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OpenBLAS/lapack-netlib/SRC/slarre.c

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#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <stdio.h>
#include <complex.h>
#ifdef complex
#undef complex
#endif
#ifdef I
#undef I
#endif
#if defined(_WIN64)
typedef long long BLASLONG;
typedef unsigned long long BLASULONG;
#else
typedef long BLASLONG;
typedef unsigned long BLASULONG;
#endif
#ifdef LAPACK_ILP64
typedef BLASLONG blasint;
#if defined(_WIN64)
#define blasabs(x) llabs(x)
#else
#define blasabs(x) labs(x)
#endif
#else
typedef int blasint;
#define blasabs(x) abs(x)
#endif
typedef blasint integer;
typedef unsigned int uinteger;
typedef char *address;
typedef short int shortint;
typedef float real;
typedef double doublereal;
typedef struct { real r, i; } complex;
typedef struct { doublereal r, i; } doublecomplex;
#ifdef _MSC_VER
static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
#else
static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
#endif
#define pCf(z) (*_pCf(z))
#define pCd(z) (*_pCd(z))
typedef blasint logical;
typedef char logical1;
typedef char integer1;
#define TRUE_ (1)
#define FALSE_ (0)
/* Extern is for use with -E */
#ifndef Extern
#define Extern extern
#endif
/* I/O stuff */
typedef int flag;
typedef int ftnlen;
typedef int ftnint;
/*external read, write*/
typedef struct
{ flag cierr;
ftnint ciunit;
flag ciend;
char *cifmt;
ftnint cirec;
} cilist;
/*internal read, write*/
typedef struct
{ flag icierr;
char *iciunit;
flag iciend;
char *icifmt;
ftnint icirlen;
ftnint icirnum;
} icilist;
/*open*/
typedef struct
{ flag oerr;
ftnint ounit;
char *ofnm;
ftnlen ofnmlen;
char *osta;
char *oacc;
char *ofm;
ftnint orl;
char *oblnk;
} olist;
/*close*/
typedef struct
{ flag cerr;
ftnint cunit;
char *csta;
} cllist;
/*rewind, backspace, endfile*/
typedef struct
{ flag aerr;
ftnint aunit;
} alist;
/* inquire */
typedef struct
{ flag inerr;
ftnint inunit;
char *infile;
ftnlen infilen;
ftnint *inex; /*parameters in standard's order*/
ftnint *inopen;
ftnint *innum;
ftnint *innamed;
char *inname;
ftnlen innamlen;
char *inacc;
ftnlen inacclen;
char *inseq;
ftnlen inseqlen;
char *indir;
ftnlen indirlen;
char *infmt;
ftnlen infmtlen;
char *inform;
ftnint informlen;
char *inunf;
ftnlen inunflen;
ftnint *inrecl;
ftnint *innrec;
char *inblank;
ftnlen inblanklen;
} inlist;
#define VOID void
union Multitype { /* for multiple entry points */
integer1 g;
shortint h;
integer i;
/* longint j; */
real r;
doublereal d;
complex c;
doublecomplex z;
};
typedef union Multitype Multitype;
struct Vardesc { /* for Namelist */
char *name;
char *addr;
ftnlen *dims;
int type;
};
typedef struct Vardesc Vardesc;
struct Namelist {
char *name;
Vardesc **vars;
int nvars;
};
typedef struct Namelist Namelist;
#define abs(x) ((x) >= 0 ? (x) : -(x))
#define dabs(x) (fabs(x))
#define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
#define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
#define dmin(a,b) (f2cmin(a,b))
#define dmax(a,b) (f2cmax(a,b))
#define bit_test(a,b) ((a) >> (b) & 1)
#define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
#define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
#define abort_() { sig_die("Fortran abort routine called", 1); }
#define c_abs(z) (cabsf(Cf(z)))
#define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
#ifdef _MSC_VER
#define c_div(c, a, b) {Cf(c)._Val[0] = (Cf(a)._Val[0]/Cf(b)._Val[0]); Cf(c)._Val[1]=(Cf(a)._Val[1]/Cf(b)._Val[1]);}
#define z_div(c, a, b) {Cd(c)._Val[0] = (Cd(a)._Val[0]/Cd(b)._Val[0]); Cd(c)._Val[1]=(Cd(a)._Val[1]/df(b)._Val[1]);}
#else
#define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
#define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
#endif
#define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
#define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
#define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
//#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
#define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
#define d_abs(x) (fabs(*(x)))
#define d_acos(x) (acos(*(x)))
#define d_asin(x) (asin(*(x)))
#define d_atan(x) (atan(*(x)))
#define d_atn2(x, y) (atan2(*(x),*(y)))
#define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
#define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
#define d_cos(x) (cos(*(x)))
#define d_cosh(x) (cosh(*(x)))
#define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
#define d_exp(x) (exp(*(x)))
#define d_imag(z) (cimag(Cd(z)))
#define r_imag(z) (cimagf(Cf(z)))
#define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
#define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
#define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
#define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
#define d_log(x) (log(*(x)))
#define d_mod(x, y) (fmod(*(x), *(y)))
#define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
#define d_nint(x) u_nint(*(x))
#define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
#define d_sign(a,b) u_sign(*(a),*(b))
#define r_sign(a,b) u_sign(*(a),*(b))
#define d_sin(x) (sin(*(x)))
#define d_sinh(x) (sinh(*(x)))
#define d_sqrt(x) (sqrt(*(x)))
#define d_tan(x) (tan(*(x)))
#define d_tanh(x) (tanh(*(x)))
#define i_abs(x) abs(*(x))
#define i_dnnt(x) ((integer)u_nint(*(x)))
