/* dlaein.f -- translated by f2c (version 20061008).
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"
#include "blaswrap.h"
/* Table of constant values */
static integer c__1 = 1;
/* Subroutine */ int dlaein_(logical *rightv, logical *noinit, integer *n,
doublereal *h__, integer *ldh, doublereal *wr, doublereal *wi,
doublereal *vr, doublereal *vi, doublereal *b, integer *ldb,
doublereal *work, doublereal *eps3, doublereal *smlnum, doublereal *
bignum, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, h_dim1, h_offset, i__1, i__2, i__3, i__4;
doublereal d__1, d__2, d__3, d__4;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j;
doublereal w, x, y;
integer i1, i2, i3;
doublereal w1, ei, ej, xi, xr, rec;
integer its, ierr;
doublereal temp, norm, vmax;
extern doublereal dnrm2_(integer *, doublereal *, integer *);
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
doublereal scale;
extern doublereal dasum_(integer *, doublereal *, integer *);
char trans[1];
doublereal vcrit, rootn, vnorm;
extern doublereal dlapy2_(doublereal *, doublereal *);
doublereal absbii, absbjj;
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int dladiv_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *), dlatrs_(
char *, char *, char *, char *, integer *, doublereal *, integer *
, doublereal *, doublereal *, doublereal *, integer *);
char normin[1];
doublereal nrmsml, growto;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DLAEIN uses inverse iteration to find a right or left eigenvector */
/* corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg */
/* matrix H. */
/* Arguments */
/* ========= */
/* RIGHTV (input) LOGICAL */
/* = .TRUE. : compute right eigenvector; */
/* = .FALSE.: compute left eigenvector. */
/* NOINIT (input) LOGICAL */
/* = .TRUE. : no initial vector supplied in (VR,VI). */
/* = .FALSE.: initial vector supplied in (VR,VI). */
/* N (input) INTEGER */
/* The order of the matrix H. N >= 0. */
/* H (input) DOUBLE PRECISION array, dimension (LDH,N) */
/* The upper Hessenberg matrix H. */
/* LDH (input) INTEGER */
/* The leading dimension of the array H. LDH >= max(1,N). */
/* WR (input) DOUBLE PRECISION */
/* WI (input) DOUBLE PRECISION */
/* The real and imaginary parts of the eigenvalue of H whose */
/* corresponding right or left eigenvector is to be computed. */
/* VR (input/output) DOUBLE PRECISION array, dimension (N) */
/* VI (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain */
/* a real starting vector for inverse iteration using the real */
/* eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI */
/* must contain the real and imaginary parts of a complex */
/* starting vector for inverse iteration using the complex */
/* eigenvalue (WR,WI); otherwise VR and VI need not be set. */
/* On exit, if WI = 0.0 (real eigenvalue), VR contains the */
/* computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), */
