/* Subroutine */ int slasd8_(integer *icompq, integer *k, real *d__, real *
z__, real *vf, real *vl, real *difl, real *difr, integer *lddifr,
real *dsigma, real *work, integer *info)
{
/* System generated locals */
integer difr_dim1, difr_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
integer i__, j;
real dj, rho;
integer iwk1, iwk2, iwk3;
real temp;
extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
integer iwk2i, iwk3i;
extern doublereal snrm2_(integer *, real *, integer *);
real diflj, difrj, dsigj;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int slasd4_(integer *, integer *, real *, real *,
real *, real *, real *, real *, integer *), xerbla_(char *,
integer *);
real dsigjp;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* October 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLASD8 finds the square roots of the roots of the secular equation, */
/* as defined by the values in DSIGMA and Z. It makes the appropriate */
/* calls to SLASD4, and stores, for each element in D, the distance */
/* to its two nearest poles (elements in DSIGMA). It also updates */
/* the arrays VF and VL, the first and last components of all the */
/* right singular vectors of the original bidiagonal matrix. */
/* SLASD8 is called from SLASD6. */
/* Arguments */
/* ========= */
/* ICOMPQ (input) INTEGER */
/* Specifies whether singular vectors are to be computed in */
/* factored form in the calling routine: */
/* = 0: Compute singular values only. */
/* = 1: Compute singular vectors in factored form as well. */
/* K (input) INTEGER */
/* The number of terms in the rational function to be solved */
/* by SLASD4. K >= 1. */
/* D (output) REAL array, dimension ( K ) */
/* On output, D contains the updated singular values. */
/* Z (input/output) REAL array, dimension ( K ) */
/* On entry, the first K elements of this array contain the */
/* components of the deflation-adjusted updating row vector. */
/* On exit, Z is updated. */
/* VF (input/output) REAL array, dimension ( K ) */
/* On entry, VF contains information passed through DBEDE8. */
/* On exit, VF contains the first K components of the first */
/* components of all right singular vectors of the bidiagonal */
/* matrix. */
/* VL (input/output) REAL array, dimension ( K ) */
/* On entry, VL contains information passed through DBEDE8. */
/* On exit, VL contains the first K components of the last */
/* components of all right singular vectors of the bidiagonal */
/* matrix. */
/* DIFL (output) REAL array, dimension ( K ) */
/* On exit, DIFL(I) = D(I) - DSIGMA(I). */
/* DIFR (output) REAL array, */
/* dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and */
/* dimension ( K ) if ICOMPQ = 0. */
/* On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not */
/* defined and will not be referenced. */
/* If ICOMPQ = 1, DIFR(1:K,2) is an array containing the */
/* normalizing factors for the right singular vector matrix. */
/* LDDIFR (input) INTEGER */
/* The leading dimension of DIFR, must be at least K. */
/* DSIGMA (input/output) REAL array, dimension ( K ) */
/* On entry, the first K elements of this array contain the old */
/* roots of the deflated updating problem. These are the poles */
/* of the secular equation. */
/* On exit, the elements of DSIGMA may be very slightly altered */
/* in value. */
