q_offset], ldq, &s[1], &n12);
if (n12 != 0) {
latime_1.ops += (real) (*n1) * 2 * *k * n12;
sgemm_("N", "N", n1, k, &n12, &c_b22, &q2[1], n1, &s[1], &n12, &c_b23,
&q[q_offset], ldq);
} else {
slaset_("A", n1, k, &c_b23, &c_b23, &q_ref(1, 1), ldq);
}
L120:
return 0;
/* End of SLAED3 */
} /* slaed3_ */
/* Subroutine */ int slaed9_(integer *k, integer *kstart, integer *kstop,
integer *n, real *d__, real *q, integer *ldq, real *rho, real *dlamda,
real *w, real *s, integer *lds, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
Courant Institute, NAG Ltd., and Rice University
September 30, 1994
Purpose
=======
SLAED9 finds the roots of the secular equation, as defined by the
values in D, Z, and RHO, between KSTART and KSTOP. It makes the
appropriate calls to SLAED4 and then stores the new matrix of
eigenvectors for use in calculating the next level of Z vectors.
Arguments
=========
K (input) INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
KSTART (input) INTEGER
KSTOP (input) INTEGER
The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
are to be computed. 1 <= KSTART <= KSTOP <= K.
N (input) INTEGER
The number of rows and columns in the Q matrix.
N >= K (delation may result in N > K).
D (output) REAL array, dimension (N)
D(I) contains the updated eigenvalues
for KSTART <= I <= KSTOP.
Q (workspace) REAL array, dimension (LDQ,N)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
RHO (input) REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
DLAMDA (input) REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation.
W (input) REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector.
S (output) REAL array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which
will be stored for subsequent Z vector calculation and
multiplied by the previously accumulated eigenvectors
to update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max( 1, K ).
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
Further Details
===============
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer q_dim1, q_offset, s_dim1, s_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
static real temp;
extern doublereal snrm2_(integer *, real *, integer *);
static integer i__, j;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slaed4_(integer *, integer *, real *, real *, real *,
real *, real *, integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int xerbla_(char *, integer *);
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define s_ref(a_1,a_2) s[(a_2)*s_dim1 + a_1]
--d__;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--dlamda;
--w;
s_dim1 = *lds;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
/* Function Body */
*info = 0;
if (*k < 0) {
*info = -1;
} else if (*kstart < 1 || *kstart > max(1,*k)) {
*info = -2;
} else if (max(1,*kstop) < *kstart || *kstop > max(1,*k)) {
*info = -3;
} else if (*n < *k) {
*info = -4;
} else if (*ldq < max(1,*k)) {
*info = -7;
} else if (*lds < max(1,*k)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLAED9", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 0) {
return 0;
}
/* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
be computed with high relative accuracy (barring over/underflow).
This is a problem on machines without a guard digit in
add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
which on any of these machines zeros out the bottommost
bit of DLAMDA(I) if it is 1; this makes the subsequent
subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
occurs. On binary machines with a guard digit (almost all
machines) it does not change DLAMDA(I) at all. On hexadecimal
and decimal machines with a guard digit, it slightly
changes the bottommost bits of DLAMDA(I). It does not account
for hexadecimal or decimal machines without guard digits
(we know of none). We use a subroutine call to compute
2*DLAMBDA(I) to prevent optimizing compilers from eliminating
this code. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
dlamda[i__] = slamc3_(&dlamda[i__], &dlamda[i__]) - dlamda[i__];
/* L10: */
}
i__1 = *kstop;
for (j = *kstart; j <= i__1; ++j) {
slaed4_(k, &j, &dlamda[1], &w[1], &q_ref(1, j), rho, &d__[j], info);
/* If the zero finder fails, the computation is terminated. */
if (*info != 0) {
goto L120;
}
/* L20: */
}
if (*k == 1 || *k == 2) {
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *k;
for (j = 1; j <= i__2; ++j) {
s_ref(j, i__) = q_ref(j, i__);
/* L30: */
}
/* L40: */
}
goto L120;
}
/* Compute updated W. */
scopy_(k, &w[1], &c__1, &s[s_offset], &c__1);
/* Initialize W(I) = Q(I,I) */
i__1 = *ldq + 1;
scopy_(k, &q[q_offset], &i__1, &w[1], &c__1);
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
w[i__] *= q_ref(i__, j) / (dlamda[i__] - dlamda[j]);
/* L50: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
w[i__] *= q_ref(i__, j) / (dlamda[i__] - dlamda[j]);
/* L60: */
}
/* L70: */
}
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__1 = sqrt(-w[i__]);
w[i__] = r_sign(&r__1, &s_ref(i__, 1));
/* L80: */
}
/* Compute eigenvectors of the modified rank-1 modification. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
q_ref(i__, j) = w[i__] / q_ref(i__, j);
/* L90: */
}
temp = snrm2_(k, &q_ref(1, j), &c__1);
i__2 = *k;
for (i__ = 1; i__ <= i__2; ++i__) {
s_ref(i__, j) = q_ref(i__, j) / temp;
/* L100: */
}
/* L110: */
}
L120:
return 0;
/* End of SLAED9 */
} /* slaed9_ */