/* VR and VI contain the real and imaginary parts of the */
/* computed complex eigenvector. The eigenvector is normalized */
/* so that the component of largest magnitude has magnitude 1; */
/* here the magnitude of a complex number (x,y) is taken to be */
/* |x| + |y|. */
/* VI is not referenced if WI = 0.0. */
/* B (workspace) DOUBLE PRECISION array, dimension (LDB,N) */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= N+1. */
/* WORK (workspace) DOUBLE PRECISION array, dimension (N) */
/* EPS3 (input) DOUBLE PRECISION */
/* A small machine-dependent value which is used to perturb */
/* close eigenvalues, and to replace zero pivots. */
/* SMLNUM (input) DOUBLE PRECISION */
/* A machine-dependent value close to the underflow threshold. */
/* BIGNUM (input) DOUBLE PRECISION */
/* A machine-dependent value close to the overflow threshold. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* = 1: inverse iteration did not converge; VR is set to the */
/* last iterate, and so is VI if WI.ne.0.0. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--vr;
--vi;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--work;
/* Function Body */
*info = 0;
/* GROWTO is the threshold used in the acceptance test for an */
/* eigenvector. */
rootn = sqrt((doublereal) (*n));
growto = .1 / rootn;
/* Computing MAX */
d__1 = 1., d__2 = *eps3 * rootn;
nrmsml = max(d__1,d__2) * *smlnum;
/* Form B = H - (WR,WI)*I (except that the subdiagonal elements and */
/* the imaginary parts of the diagonal elements are not stored). */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
b[i__ + j * b_dim1] = h__[i__ + j * h_dim1];
/* L10: */
}
b[j + j * b_dim1] = h__[j + j * h_dim1] - *wr;
/* L20: */
}
if (*wi == 0.) {
/* Real eigenvalue. */
if (*noinit) {
/* Set initial vector. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
vr[i__] = *eps3;
/* L30: */
}
} else {
/* Scale supplied initial vector. */
vnorm = dnrm2_(n, &vr[1], &c__1);
d__1 = *eps3 * rootn / max(vnorm,nrmsml);
dscal_(n, &d__1, &vr[1], &c__1);
}
if (*rightv) {
/* LU decomposition with partial pivoting of B, replacing zero */
/* pivots by EPS3. */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
ei = h__[i__ + 1 + i__ * h_dim1];
if ((d__1 = b[i__ + i__ * b_dim1], abs(d__1)) < abs(ei)) {
/* Interchange rows and eliminate. */
x = b[i__ + i__ * b_dim1] / ei;
b[i__ + i__ * b_dim1] = ei;
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
temp = b[i__ + 1 + j * b_dim1];
b[i__ + 1 + j * b_dim1] = b[i__ + j * b_dim1] - x *
temp;
b[i__ + j * b_dim1] = temp;
/* L40: */
}
} else {
/* Eliminate without interchange. */
if (b[i__ + i__ * b_dim1] == 0.) {
b[i__ + i__ * b_dim1] = *eps3;
}
x = ei / b[i__ + i__ * b_dim1];
if (x != 0.) {
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
b[i__ + 1 + j * b_dim1] -= x * b[i__ + j * b_dim1]
;
/* L50: */
}
}
}
/* L60: */
}
if (b[*n + *n * b_dim1] == 0.) {