/* WORK (workspace) REAL array, dimension at least 3 * K */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an singular value did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--z__;
--vf;
--vl;
--difl;
difr_dim1 = *lddifr;
difr_offset = 1 + difr_dim1;
difr -= difr_offset;
--dsigma;
--work;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*k < 1) {
*info = -2;
} else if (*lddifr < *k) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASD8", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 1) {
d__[1] = dabs(z__[1]);
difl[1] = d__[1];
if (*icompq == 1) {
difl[2] = 1.f;
difr[(difr_dim1 << 1) + 1] = 1.f;
}
return 0;
}
/* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(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 DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), */
/* which on any of these machines zeros out the bottommost */
/* bit of DSIGMA(I) if it is 1; this makes the subsequent */
/* subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation */
/* occurs. On binary machines with a guard digit (almost all */
/* machines) it does not change DSIGMA(I) at all. On hexadecimal */
/* and decimal machines with a guard digit, it slightly */
/* changes the bottommost bits of DSIGMA(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__) {
dsigma[i__] = slamc3_(&dsigma[i__], &dsigma[i__]) - dsigma[i__];
/* L10: */
}
/* Book keeping. */
iwk1 = 1;
iwk2 = iwk1 + *k;
iwk3 = iwk2 + *k;
iwk2i = iwk2 - 1;
iwk3i = iwk3 - 1;
/* Normalize Z. */
rho = snrm2_(k, &z__[1], &c__1);
slascl_("G", &c__0, &c__0, &rho, &c_b8, k, &c__1, &z__[1], k, info);
rho *= rho;
/* Initialize WORK(IWK3). */
slaset_("A", k, &c__1, &c_b8, &c_b8, &work[iwk3], k);
/* Compute the updated singular values, the arrays DIFL, DIFR, */
/* and the updated Z. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
slasd4_(k, &j, &dsigma[1], &z__[1], &work[iwk1], &rho, &d__[j], &work[
iwk2], info);
/* If the root finder fails, the computation is terminated. */
if (*info != 0) {
return 0;
}
work[iwk3i + j] = work[iwk3i + j] * work[j] * work[iwk2i + j];
difl[j] = -work[j];
difr[j + difr_dim1] = -work[j + 1];
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[iwk3i + i__] = work[iwk3i + i__] * work[i__] * work[iwk2i +
i__] / (dsigma[i__] - dsigma[j]) / (dsigma[i__] + dsigma[
j]);
/* L20: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
work[iwk3i + i__] = work[iwk3i + i__] * work[i__] * work[iwk2i +
i__] / (dsigma[i__] - dsigma[j]) / (dsigma[i__] + dsigma[
j]);
/* L30: */
}
/* L40: */
}
/* Compute updated Z. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__2 = sqrt((r__1 = work[iwk3i + i__], dabs(r__1)));
z__[i__] = r_sign(&r__2, &z__[i__]);
/* L50: */
}
/* Update VF and VL. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
diflj = difl[j];
dj = d__[j];
dsigj = -dsigma[j];
if (j < *k) {
difrj = -difr[j + difr_dim1];
dsigjp = -dsigma[j + 1];
}
work[j] = -z__[j] / diflj / (dsigma[j] + dj);
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[i__] = z__[i__] / (slamc3_(&dsigma[i__], &dsigj) - diflj) / (
dsigma[i__] + dj);
/* L60: */
}
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
work[i__] = z__[i__] / (slamc3_(&dsigma[i__], &dsigjp) + difrj) /
(dsigma[i__] + dj);
/* L70: */
}
temp = snrm2_(k, &work[1], &c__1);
work[iwk2i + j] = sdot_(k, &work[1], &c__1, &vf[1], &c__1) / temp;
work[iwk3i + j] = sdot_(k, &work[1], &c__1, &vl[1], &c__1) / temp;
if (*icompq == 1) {
difr[j + (difr_dim1 << 1)] = temp;
}
/* L80: */
}
scopy_(k, &work[iwk2], &c__1, &vf[1], &c__1);