#define i_len(s, n) (n)
#define i_nint(x) ((integer)u_nint(*(x)))
#define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
#define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
#define pow_si(B,E) spow_ui(*(B),*(E))
#define pow_ri(B,E) spow_ui(*(B),*(E))
#define pow_di(B,E) dpow_ui(*(B),*(E))
#define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
#define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
#define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
#define s_cat(lpp, rpp, rnp, np, llp) { ftnlen i, nc, ll; char *f__rp, *lp; ll = (llp); lp = (lpp); for(i=0; i < (int)*(np); ++i) { nc = ll; if((rnp)[i] < nc) nc = (rnp)[i]; ll -= nc; f__rp = (rpp)[i]; while(--nc >= 0) *lp++ = *(f__rp)++; } while(--ll >= 0) *lp++ = ' '; }
#define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
#define s_copy(A,B,C,D) { int __i,__m; for (__i=0, __m=f2cmin((C),(D)); __i<__m && (B)[__i] != 0; ++__i) (A)[__i] = (B)[__i]; }
#define sig_die(s, kill) { exit(1); }
#define s_stop(s, n) {exit(0);}
static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
#define z_abs(z) (cabs(Cd(z)))
#define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
#define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
#define myexit_() break;
#define mycycle() continue;
#define myceiling(w) {ceil(w)}
#define myhuge(w) {HUGE_VAL}
//#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
#define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
/* procedure parameter types for -A and -C++ */
#ifdef __cplusplus
typedef logical (*L_fp)(...);
#else
typedef logical (*L_fp)();
#endif
static float spow_ui(float x, integer n) {
float pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
static double dpow_ui(double x, integer n) {
double pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#ifdef _MSC_VER
static _Fcomplex cpow_ui(complex x, integer n) {
complex pow={1.0,0.0}; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
for(u = n; ; ) {
if(u & 01) pow.r *= x.r, pow.i *= x.i;
if(u >>= 1) x.r *= x.r, x.i *= x.i;
else break;
}
}
_Fcomplex p={pow.r, pow.i};
return p;
}
#else
static _Complex float cpow_ui(_Complex float x, integer n) {
_Complex float pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#endif
#ifdef _MSC_VER
static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
_Dcomplex pow={1.0,0.0}; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
for(u = n; ; ) {
if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
else break;
}
}
_Dcomplex p = {pow._Val[0], pow._Val[1]};
return p;
}
#else
static _Complex double zpow_ui(_Complex double x, integer n) {
_Complex double pow=1.0; unsigned long int u;
if(n != 0) {
if(n < 0) n = -n, x = 1/x;
for(u = n; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
#endif
static integer pow_ii(integer x, integer n) {
integer pow; unsigned long int u;
if (n <= 0) {
if (n == 0 || x == 1) pow = 1;
else if (x != -1) pow = x == 0 ? 1/x : 0;
else n = -n;
}
if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
u = n;
for(pow = 1; ; ) {
if(u & 01) pow *= x;
if(u >>= 1) x *= x;
else break;
}
}
return pow;
}
static integer dmaxloc_(double *w, integer s, integer e, integer *n)
{
double m; integer i, mi;
for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
if (w[i-1]>m) mi=i ,m=w[i-1];
return mi-s+1;
}
static integer smaxloc_(float *w, integer s, integer e, integer *n)
{
float m; integer i, mi;
for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
if (w[i-1]>m) mi=i ,m=w[i-1];
return mi-s+1;
}
static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Fcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
}
}
pCf(z) = zdotc;
}
#else
_Complex float zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
}
}
pCf(z) = zdotc;
}
#endif
static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Dcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
}
}
pCd(z) = zdotc;
}
#else
_Complex double zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
}
}
pCd(z) = zdotc;
}
#endif
static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Fcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
}
}
pCf(z) = zdotc;
}
#else
_Complex float zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cf(&x[i]) * Cf(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
}
}
pCf(z) = zdotc;
}
#endif
static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
integer n = *n_, incx = *incx_, incy = *incy_, i;
#ifdef _MSC_VER
_Dcomplex zdotc = {0.0, 0.0};
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
}
}
pCd(z) = zdotc;
}
#else
_Complex double zdotc = 0.0;
if (incx == 1 && incy == 1) {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cd(&x[i]) * Cd(&y[i]);
}
} else {
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
}
}
pCd(z) = zdotc;
}
#endif
/* -- translated by f2c (version 20000121).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
/* Table of constant values */
static integer c__1 = 1;
static integer c__2 = 2;
/* > \brief \b SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each un
reduced block Ti, finds base representations and eigenvalues. */
/* =========== DOCUMENTATION =========== */
/* Online html documentation available at */
/* http://www.netlib.org/lapack/explore-html/ */
/* > \htmlonly */
/* > Download SLARRE + dependencies */
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slarre.