b[*n + *n * b_dim1] = *eps3;
}
*(unsigned char *)trans = 'N';
} else {
/* UL decomposition with partial pivoting of B, replacing zero */
/* pivots by EPS3. */
for (j = *n; j >= 2; --j) {
ej = h__[j + (j - 1) * h_dim1];
if ((d__1 = b[j + j * b_dim1], abs(d__1)) < abs(ej)) {
/* Interchange columns and eliminate. */
x = b[j + j * b_dim1] / ej;
b[j + j * b_dim1] = ej;
i__1 = j - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
temp = b[i__ + (j - 1) * b_dim1];
b[i__ + (j - 1) * b_dim1] = b[i__ + j * b_dim1] - x *
temp;
b[i__ + j * b_dim1] = temp;
/* L70: */
}
} else {
/* Eliminate without interchange. */
if (b[j + j * b_dim1] == 0.) {
b[j + j * b_dim1] = *eps3;
}
x = ej / b[j + j * b_dim1];
if (x != 0.) {
i__1 = j - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
b[i__ + (j - 1) * b_dim1] -= x * b[i__ + j *
b_dim1];
/* L80: */
}
}
}
/* L90: */
}
if (b[b_dim1 + 1] == 0.) {
b[b_dim1 + 1] = *eps3;
}
*(unsigned char *)trans = 'T';
}
*(unsigned char *)normin = 'N';
i__1 = *n;
for (its = 1; its <= i__1; ++its) {
/* Solve U*x = scale*v for a right eigenvector */
/* or U'*x = scale*v for a left eigenvector, */
/* overwriting x on v. */
dlatrs_("Upper", trans, "Nonunit", normin, n, &b[b_offset], ldb, &
vr[1], &scale, &work[1], &ierr);
*(unsigned char *)normin = 'Y';
/* Test for sufficient growth in the norm of v. */
vnorm = dasum_(n, &vr[1], &c__1);
if (vnorm >= growto * scale) {
goto L120;
}
/* Choose new orthogonal starting vector and try again. */
temp = *eps3 / (rootn + 1.);
vr[1] = *eps3;
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
vr[i__] = temp;
/* L100: */
}
vr[*n - its + 1] -= *eps3 * rootn;
/* L110: */
}
/* Failure to find eigenvector in N iterations. */
*info = 1;
L120:
/* Normalize eigenvector. */
i__ = idamax_(n, &vr[1], &c__1);
d__2 = 1. / (d__1 = vr[i__], abs(d__1));
dscal_(n, &d__2, &vr[1], &c__1);
} else {
/* Complex eigenvalue. */
if (*noinit) {
/* Set initial vector. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
vr[i__] = *eps3;
vi[i__] = 0.;
/* L130: */
}
} else {
/* Scale supplied initial vector. */
d__1 = dnrm2_(n, &vr[1], &c__1);
d__2 = dnrm2_(n, &vi[1], &c__1);
norm = dlapy2_(&d__1, &d__2);
rec = *eps3 * rootn / max(norm,nrmsml);
dscal_(n, &rec, &vr[1], &c__1);
dscal_(n, &rec, &vi[1], &c__1);
}
if (*rightv) {
/* LU decomposition with partial pivoting of B, replacing zero */
/* pivots by EPS3. */
/* The imaginary part of the (i,j)-th element of U is stored in */
/* B(j+1,i). */
b[b_dim1 + 2] = -(*wi);
i__1 = *n;
for (i__ = 2; i__ <= i__1; ++i__) {
b[i__ + 1 + b_dim1] = 0.;
/* L140: */
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
absbii = dlapy2_(&b[i__ + i__ * b_dim1], &b[i__ + 1 + i__ *
b_dim1]);
ei = h__[i__ + 1 + i__ * h_dim1];
if (absbii < abs(ei)) {
/* Interchange rows and eliminate. */
xr = b[i__ + i__ * b_dim1] / ei;
xi = b[i__ + 1 + i__ * b_dim1] / ei;
b[i__ + i__ * b_dim1] = ei;