scopy_(k, &work[iwk3], &c__1, &vl[1], &c__1);
return 0;
/* End of SLASD8 */
} /* slasd8_ */
/* Subroutine */ int slasd3_(integer *nl, integer *nr, integer *sqre, integer
*k, real *d__, real *q, integer *ldq, real *dsigma, real *u, integer *
ldu, real *u2, integer *ldu2, real *vt, integer *ldvt, real *vt2,
integer *ldvt2, integer *idxc, integer *ctot, real *z__, integer *
info)
{
/* System generated locals */
integer q_dim1, q_offset, u_dim1, u_offset, u2_dim1, u2_offset, vt_dim1,
vt_offset, vt2_dim1, vt2_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
integer i__, j, m, n, jc;
real rho;
integer nlp1, nlp2, nrp1;
real temp;
extern doublereal snrm2_(integer *, real *, integer *);
integer ctemp;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
integer ktemp;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int slasd4_(integer *, integer *, real *, real *,
real *, real *, real *, real *, integer *), xerbla_(char *,
integer *), slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *,
real *, integer *);
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLASD3 finds all the square roots of the roots of the secular */
/* equation, as defined by the values in D and Z. It makes the */
/* appropriate calls to SLASD4 and then updates the singular */
/* vectors by matrix multiplication. */
/* 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 XMP, Cray YMP, Cray C 90, or Cray 2. */
/* It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. */
/* SLASD3 is called from SLASD1. */
/* Arguments */
/* ========= */
/* NL (input) INTEGER */
/* The row dimension of the upper block. NL >= 1. */
/* NR (input) INTEGER */
/* The row dimension of the lower block. NR >= 1. */
/* SQRE (input) INTEGER */
/* = 0: the lower block is an NR-by-NR square matrix. */
/* = 1: the lower block is an NR-by-(NR+1) rectangular matrix. */
/* The bidiagonal matrix has N = NL + NR + 1 rows and */
/* M = N + SQRE >= N columns. */
/* K (input) INTEGER */
/* The size of the secular equation, 1 =< K = < N. */
/* D (output) REAL array, dimension(K) */
/* On exit the square roots of the roots of the secular equation, */
/* in ascending order. */
/* Q (workspace) REAL array, */
/* dimension at least (LDQ,K). */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= K. */
/* DSIGMA (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. */
/* U (output) REAL array, dimension (LDU, N) */
/* The last N - K columns of this matrix contain the deflated */
/* left singular vectors. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= N. */
/* U2 (input) REAL array, dimension (LDU2, N) */
/* The first K columns of this matrix contain the non-deflated */
/* left singular vectors for the split problem. */
/* LDU2 (input) INTEGER */
/* The leading dimension of the array U2. LDU2 >= N. */
/* VT (output) REAL array, dimension (LDVT, M) */
/* The last M - K columns of VT' contain the deflated */
/* right singular vectors. */
/* LDVT (input) INTEGER */
/* The leading dimension of the array VT. LDVT >= N. */
/* VT2 (input/output) REAL array, dimension (LDVT2, N) */
/* The first K columns of VT2' contain the non-deflated */
/* right singular vectors for the split problem. */
/* LDVT2 (input) INTEGER */
/* The leading dimension of the array VT2. LDVT2 >= N. */
/* IDXC (input) INTEGER array, dimension (N) */
/* The permutation used to arrange the columns of U (and rows of */