f"> */
/* > [TGZ]</a> */
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarre.
f"> */
/* > [ZIP]</a> */
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarre.
f"> */
/* > [TXT]</a> */
/* > \endhtmlonly */
/* Definition: */
/* =========== */
/* SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, */
/* RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, */
/* W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, */
/* WORK, IWORK, INFO ) */
/* CHARACTER RANGE */
/* INTEGER IL, INFO, IU, M, N, NSPLIT */
/* REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU */
/* INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), */
/* $ INDEXW( * ) */
/* REAL D( * ), E( * ), E2( * ), GERS( * ), */
/* $ W( * ),WERR( * ), WGAP( * ), WORK( * ) */
/* > \par Purpose: */
/* ============= */
/* > */
/* > \verbatim */
/* > */
/* > To find the desired eigenvalues of a given real symmetric */
/* > tridiagonal matrix T, SLARRE sets any "small" off-diagonal */
/* > elements to zero, and for each unreduced block T_i, it finds */
/* > (a) a suitable shift at one end of the block's spectrum, */
/* > (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and */
/* > (c) eigenvalues of each L_i D_i L_i^T. */
/* > The representations and eigenvalues found are then used by */
/* > SSTEMR to compute the eigenvectors of T. */
/* > The accuracy varies depending on whether bisection is used to */
/* > find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to */
/* > conpute all and then discard any unwanted one. */
/* > As an added benefit, SLARRE also outputs the n */
/* > Gerschgorin intervals for the matrices L_i D_i L_i^T. */
/* > \endverbatim */
/* Arguments: */
/* ========== */
/* > \param[in] RANGE */
/* > \verbatim */
/* > RANGE is CHARACTER*1 */
/* > = 'A': ("All") all eigenvalues will be found. */
/* > = 'V': ("Value") all eigenvalues in the half-open interval */
/* > (VL, VU] will be found. */
/* > = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */
/* > entire matrix) will be found. */
/* > \endverbatim */
/* > */
/* > \param[in] N */
/* > \verbatim */
/* > N is INTEGER */
/* > The order of the matrix. N > 0. */
/* > \endverbatim */
/* > */
/* > \param[in,out] VL */
/* > \verbatim */
/* > VL is REAL */
/* > If RANGE='V', the lower bound for the eigenvalues. */
/* > Eigenvalues less than or equal to VL, or greater than VU, */
/* > will not be returned. VL < VU. */
/* > If RANGE='I' or ='A', SLARRE computes bounds on the desired */
/* > part of the spectrum. */
/* > \endverbatim */
/* > */
/* > \param[in,out] VU */
/* > \verbatim */
/* > VU is REAL */
/* > If RANGE='V', the upper bound for the eigenvalues. */
/* > Eigenvalues less than or equal to VL, or greater than VU, */
/* > will not be returned. VL < VU. */
/* > If RANGE='I' or ='A', SLARRE computes bounds on the desired */
/* > part of the spectrum. */
/* > \endverbatim */
/* > */
/* > \param[in] IL */
/* > \verbatim */
/* > IL is INTEGER */
/* > If RANGE='I', the index of the */
/* > smallest eigenvalue to be returned. */
/* > 1 <= IL <= IU <= N. */
/* > \endverbatim */
/* > */
/* > \param[in] IU */
/* > \verbatim */
/* > IU is INTEGER */
/* > If RANGE='I', the index of the */
/* > largest eigenvalue to be returned. */
/* > 1 <= IL <= IU <= N. */
/* > \endverbatim */
/* > */
/* > \param[in,out] D */
/* > \verbatim */
/* > D is REAL array, dimension (N) */
/* > On entry, the N diagonal elements of the tridiagonal */
/* > matrix T. */
/* > On exit, the N diagonal elements of the diagonal */
/* > matrices D_i. */
/* > \endverbatim */
/* > */
/* > \param[in,out] E */
/* > \verbatim */
/* > E is REAL array, dimension (N) */
/* > On entry, the first (N-1) entries contain the subdiagonal */
/* > elements of the tridiagonal matrix T; E(N) need not be set. */
/* > On exit, E contains the subdiagonal elements of the unit */
/* > bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), */
/* > 1 <= I <= NSPLIT, contain the base points sigma_i on output. */
/* > \endverbatim */
/* > */
/* > \param[in,out] E2 */
/* > \verbatim */
/* > E2 is REAL array, dimension (N) */
/* > On entry, the first (N-1) entries contain the SQUARES of the */
/* > subdiagonal elements of the tridiagonal matrix T; */
/* > E2(N) need not be set. */
/* > On exit, the entries E2( ISPLIT( I ) ), */
/* > 1 <= I <= NSPLIT, have been set to zero */
/* > \endverbatim */
/* > */
/* > \param[in] RTOL1 */
/* > \verbatim */
/* > RTOL1 is REAL */
/* > \endverbatim */
/* > */
/* > \param[in] RTOL2 */
/* > \verbatim */
/* > RTOL2 is REAL */