b[i__ + 1 + i__ * b_dim1] = 0.;
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
temp = b[i__ + 1 + j * b_dim1];
b[i__ + 1 + j * b_dim1] = b[i__ + j * b_dim1] - xr *
temp;
b[j + 1 + (i__ + 1) * b_dim1] = b[j + 1 + i__ *
b_dim1] - xi * temp;
b[i__ + j * b_dim1] = temp;
b[j + 1 + i__ * b_dim1] = 0.;
/* L150: */
}
b[i__ + 2 + i__ * b_dim1] = -(*wi);
b[i__ + 1 + (i__ + 1) * b_dim1] -= xi * *wi;
b[i__ + 2 + (i__ + 1) * b_dim1] += xr * *wi;
} else {
/* Eliminate without interchanging rows. */
if (absbii == 0.) {
b[i__ + i__ * b_dim1] = *eps3;
b[i__ + 1 + i__ * b_dim1] = 0.;
absbii = *eps3;
}
ei = ei / absbii / absbii;
xr = b[i__ + i__ * b_dim1] * ei;
xi = -b[i__ + 1 + i__ * b_dim1] * ei;
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
b[i__ + 1 + j * b_dim1] = b[i__ + 1 + j * b_dim1] -
xr * b[i__ + j * b_dim1] + xi * b[j + 1 + i__
* b_dim1];
b[j + 1 + (i__ + 1) * b_dim1] = -xr * b[j + 1 + i__ *
b_dim1] - xi * b[i__ + j * b_dim1];
/* L160: */
}
b[i__ + 2 + (i__ + 1) * b_dim1] -= *wi;
}
/* Compute 1-norm of offdiagonal elements of i-th row. */
i__2 = *n - i__;
i__3 = *n - i__;
work[i__] = dasum_(&i__2, &b[i__ + (i__ + 1) * b_dim1], ldb)
+ dasum_(&i__3, &b[i__ + 2 + i__ * b_dim1], &c__1);
/* L170: */
}
if (b[*n + *n * b_dim1] == 0. && b[*n + 1 + *n * b_dim1] == 0.) {
b[*n + *n * b_dim1] = *eps3;
}
work[*n] = 0.;
i1 = *n;
i2 = 1;
i3 = -1;
} else {
/* UL decomposition with partial pivoting of conjg(B), */
/* replacing zero pivots by EPS3. */
/* The imaginary part of the (i,j)-th element of U is stored in */
/* B(j+1,i). */
b[*n + 1 + *n * b_dim1] = *wi;
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
b[*n + 1 + j * b_dim1] = 0.;
/* L180: */
}
for (j = *n; j >= 2; --j) {
ej = h__[j + (j - 1) * h_dim1];
absbjj = dlapy2_(&b[j + j * b_dim1], &b[j + 1 + j * b_dim1]);
if (absbjj < abs(ej)) {
/* Interchange columns and eliminate */
xr = b[j + j * b_dim1] / ej;
xi = b[j + 1 + j * b_dim1] / ej;
b[j + j * b_dim1] = ej;
b[j + 1 + j * b_dim1] = 0.;
i__1 = j - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
temp = b[i__ + (j - 1) * b_dim1];
b[i__ + (j - 1) * b_dim1] = b[i__ + j * b_dim1] - xr *
temp;
b[j + i__ * b_dim1] = b[j + 1 + i__ * b_dim1] - xi *
temp;
b[i__ + j * b_dim1] = temp;
b[j + 1 + i__ * b_dim1] = 0.;
/* L190: */
}
b[j + 1 + (j - 1) * b_dim1] = *wi;
b[j - 1 + (j - 1) * b_dim1] += xi * *wi;
b[j + (j - 1) * b_dim1] -= xr * *wi;
} else {
/* Eliminate without interchange. */
if (absbjj == 0.) {
b[j + j * b_dim1] = *eps3;
b[j + 1 + j * b_dim1] = 0.;
absbjj = *eps3;
}
ej = ej / absbjj / absbjj;
xr = b[j + j * b_dim1] * ej;
xi = -b[j + 1 + j * b_dim1] * ej;
i__1 = j - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
b[i__ + (j - 1) * b_dim1] = b[i__ + (j - 1) * b_dim1]
- xr * b[i__ + j * b_dim1] + xi * b[j + 1 +
i__ * b_dim1];
b[j + i__ * b_dim1] = -xr * b[j + 1 + i__ * b_dim1] -
xi * b[i__ + j * b_dim1];
/* L200: */
}