/* VT) into three groups: the first group contains non-zero */
/* entries only at and above (or before) NL +1; the second */
/* contains non-zero entries only at and below (or after) NL+2; */
/* and the third is dense. The first column of U and the row of */
/* VT are treated separately, however. */
/* The rows of the singular vectors found by SLASD4 */
/* must be likewise permuted before the matrix multiplies can */
/* take place. */
/* CTOT (input) INTEGER array, dimension (4) */
/* A count of the total number of the various types of columns */
/* in U (or rows in VT), as described in IDXC. The fourth column */
/* type is any column which has been deflated. */
/* Z (input/output) REAL array, dimension (K) */
/* The first K elements of this array contain the components */
/* of the deflation-adjusted updating row vector. */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, an singular value did not converge */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. 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;
--dsigma;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
u2_dim1 = *ldu2;
u2_offset = 1 + u2_dim1;
u2 -= u2_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
vt2_dim1 = *ldvt2;
vt2_offset = 1 + vt2_dim1;
vt2 -= vt2_offset;
--idxc;
--ctot;
--z__;
/* Function Body */
*info = 0;
if (*nl < 1) {
*info = -1;
} else if (*nr < 1) {
*info = -2;
} else if (*sqre != 1 && *sqre != 0) {
*info = -3;
}
n = *nl + *nr + 1;
m = n + *sqre;
nlp1 = *nl + 1;
nlp2 = *nl + 2;
if (*k < 1 || *k > n) {
*info = -4;
} else if (*ldq < *k) {
*info = -7;
} else if (*ldu < n) {
*info = -10;
} else if (*ldu2 < n) {
*info = -12;
} else if (*ldvt < m) {
*info = -14;
} else if (*ldvt2 < m) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASD3", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 1) {
d__[1] = dabs(z__[1]);
scopy_(&m, &vt2[vt2_dim1 + 1], ldvt2, &vt[vt_dim1 + 1], ldvt);
if (z__[1] > 0.f) {
scopy_(&n, &u2[u2_dim1 + 1], &c__1, &u[u_dim1 + 1], &c__1);
} else {
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
u[i__ + u_dim1] = -u2[i__ + u2_dim1];
/* L10: */
}
}
return 0;
}
/* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(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 DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), */
/* which on any of these machines zeros out the bottommost */
/* bit of DSIGMA(I) if it is 1; this makes the subsequent */
/* subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation */
/* occurs. On binary machines with a guard digit (almost all */
/* machines) it does not change DSIGMA(I) at all. On hexadecimal */
/* and decimal machines with a guard digit, it slightly */
/* changes the bottommost bits of DSIGMA(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*DSIGMA(I) to prevent optimizing compilers from eliminating */
/* this code. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
dsigma[i__] = slamc3_(&dsigma[i__], &dsigma[i__]) - dsigma[i__];
/* L20: */
}
/* Keep a copy of Z. */
scopy_(k, &z__[1], &c__1, &q[q_offset], &c__1);
/* Normalize Z. */
rho = snrm2_(k, &z__[1], &c__1);
slascl_("G", &c__0, &c__0, &rho, &c_b13, k, &c__1, &z__[1], k, info);
rho *= rho;
/* Find the new singular values. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
slasd4_(k, &j, &dsigma[1], &z__[1], &u[j * u_dim1 + 1], &rho, &d__[j],
&vt[j * vt_dim1 + 1], info);
/* If the zero finder fails, the computation is terminated. */
if (*info != 0) {