/* > Parameters for bisection. */
/* > An interval [LEFT,RIGHT] has converged if */
/* > RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
/* > \endverbatim */
/* > */
/* > \param[in] SPLTOL */
/* > \verbatim */
/* > SPLTOL is REAL */
/* > The threshold for splitting. */
/* > \endverbatim */
/* > */
/* > \param[out] NSPLIT */
/* > \verbatim */
/* > NSPLIT is INTEGER */
/* > The number of blocks T splits into. 1 <= NSPLIT <= N. */
/* > \endverbatim */
/* > */
/* > \param[out] ISPLIT */
/* > \verbatim */
/* > ISPLIT is INTEGER array, dimension (N) */
/* > The splitting points, at which T breaks up into blocks. */
/* > The first block consists of rows/columns 1 to ISPLIT(1), */
/* > the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */
/* > etc., and the NSPLIT-th consists of rows/columns */
/* > ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */
/* > \endverbatim */
/* > */
/* > \param[out] M */
/* > \verbatim */
/* > M is INTEGER */
/* > The total number of eigenvalues (of all L_i D_i L_i^T) */
/* > found. */
/* > \endverbatim */
/* > */
/* > \param[out] W */
/* > \verbatim */
/* > W is REAL array, dimension (N) */
/* > The first M elements contain the eigenvalues. The */
/* > eigenvalues of each of the blocks, L_i D_i L_i^T, are */
/* > sorted in ascending order ( SLARRE may use the */
/* > remaining N-M elements as workspace). */
/* > \endverbatim */
/* > */
/* > \param[out] WERR */
/* > \verbatim */
/* > WERR is REAL array, dimension (N) */
/* > The error bound on the corresponding eigenvalue in W. */
/* > \endverbatim */
/* > */
/* > \param[out] WGAP */
/* > \verbatim */
/* > WGAP is REAL array, dimension (N) */
/* > The separation from the right neighbor eigenvalue in W. */
/* > The gap is only with respect to the eigenvalues of the same block */
/* > as each block has its own representation tree. */
/* > Exception: at the right end of a block we store the left gap */
/* > \endverbatim */
/* > */
/* > \param[out] IBLOCK */
/* > \verbatim */
/* > IBLOCK is INTEGER array, dimension (N) */
/* > The indices of the blocks (submatrices) associated with the */
/* > corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue */
/* > W(i) belongs to the first block from the top, =2 if W(i) */
/* > belongs to the second block, etc. */
/* > \endverbatim */
/* > */
/* > \param[out] INDEXW */
/* > \verbatim */
/* > INDEXW is INTEGER array, dimension (N) */
/* > The indices of the eigenvalues within each block (submatrix); */
/* > for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the */
/* > i-th eigenvalue W(i) is the 10-th eigenvalue in block 2 */
/* > \endverbatim */
/* > */
/* > \param[out] GERS */
/* > \verbatim */
/* > GERS is REAL array, dimension (2*N) */
/* > The N Gerschgorin intervals (the i-th Gerschgorin interval */
/* > is (GERS(2*i-1), GERS(2*i)). */
/* > \endverbatim */
/* > */
/* > \param[out] PIVMIN */
/* > \verbatim */
/* > PIVMIN is REAL */
/* > The minimum pivot in the Sturm sequence for T. */
/* > \endverbatim */
/* > */
/* > \param[out] WORK */
/* > \verbatim */
/* > WORK is REAL array, dimension (6*N) */
/* > Workspace. */
/* > \endverbatim */
/* > */
/* > \param[out] IWORK */
/* > \verbatim */
/* > IWORK is INTEGER array, dimension (5*N) */
/* > Workspace. */
/* > \endverbatim */
/* > */
/* > \param[out] INFO */
/* > \verbatim */
/* > INFO is INTEGER */
/* > = 0: successful exit */
/* > > 0: A problem occurred in SLARRE. */
/* > < 0: One of the called subroutines signaled an internal problem. */
/* > Needs inspection of the corresponding parameter IINFO */
/* > for further information. */
/* > */
/* > =-1: Problem in SLARRD. */
/* > = 2: No base representation could be found in MAXTRY iterations. */
/* > Increasing MAXTRY and recompilation might be a remedy. */
/* > =-3: Problem in SLARRB when computing the refined root */
/* > representation for SLASQ2. */
/* > =-4: Problem in SLARRB when preforming bisection on the */
/* > desired part of the spectrum. */
/* > =-5: Problem in SLASQ2. */
/* > =-6: Problem in SLASQ2. */
/* > \endverbatim */
/* Authors: */
/* ======== */
/* > \author Univ. of Tennessee */
/* > \author Univ. of California Berkeley */
/* > \author Univ. of Colorado Denver */
/* > \author NAG Ltd. */
/* > \date June 2016 */
/* > \ingroup OTHERauxiliary */
/* > \par Further Details: */
/* ===================== */
/* > */
/* > \verbatim */
/* > */
/* > The base representations are required to suffer very little */
/* > element growth and consequently define all their eigenvalues to */
/* > high relative accuracy. */