b[j + (j - 1) * b_dim1] += *wi;
}
/* Compute 1-norm of offdiagonal elements of j-th column. */
i__1 = j - 1;
i__2 = j - 1;
work[j] = dasum_(&i__1, &b[j * b_dim1 + 1], &c__1) + dasum_(&
i__2, &b[j + 1 + b_dim1], ldb);
/* L210: */
}
if (b[b_dim1 + 1] == 0. && b[b_dim1 + 2] == 0.) {
b[b_dim1 + 1] = *eps3;
}
work[1] = 0.;
i1 = 1;
i2 = *n;
i3 = 1;
}
i__1 = *n;
for (its = 1; its <= i__1; ++its) {
scale = 1.;
vmax = 1.;
vcrit = *bignum;
/* Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, */
/* or U'*(xr,xi) = scale*(vr,vi) for a left eigenvector, */
/* overwriting (xr,xi) on (vr,vi). */
i__2 = i2;
i__3 = i3;
for (i__ = i1; i__3 < 0 ? i__ >= i__2 : i__ <= i__2; i__ += i__3)
{
if (work[i__] > vcrit) {
rec = 1. / vmax;
dscal_(n, &rec, &vr[1], &c__1);
dscal_(n, &rec, &vi[1], &c__1);
scale *= rec;
vmax = 1.;
vcrit = *bignum;
}
xr = vr[i__];
xi = vi[i__];
if (*rightv) {
i__4 = *n;
for (j = i__ + 1; j <= i__4; ++j) {
xr = xr - b[i__ + j * b_dim1] * vr[j] + b[j + 1 + i__
* b_dim1] * vi[j];
xi = xi - b[i__ + j * b_dim1] * vi[j] - b[j + 1 + i__
* b_dim1] * vr[j];
/* L220: */
}
} else {
i__4 = i__ - 1;
for (j = 1; j <= i__4; ++j) {
xr = xr - b[j + i__ * b_dim1] * vr[j] + b[i__ + 1 + j
* b_dim1] * vi[j];
xi = xi - b[j + i__ * b_dim1] * vi[j] - b[i__ + 1 + j
* b_dim1] * vr[j];
/* L230: */
}
}
w = (d__1 = b[i__ + i__ * b_dim1], abs(d__1)) + (d__2 = b[i__
+ 1 + i__ * b_dim1], abs(d__2));
if (w > *smlnum) {
if (w < 1.) {
w1 = abs(xr) + abs(xi);
if (w1 > w * *bignum) {
rec = 1. / w1;
dscal_(n, &rec, &vr[1], &c__1);
dscal_(n, &rec, &vi[1], &c__1);
xr = vr[i__];
xi = vi[i__];
scale *= rec;
vmax *= rec;
}
}
/* Divide by diagonal element of B. */
dladiv_(&xr, &xi, &b[i__ + i__ * b_dim1], &b[i__ + 1 +
i__ * b_dim1], &vr[i__], &vi[i__]);
/* Computing MAX */
d__3 = (d__1 = vr[i__], abs(d__1)) + (d__2 = vi[i__], abs(
d__2));
vmax = max(d__3,vmax);
vcrit = *bignum / vmax;
} else {
i__4 = *n;
for (j = 1; j <= i__4; ++j) {
vr[j] = 0.;
vi[j] = 0.;
/* L240: */
}
vr[i__] = 1.;
vi[i__] = 1.;
scale = 0.;
vmax = 1.;
vcrit = *bignum;
}
/* L250: */
}
/* Test for sufficient growth in the norm of (VR,VI). */
vnorm = dasum_(n, &vr[1], &c__1) + dasum_(n, &vi[1], &c__1);
if (vnorm >= growto * scale) {
goto L280;
}
/* Choose a new orthogonal starting vector and try again. */
y = *eps3 / (rootn + 1.);
vr[1] = *eps3;
vi[1] = 0.;
i__3 = *n;
for (i__ = 2; i__ <= i__3; ++i__) {
vr[i__] = y;
vi[i__] = 0.;
/* L260: */
}
vr[*n - its + 1] -= *eps3 * rootn;
/* L270: */
}
/* Failure to find eigenvector in N iterations */
*info = 1;
L280:
/* Normalize eigenvector. */
vnorm = 0.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
d__3 = vnorm, d__4 = (d__1 = vr[i__], abs(d__1)) + (d__2 = vi[i__]
, abs(d__2));
vnorm = max(d__3,d__4);
/* L290: */
}
d__1 = 1. / vnorm;
dscal_(n, &d__1, &vr[1], &c__1);
d__1 = 1. / vnorm;
dscal_(n, &d__1, &vi[1], &c__1);
}
return 0;
/* End of DLAEIN */
} /* dlaein_ */