return 0;
}
/* L30: */
}
/* Compute updated Z. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
z__[i__] = u[i__ + *k * u_dim1] * vt[i__ + *k * vt_dim1];
i__2 = i__ - 1;
for (j = 1; j <= i__2; ++j) {
z__[i__] *= u[i__ + j * u_dim1] * vt[i__ + j * vt_dim1] / (dsigma[
i__] - dsigma[j]) / (dsigma[i__] + dsigma[j]);
/* L40: */
}
i__2 = *k - 1;
for (j = i__; j <= i__2; ++j) {
z__[i__] *= u[i__ + j * u_dim1] * vt[i__ + j * vt_dim1] / (dsigma[
i__] - dsigma[j + 1]) / (dsigma[i__] + dsigma[j + 1]);
/* L50: */
}
r__2 = sqrt((r__1 = z__[i__], dabs(r__1)));
z__[i__] = r_sign(&r__2, &q[i__ + q_dim1]);
/* L60: */
}
/* Compute left singular vectors of the modified diagonal matrix, */
/* and store related information for the right singular vectors. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
vt[i__ * vt_dim1 + 1] = z__[1] / u[i__ * u_dim1 + 1] / vt[i__ *
vt_dim1 + 1];
u[i__ * u_dim1 + 1] = -1.f;
i__2 = *k;
for (j = 2; j <= i__2; ++j) {
vt[j + i__ * vt_dim1] = z__[j] / u[j + i__ * u_dim1] / vt[j + i__
* vt_dim1];
u[j + i__ * u_dim1] = dsigma[j] * vt[j + i__ * vt_dim1];
/* L70: */
}
temp = snrm2_(k, &u[i__ * u_dim1 + 1], &c__1);
q[i__ * q_dim1 + 1] = u[i__ * u_dim1 + 1] / temp;
i__2 = *k;
for (j = 2; j <= i__2; ++j) {
jc = idxc[j];
q[j + i__ * q_dim1] = u[jc + i__ * u_dim1] / temp;
/* L80: */
}
/* L90: */
}
/* Update the left singular vector matrix. */
if (*k == 2) {
sgemm_("N", "N", &n, k, k, &c_b13, &u2[u2_offset], ldu2, &q[q_offset],
ldq, &c_b26, &u[u_offset], ldu);
goto L100;
}
if (ctot[1] > 0) {
sgemm_("N", "N", nl, k, &ctot[1], &c_b13, &u2[(u2_dim1 << 1) + 1],
ldu2, &q[q_dim1 + 2], ldq, &c_b26, &u[u_dim1 + 1], ldu);
if (ctot[3] > 0) {
ktemp = ctot[1] + 2 + ctot[2];
sgemm_("N", "N", nl, k, &ctot[3], &c_b13, &u2[ktemp * u2_dim1 + 1]
, ldu2, &q[ktemp + q_dim1], ldq, &c_b13, &u[u_dim1 + 1],
ldu);
}
} else if (ctot[3] > 0) {
ktemp = ctot[1] + 2 + ctot[2];
sgemm_("N", "N", nl, k, &ctot[3], &c_b13, &u2[ktemp * u2_dim1 + 1],
ldu2, &q[ktemp + q_dim1], ldq, &c_b26, &u[u_dim1 + 1], ldu);
} else {
slacpy_("F", nl, k, &u2[u2_offset], ldu2, &u[u_offset], ldu);
}
scopy_(k, &q[q_dim1 + 1], ldq, &u[nlp1 + u_dim1], ldu);
ktemp = ctot[1] + 2;
ctemp = ctot[2] + ctot[3];
sgemm_("N", "N", nr, k, &ctemp, &c_b13, &u2[nlp2 + ktemp * u2_dim1], ldu2,
&q[ktemp + q_dim1], ldq, &c_b26, &u[nlp2 + u_dim1], ldu);
/* Generate the right singular vectors. */
L100:
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
temp = snrm2_(k, &vt[i__ * vt_dim1 + 1], &c__1);
q[i__ + q_dim1] = vt[i__ * vt_dim1 + 1] / temp;
i__2 = *k;
for (j = 2; j <= i__2; ++j) {
jc = idxc[j];
q[i__ + j * q_dim1] = vt[jc + i__ * vt_dim1] / temp;
/* L110: */
}
/* L120: */
}
/* Update the right singular vector matrix. */
if (*k == 2) {
sgemm_("N", "N", k, &m, k, &c_b13, &q[q_offset], ldq, &vt2[vt2_offset]
, ldvt2, &c_b26, &vt[vt_offset], ldvt);
return 0;
}
ktemp = ctot[1] + 1;
sgemm_("N", "N", k, &nlp1, &ktemp, &c_b13, &q[q_dim1 + 1], ldq, &vt2[
vt2_dim1 + 1], ldvt2, &c_b26, &vt[vt_dim1 + 1], ldvt);
ktemp = ctot[1] + 2 + ctot[2];
if (ktemp <= *ldvt2) {
sgemm_("N", "N", k, &nlp1, &ctot[3], &c_b13, &q[ktemp * q_dim1 + 1],
ldq, &vt2[ktemp + vt2_dim1], ldvt2, &c_b13, &vt[vt_dim1 + 1],
ldvt);
}
ktemp = ctot[1] + 1;
nrp1 = *nr + *sqre;
if (ktemp > 1) {
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
q[i__ + ktemp * q_dim1] = q[i__ + q_dim1];
/* L130: */
}
i__1 = m;
for (i__ = nlp2; i__ <= i__1; ++i__) {