/* > \endverbatim */
/* > \par Contributors: */
/* ================== */
/* > */
/* > Beresford Parlett, University of California, Berkeley, USA \n */
/* > Jim Demmel, University of California, Berkeley, USA \n */
/* > Inderjit Dhillon, University of Texas, Austin, USA \n */
/* > Osni Marques, LBNL/NERSC, USA \n */
/* > Christof Voemel, University of California, Berkeley, USA \n */
/* > */
/* ===================================================================== */
/* Subroutine */ void slarre_(char *range, integer *n, real *vl, real *vu,
integer *il, integer *iu, real *d__, real *e, real *e2, real *rtol1,
real *rtol2, real *spltol, integer *nsplit, integer *isplit, integer *
m, real *w, real *werr, real *wgap, integer *iblock, integer *indexw,
real *gers, real *pivmin, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer i__1, i__2;
real r__1, r__2, r__3;
/* Local variables */
real eabs;
integer iend, jblk;
real eold;
integer indl;
real dmax__, emax;
integer wend, idum, indu;
real rtol;
integer i__, j, iseed[4];
real avgap, sigma;
extern logical lsame_(char *, char *);
integer iinfo;
logical norep;
extern /* Subroutine */ void scopy_(integer *, real *, integer *, real *,
integer *);
real s1, s2;
extern /* Subroutine */ void slasq2_(integer *, real *, integer *);
integer mb;
real gl;
integer in, mm;
real gu;
integer ibegin;
logical forceb;
integer irange;
real sgndef;
extern real slamch_(char *);
integer wbegin;
real safmin, spdiam;
extern /* Subroutine */ void slarra_(integer *, real *, real *, real *,
real *, real *, integer *, integer *, integer *);
logical usedqd;
real clwdth, isleft;
extern /* Subroutine */ void slarrb_(integer *, real *, real *, integer *,
integer *, real *, real *, integer *, real *, real *, real *,
real *, integer *, real *, real *, integer *, integer *), slarrc_(
char *, integer *, real *, real *, real *, real *, real *,
integer *, integer *, integer *, integer *), slarrd_(char
*, char *, integer *, real *, real *, integer *, integer *, real *
, real *, real *, real *, real *, real *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *,
real *, integer *, integer *), slarrk_(integer *,
integer *, real *, real *, real *, real *, real *, real *, real *,
real *, integer *);
real isrght, bsrtol, dpivot;
extern /* Subroutine */ void slarnv_(integer *, integer *, integer *, real
*);
integer cnt;
real eps, tau, tmp, rtl;
integer cnt1, cnt2;
real tmp1;
/* -- LAPACK auxiliary routine (version 3.8.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* June 2016 */
/* ===================================================================== */
/* Parameter adjustments */
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Quick return if possible */
if (*n <= 0) {
return;
}
/* Decode RANGE */
if (lsame_(range, "A")) {
irange = 1;
} else if (lsame_(range, "V")) {
irange = 3;
} else if (lsame_(range, "I")) {
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = slamch_("S");
eps = slamch_("P");
/* Set parameters */
rtl = eps * 100.f;
/* If one were ever to ask for less initial precision in BSRTOL, */
/* one should keep in mind that for the subset case, the extremal */
/* eigenvalues must be at least as accurate as the current setting */
/* (eigenvalues in the middle need not as much accuracy) */
bsrtol = sqrt(eps) * 5e-4f;
/* Treat case of 1x1 matrix for quick return */
if (*n == 1) {
if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu ||
irange == 2 && *il == 1 && *iu == 1) {
*m = 1;
w[1] = d__[1];
/* The computation error of the eigenvalue is zero */
werr[1] = 0.f;
wgap[1] = 0.f;
iblock[1] = 1;
indexw[1] = 1;
gers[1] = d__[1];
gers[2] = d__[1];
}
/* store the shift for the initial RRR, which is zero in this case */
e[1] = 0.f;
return;
}
/* General case: tridiagonal matrix of order > 1 */
/* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. */
/* Compute maximum off-diagonal entry and pivmin. */
gl = d__[1];
gu = d__[1];
eold = 0.f;
emax = 0.f;
e[*n] = 0.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
werr[i__] = 0.f;
wgap[i__] = 0.f;
eabs = (r__1 = e[i__], abs(r__1));
if (eabs >= emax) {
emax = eabs;
}
tmp1 = eabs + eold;
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
/* Computing MIN */
r__1 = gl, r__2 = gers[(i__ << 1) - 1];
gl = f2cmin(r__1,r__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
r__1 = gu, r__2 = gers[i__ * 2];
gu = f2cmax(r__1,r__2);
eold = eabs;
/* L5: */
}
/* The minimum pivot allowed in the Sturm sequence for T */
/* Computing MAX */
/* Computing 2nd power */
r__3 = emax;
r__1 = 1.f, r__2 = r__3 * r__3;
*pivmin = safmin * f2cmax(r__1,r__2);
/* Compute spectral diameter. The Gerschgorin bounds give an */
/* estimate that is wrong by at most a factor of SQRT(2) */
spdiam = gu - gl;
/* Compute splitting points */
slarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
iinfo);
/* Can force use of bisection instead of faster DQDS. */
/* Option left in the code for future multisection work. */
forceb = FALSE_;
/* Initialize USEDQD, DQDS should be used for ALLRNG unless someone */
/* explicitly wants bisection. */
usedqd = irange == 1 && ! forceb;
if (irange == 1 && ! forceb) {
/* Set interval [VL,VU] that contains all eigenvalues */
*vl = gl;
*vu = gu;
} else {
/* We call SLARRD to find crude approximations to the eigenvalues */
/* in the desired range. In case IRANGE = INDRNG, we also obtain the */
/* interval (VL,VU] that contains all the wanted eigenvalues. */
/* An interval [LEFT,RIGHT] has converged if */
/* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) */
/* SLARRD needs a WORK of size 4*N, IWORK of size 3*N */
slarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
if (iinfo != 0) {
*info = -1;
return;
}
/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
i__1 = *n;
for (i__ = mm + 1; i__ <= i__1; ++i__) {
w[i__] = 0.f;
werr[i__] = 0.f;
iblock[i__] = 0;
indexw[i__] = 0;
/* L14: */
}
}
/* ** */
/* Loop over unreduced blocks */
ibegin = 1;
wbegin = 1;
i__1 = *nsplit;
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
in = iend - ibegin + 1;
/* 1 X 1 block */
if (in == 1) {
if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin]
<= *vu || irange == 2 && iblock[wbegin] == jblk) {
++(*m);
w[*m] = d__[ibegin];
werr[*m] = 0.f;
/* The gap for a single block doesn't matter for the later */
/* algorithm and is assigned an arbitrary large value */
wgap[*m] = 0.f;
iblock[*m] = jblk;
indexw[*m] = 1;
++wbegin;
}
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.f;
ibegin = iend + 1;
goto L170;
}
/* Blocks of size larger than 1x1 */
/* E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
e[iend] = 0.f;
/* Find local outer bounds GL,GU for the block */
gl = d__[ibegin];
gu = d__[ibegin];
i__2 = iend;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing MIN */
r__1 = gers[(i__ << 1) - 1];
gl = f2cmin(r__1,gl);
/* Computing MAX */
r__1 = gers[i__ * 2];
gu = f2cmax(r__1,gu);
/* L15: */
}
spdiam = gu - gl;
if (! (irange == 1 && ! forceb)) {
/* Count the number of eigenvalues in the current block. */
mb = 0;
i__2 = mm;
for (i__ = wbegin; i__ <= i__2; ++i__) {
if (iblock[i__] == jblk) {
++mb;
} else {
goto L21;
}
/* L20: */
}
L21:
if (mb == 0) {
/* No eigenvalue in the current block lies in the desired range */
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.f;
ibegin = iend + 1;
goto L170;
} else {
/* Decide whether dqds or bisection is more efficient */
usedqd = (real) mb > in * .5f && ! forceb;
wend = wbegin + mb - 1;
/* Calculate gaps for the current block */
/* In later stages, when representations for individual */
/* eigenvalues are different, we use SIGMA = E( IEND ). */
sigma = 0.f;
i__2 = wend - 1;
for (i__ = wbegin; i__ <= i__2; ++i__) {
/* Computing MAX */
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = f2cmax(r__1,r__2);
/* L30: */
}
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = f2cmax(r__1,r__2);
/* Find local index of the first and last desired evalue. */
indl = indexw[wbegin];
indu = indexw[wend];
}
}
if (irange == 1 && ! forceb || usedqd) {
/* Case of DQDS */
/* Find approximations to the extremal eigenvalues of the block */
slarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return;
}
/* Computing MAX */
r__2 = gl, r__3 = tmp - tmp1 - eps * 100.f * (r__1 = tmp - tmp1,
abs(r__1));
isleft = f2cmax(r__2,r__3);
slarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return;
}
/* Computing MIN */
r__2 = gu, r__3 = tmp + tmp1 + eps * 100.f * (r__1 = tmp + tmp1,
abs(r__1));
isrght = f2cmin(r__2,r__3);
/* Improve the estimate of the spectral diameter */
spdiam = isrght - isleft;
} else {
/* Case of bisection */
/* Find approximations to the wanted extremal eigenvalues */
/* Computing MAX */
r__2 = gl, r__3 = w[wbegin] - werr[wbegin] - eps * 100.f * (r__1 =
w[wbegin] - werr[wbegin], abs(r__1));
isleft = f2cmax(r__2,r__3);
/* Computing MIN */
r__2 = gu, r__3 = w[wend] + werr[wend] + eps * 100.f * (r__1 = w[
wend] + werr[wend], abs(r__1));
isrght = f2cmin(r__2,r__3);
}
/* Decide whether the base representation for the current block */
/* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I */