vt2[ktemp + i__ * vt2_dim1] = vt2[i__ * vt2_dim1 + 1];
/* L140: */
}
}
ctemp = ctot[2] + 1 + ctot[3];
sgemm_("N", "N", k, &nrp1, &ctemp, &c_b13, &q[ktemp * q_dim1 + 1], ldq, &
vt2[ktemp + nlp2 * vt2_dim1], ldvt2, &c_b26, &vt[nlp2 * vt_dim1 +
1], ldvt);
return 0;
/* End of SLASD3 */
} /* slasd3_ */
/* Subroutine */ int slasd8_(integer *icompq, integer *k, real *d__, real *
z__, real *vf, real *vl, real *difl, real *difr, integer *lddifr,
real *dsigma, real *work, integer *info)
{
/* System generated locals */
integer difr_dim1, difr_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
static real temp;
extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
static integer iwk2i, iwk3i;
extern doublereal snrm2_(integer *, real *, integer *);
static integer i__, j;
static real diflj, difrj, dsigj;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
extern doublereal slamc3_(real *, real *);
extern /* Subroutine */ int slasd4_(integer *, integer *, real *, real *,
real *, real *, real *, real *, integer *);
static real dj;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real dsigjp;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
static real rho;
static integer iwk1, iwk2, iwk3;
#define difr_ref(a_1,a_2) difr[(a_2)*difr_dim1 + a_1]
/* -- LAPACK auxiliary routine (instrumented to count ops, version 3.0) --
Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
Courant Institute, NAG Ltd., and Rice University
June 30, 1999
Purpose
=======
SLASD8 finds the square roots of the roots of the secular equation,
as defined by the values in DSIGMA and Z. It makes the appropriate
calls to SLASD4, and stores, for each element in D, the distance
to its two nearest poles (elements in DSIGMA). It also updates
the arrays VF and VL, the first and last components of all the
right singular vectors of the original bidiagonal matrix.
SLASD8 is called from SLASD6.
Arguments
=========
ICOMPQ (input) INTEGER
Specifies whether singular vectors are to be computed in
factored form in the calling routine:
= 0: Compute singular values only.
= 1: Compute singular vectors in factored form as well.
K (input) INTEGER
The number of terms in the rational function to be solved
by SLASD4. K >= 1.
D (output) REAL array, dimension ( K )
On output, D contains the updated singular values.
Z (input) REAL array, dimension ( K )
The first K elements of this array contain the components
of the deflation-adjusted updating row vector.
VF (input/output) REAL array, dimension ( K )
On entry, VF contains information passed through DBEDE8.
On exit, VF contains the first K components of the first
components of all right singular vectors of the bidiagonal
matrix.
VL (input/output) REAL array, dimension ( K )
On entry, VL contains information passed through DBEDE8.
On exit, VL contains the first K components of the last
components of all right singular vectors of the bidiagonal
matrix.
DIFL (output) REAL array, dimension ( K )
On exit, DIFL(I) = D(I) - DSIGMA(I).
DIFR (output) REAL array,
dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
dimension ( K ) if ICOMPQ = 0.
On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
defined and will not be referenced.
If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
normalizing factors for the right singular vector matrix.
LDDIFR (input) INTEGER
The leading dimension of DIFR, must be at least K.
DSIGMA (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.