/* should be on the left or the right end of the current block. */
/* The strategy is to shift to the end which is "more populated" */
/* Furthermore, decide whether to use DQDS for the computation of */
/* the eigenvalue approximations at the end of SLARRE or bisection. */
/* dqds is chosen if all eigenvalues are desired or the number of */
/* eigenvalues to be computed is large compared to the blocksize. */
if (irange == 1 && ! forceb) {
/* If all the eigenvalues have to be computed, we use dqd */
usedqd = TRUE_;
/* INDL is the local index of the first eigenvalue to compute */
indl = 1;
indu = in;
/* MB = number of eigenvalues to compute */
mb = in;
wend = wbegin + mb - 1;
/* Define 1/4 and 3/4 points of the spectrum */
s1 = isleft + spdiam * .25f;
s2 = isrght - spdiam * .25f;
} else {
/* SLARRD has computed IBLOCK and INDEXW for each eigenvalue */
/* approximation. */
/* choose sigma */
if (usedqd) {
s1 = isleft + spdiam * .25f;
s2 = isrght - spdiam * .25f;
} else {
tmp = f2cmin(isrght,*vu) - f2cmax(isleft,*vl);
s1 = f2cmax(isleft,*vl) + tmp * .25f;
s2 = f2cmin(isrght,*vu) - tmp * .25f;
}
}
/* Compute the negcount at the 1/4 and 3/4 points */
if (mb > 1) {
slarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
cnt, &cnt1, &cnt2, &iinfo);
}
if (mb == 1) {
sigma = gl;
sgndef = 1.f;
} else if (cnt1 - indl >= indu - cnt2) {
if (irange == 1 && ! forceb) {
sigma = f2cmax(isleft,gl);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get pos def matrix */
/* for dqds */
sigma = isleft;
} else {
/* use approximation of the first desired eigenvalue of the */
/* block as shift */
sigma = f2cmax(isleft,*vl);
}
sgndef = 1.f;
} else {
if (irange == 1 && ! forceb) {
sigma = f2cmin(isrght,gu);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get neg def matrix */
/* for dqds */
sigma = isrght;
} else {
/* use approximation of the first desired eigenvalue of the */
/* block as shift */
sigma = f2cmin(isrght,*vu);
}
sgndef = -1.f;
}
/* An initial SIGMA has been chosen that will be used for computing */
/* T - SIGMA I = L D L^T */
/* Define the increment TAU of the shift in case the initial shift */
/* needs to be refined to obtain a factorization with not too much */
/* element growth. */
if (usedqd) {
/* The initial SIGMA was to the outer end of the spectrum */
/* the matrix is definite and we need not retreat. */
tau = spdiam * eps * *n + *pivmin * 2.f;
/* Computing MAX */
r__1 = tau, r__2 = eps * 2.f * abs(sigma);
tau = f2cmax(r__1,r__2);
} else {
if (mb > 1) {
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
avgap = (r__1 = clwdth / (real) (wend - wbegin), abs(r__1));
if (sgndef == 1.f) {
/* Computing MAX */
r__1 = wgap[wbegin];
tau = f2cmax(r__1,avgap) * .5f;
/* Computing MAX */
r__1 = tau, r__2 = werr[wbegin];
tau = f2cmax(r__1,r__2);
} else {
/* Computing MAX */
r__1 = wgap[wend - 1];
tau = f2cmax(r__1,avgap) * .5f;
/* Computing MAX */
r__1 = tau, r__2 = werr[wend];
tau = f2cmax(r__1,r__2);
}
} else {
tau = werr[wbegin];
}
}
for (idum = 1; idum <= 6; ++idum) {
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I. */
/* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of */
/* pivots in WORK(2*IN+1:3*IN) */
dpivot = d__[ibegin] - sigma;
work[1] = dpivot;
dmax__ = abs(work[1]);
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(in << 1) + i__] = 1.f / work[i__];
tmp = e[j] * work[(in << 1) + i__];
work[in + i__] = tmp;
dpivot = d__[j + 1] - sigma - tmp * e[j];
work[i__ + 1] = dpivot;
/* Computing MAX */
r__1 = dmax__, r__2 = abs(dpivot);
dmax__ = f2cmax(r__1,r__2);
++j;
/* L70: */
}
/* check for element growth */
if (dmax__ > spdiam * 64.f) {
norep = TRUE_;
} else {
norep = FALSE_;
}
if (usedqd && ! norep) {
/* Ensure the definiteness of the representation */
/* All entries of D (of L D L^T) must have the same sign */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
tmp = sgndef * work[i__];
if (tmp < 0.f) {
norep = TRUE_;
}
/* L71: */
}
}
if (norep) {
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin */
/* shift which makes the matrix definite. So we should end up */
/* here really only in the case of IRANGE = VALRNG or INDRNG. */
if (idum == 5) {
if (sgndef == 1.f) {
/* The fudged Gerschgorin shift should succeed */
sigma = gl - spdiam * 2.f * eps * *n - *pivmin * 4.f;
} else {
sigma = gu + spdiam * 2.f * eps * *n + *pivmin * 4.f;
}
} else {
sigma -= sgndef * tau;
tau *= 2.f;
}
} else {
/* an initial RRR is found */
goto L83;
}
/* L80: */
}
/* if the program reaches this point, no base representation could be */
/* found in MAXTRY iterations. */
*info = 2;
return;
L83:
/* At this point, we have found an initial base representation */
/* T - SIGMA I = L D L^T with not too much element growth. */
/* Store the shift. */
e[iend] = sigma;
/* Store D and L. */
scopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
scopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
if (mb > 1) {
/* Perturb each entry of the base representation by a small */
/* (but random) relative amount to overcome difficulties with */
/* glued matrices. */
for (i__ = 1; i__ <= 4; ++i__) {
iseed[i__ - 1] = 1;
/* L122: */
}
i__2 = (in << 1) - 1;
slarnv_(&c__2, iseed, &i__2, &work[1]);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
d__[ibegin + i__ - 1] *= eps * 4.f * work[i__] + 1.f;
e[ibegin + i__ - 1] *= eps * 4.f * work[in + i__] + 1.f;
/* L125: */
}
d__[iend] *= eps * 4.f * work[in] + 1.f;
}
/* Don't update the Gerschgorin intervals because keeping track */
/* of the updates would be too much work in SLARRV. */
/* We update W instead and use it to locate the proper Gerschgorin */
/* intervals. */
/* Compute the required eigenvalues of L D L' by bisection or dqds */
if (! usedqd) {
/* If SLARRD has been used, shift the eigenvalue approximations */
/* according to their representation. This is necessary for */
/* a uniform SLARRV since dqds computes eigenvalues of the */
/* shifted representation. In SLARRV, W will always hold the */
/* UNshifted eigenvalue approximation. */
i__2 = wend;
for (j = wbegin; j <= i__2; ++j) {
w[j] -= sigma;
werr[j] += (r__1 = w[j], abs(r__1)) * eps;
/* L134: */
}
/* call SLARRB to reduce eigenvalue error of the approximations */
/* from SLARRD */
i__2 = iend - 1;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing 2nd power */
r__1 = e[i__];
work[i__] = d__[i__] * (r__1 * r__1);
/* L135: */
}
/* use bisection to find EV from INDL to INDU */
i__2 = indl - 1;
slarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
iinfo);
if (iinfo != 0) {
*info = -4;
return;
}
/* SLARRB computes all gaps correctly except for the last one */
/* Record distance to VU/GU */
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = f2cmax(r__1,r__2);
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
iblock[*m] = jblk;
indexw[*m] = i__;
/* L138: */
}
} else {
/* Call dqds to get all eigs (and then possibly delete unwanted */
/* eigenvalues). */
/* Note that dqds finds the eigenvalues of the L D L^T representation */
/* of T to high relative accuracy. High relative accuracy */
/* might be lost when the shift of the RRR is subtracted to obtain */
/* the eigenvalues of T. However, T is not guaranteed to define its */
/* eigenvalues to high relative accuracy anyway. */
/* Set RTOL to the order of the tolerance used in SLASQ2 */
/* This is an ESTIMATED error, the worst case bound is 4*N*EPS */
/* which is usually too large and requires unnecessary work to be */
/* done by bisection when computing the eigenvectors */
rtol = log((real) in) * 4.f * eps;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(i__ << 1) - 1] = (r__1 = d__[j], abs(r__1));
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
++j;
/* L140: */
}
work[(in << 1) - 1] = (r__1 = d__[iend], abs(r__1));
work[in * 2] = 0.f;
slasq2_(&in, &work[1], &iinfo);
if (iinfo != 0) {
/* If IINFO = -5 then an index is part of a tight cluster */
/* and should be changed. The index is in IWORK(1) and the */
/* gap is in WORK(N+1) */
*info = -5;
return;
} else {
/* Test that all eigenvalues are positive as expected */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] < 0.f) {
*info = -6;
return;
}
/* L149: */
}
}
if (sgndef > 0.f) {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = work[in - i__ + 1];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L150: */
}
} else {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = -work[i__];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L160: */
}
}
i__2 = *m;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* the value of RTOL below should be the tolerance in SLASQ2 */
werr[i__] = rtol * (r__1 = w[i__], abs(r__1));
/* L165: */
}
i__2 = *m - 1;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* compute the right gap between the intervals */
/* Computing MAX */
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = f2cmax(r__1,r__2);
/* L166: */
}
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[*m] + werr[*m]);
wgap[*m] = f2cmax(r__1,r__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return;
/* end of SLARRE */
} /* slarre_ */
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