WORK (workspace) REAL array, dimension at least 3 * K
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, an singular value did not converge
Further Details
===============
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--z__;
--vf;
--vl;
--difl;
difr_dim1 = *lddifr;
difr_offset = 1 + difr_dim1 * 1;
difr -= difr_offset;
--dsigma;
--work;
/* Function Body */
*info = 0;
if (*icompq < 0 || *icompq > 1) {
*info = -1;
} else if (*k < 1) {
*info = -2;
} else if (*lddifr < *k) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASD8", &i__1);
return 0;
}
/* Quick return if possible */
if (*k == 1) {
d__[1] = dabs(z__[1]);
difl[1] = d__[1];
if (*icompq == 1) {
difl[2] = 1.f;
difr_ref(1, 2) = 1.f;
}
return 0;
}
/* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(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 DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
which on any of these machines zeros out the bottommost
bit of DSIGMA(I) if it is 1; this makes the subsequent
subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
occurs. On binary machines with a guard digit (almost all
machines) it does not change DSIGMA(I) at all. On hexadecimal
and decimal machines with a guard digit, it slightly
changes the bottommost bits of DSIGMA(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 += (real) (*k << 1);
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
dsigma[i__] = slamc3_(&dsigma[i__], &dsigma[i__]) - dsigma[i__];
/* L10: */
}
/* Book keeping. */
iwk1 = 1;
iwk2 = iwk1 + *k;
iwk3 = iwk2 + *k;
iwk2i = iwk2 - 1;
iwk3i = iwk3 - 1;
/* Normalize Z. */
latime_1.ops += (real) (*k * 3 + 1);
rho = snrm2_(k, &z__[1], &c__1);
slascl_("G", &c__0, &c__0, &rho, &c_b8, k, &c__1, &z__[1], k, info);
rho *= rho;
/* Initialize WORK(IWK3). */
slaset_("A", k, &c__1, &c_b8, &c_b8, &work[iwk3], k);
/* Compute the updated singular values, the arrays DIFL, DIFR,
and the updated Z. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
slasd4_(k, &j, &dsigma[1], &z__[1], &work[iwk1], &rho, &d__[j], &work[
iwk2], info);
/* If the root finder fails, the computation is terminated. */
if (*info != 0) {
return 0;
}
latime_1.ops += 2.f;
work[iwk3i + j] = work[iwk3i + j] * work[j] * work[iwk2i + j];
difl[j] = -work[j];
difr_ref(j, 1) = -work[j + 1];
latime_1.ops += (real) ((j - 1) * 6);
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[iwk3i + i__] = work[iwk3i + i__] * work[i__] * work[iwk2i +
i__] / (dsigma[i__] - dsigma[j]) / (dsigma[i__] + dsigma[
j]);
/* L20: */
}
latime_1.ops += (real) ((*k - j) * 6);
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
work[iwk3i + i__] = work[iwk3i + i__] * work[i__] * work[iwk2i +
i__] / (dsigma[i__] - dsigma[j]) / (dsigma[i__] + dsigma[
j]);
/* L30: */
}
/* L40: */
}
/* Compute updated Z. */
latime_1.ops += (real) (*k);
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
r__2 = sqrt((r__1 = work[iwk3i + i__], dabs(r__1)));
z__[i__] = r_sign(&r__2, &z__[i__]);
/* L50: */
}
/* Update VF and VL. */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
diflj = difl[j];
dj = d__[j];
dsigj = -dsigma[j];
if (j < *k) {
difrj = -difr_ref(j, 1);
dsigjp = -dsigma[j + 1];
}
latime_1.ops += 3.f;
work[j] = -z__[j] / diflj / (dsigma[j] + dj);
latime_1.ops += (real) ((j - 1) * 5);
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[i__] = z__[i__] / (slamc3_(&dsigma[i__], &dsigj) - diflj) / (
dsigma[i__] + dj);
/* L60: */
}
latime_1.ops += (real) ((*k - j) * 5);
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
work[i__] = z__[i__] / (slamc3_(&dsigma[i__], &dsigjp) + difrj) /
(dsigma[i__] + dj);
/* L70: */
}
latime_1.ops += (real) (*k * 6);
temp = snrm2_(k, &work[1], &c__1);
work[iwk2i + j] = sdot_(k, &work[1], &c__1, &vf[1], &c__1) / temp;
work[iwk3i + j] = sdot_(k, &work[1], &c__1, &vl[1], &c__1) / temp;
if (*icompq == 1) {
difr_ref(j, 2) = temp;
}
/* L80: */
}
scopy_(k, &work[iwk2], &c__1, &vf[1], &c__1);
scopy_(k, &work[iwk3], &c__1, &vl[1], &c__1);
return 0;
/* End of SLASD8 */
} /* slasd8_ */
