14 typedef long long BLASLONG;
15 typedef unsigned long long BLASULONG;
17 typedef long BLASLONG;
18 typedef unsigned long BLASULONG;
22 typedef BLASLONG blasint;
24 #define blasabs(x) llabs(x)
26 #define blasabs(x) labs(x)
30 #define blasabs(x) abs(x)
33 typedef blasint integer;
35 typedef unsigned int uinteger;
36 typedef char *address;
37 typedef short int shortint;
39 typedef double doublereal;
40 typedef struct { real r, i; } complex;
41 typedef struct { doublereal r, i; } doublecomplex;
43 static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
44 static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
45 static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
46 static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
48 static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
49 static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
50 static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
51 static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
53 #define pCf(z) (*_pCf(z))
54 #define pCd(z) (*_pCd(z))
56 typedef short int shortlogical;
57 typedef char logical1;
58 typedef char integer1;
63 /* Extern is for use with -E */
74 /*external read, write*/
83 /*internal read, write*/
113 /*rewind, backspace, endfile*/
125 ftnint *inex; /*parameters in standard's order*/
151 union Multitype { /* for multiple entry points */
162 typedef union Multitype Multitype;
164 struct Vardesc { /* for Namelist */
170 typedef struct Vardesc Vardesc;
177 typedef struct Namelist Namelist;
179 #define abs(x) ((x) >= 0 ? (x) : -(x))
180 #define dabs(x) (fabs(x))
181 #define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
182 #define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
183 #define dmin(a,b) (f2cmin(a,b))
184 #define dmax(a,b) (f2cmax(a,b))
185 #define bit_test(a,b) ((a) >> (b) & 1)
186 #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
187 #define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
189 #define abort_() { sig_die("Fortran abort routine called", 1); }
190 #define c_abs(z) (cabsf(Cf(z)))
191 #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
193 #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]);}
194 #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]);}
196 #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
197 #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
199 #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
200 #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
201 #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
202 //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
203 #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
204 #define d_abs(x) (fabs(*(x)))
205 #define d_acos(x) (acos(*(x)))
206 #define d_asin(x) (asin(*(x)))
207 #define d_atan(x) (atan(*(x)))
208 #define d_atn2(x, y) (atan2(*(x),*(y)))
209 #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
210 #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
211 #define d_cos(x) (cos(*(x)))
212 #define d_cosh(x) (cosh(*(x)))
213 #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
214 #define d_exp(x) (exp(*(x)))
215 #define d_imag(z) (cimag(Cd(z)))
216 #define r_imag(z) (cimagf(Cf(z)))
217 #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
218 #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
219 #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
220 #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
221 #define d_log(x) (log(*(x)))
222 #define d_mod(x, y) (fmod(*(x), *(y)))
223 #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
224 #define d_nint(x) u_nint(*(x))
225 #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
226 #define d_sign(a,b) u_sign(*(a),*(b))
227 #define r_sign(a,b) u_sign(*(a),*(b))
228 #define d_sin(x) (sin(*(x)))
229 #define d_sinh(x) (sinh(*(x)))
230 #define d_sqrt(x) (sqrt(*(x)))
231 #define d_tan(x) (tan(*(x)))
232 #define d_tanh(x) (tanh(*(x)))
233 #define i_abs(x) abs(*(x))
234 #define i_dnnt(x) ((integer)u_nint(*(x)))
235 #define i_len(s, n) (n)
236 #define i_nint(x) ((integer)u_nint(*(x)))
237 #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
238 #define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
239 #define pow_si(B,E) spow_ui(*(B),*(E))
240 #define pow_ri(B,E) spow_ui(*(B),*(E))
241 #define pow_di(B,E) dpow_ui(*(B),*(E))
242 #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
243 #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
244 #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
245 #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++ = ' '; }
246 #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
247 #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]; }
248 #define sig_die(s, kill) { exit(1); }
249 #define s_stop(s, n) {exit(0);}
250 static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
251 #define z_abs(z) (cabs(Cd(z)))
252 #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
253 #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
254 #define myexit_() break;
255 #define mycycle() continue;
256 #define myceiling(w) {ceil(w)}
257 #define myhuge(w) {HUGE_VAL}
258 //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
259 #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
261 /* procedure parameter types for -A and -C++ */
263 #define F2C_proc_par_types 1
265 typedef logical (*L_fp)(...);
267 typedef logical (*L_fp)();
270 static float spow_ui(float x, integer n) {
271 float pow=1.0; unsigned long int u;
273 if(n < 0) n = -n, x = 1/x;
282 static double dpow_ui(double x, integer n) {
283 double pow=1.0; unsigned long int u;
285 if(n < 0) n = -n, x = 1/x;
295 static _Fcomplex cpow_ui(complex x, integer n) {
296 complex pow={1.0,0.0}; unsigned long int u;
298 if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
300 if(u & 01) pow.r *= x.r, pow.i *= x.i;
301 if(u >>= 1) x.r *= x.r, x.i *= x.i;
305 _Fcomplex p={pow.r, pow.i};
309 static _Complex float cpow_ui(_Complex float x, integer n) {
310 _Complex float pow=1.0; unsigned long int u;
312 if(n < 0) n = -n, x = 1/x;
323 static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
324 _Dcomplex pow={1.0,0.0}; unsigned long int u;
326 if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
328 if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
329 if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
333 _Dcomplex p = {pow._Val[0], pow._Val[1]};
337 static _Complex double zpow_ui(_Complex double x, integer n) {
338 _Complex double pow=1.0; unsigned long int u;
340 if(n < 0) n = -n, x = 1/x;
350 static integer pow_ii(integer x, integer n) {
351 integer pow; unsigned long int u;
353 if (n == 0 || x == 1) pow = 1;
354 else if (x != -1) pow = x == 0 ? 1/x : 0;
357 if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
367 static integer dmaxloc_(double *w, integer s, integer e, integer *n)
369 double m; integer i, mi;
370 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
371 if (w[i-1]>m) mi=i ,m=w[i-1];
374 static integer smaxloc_(float *w, integer s, integer e, integer *n)
376 float m; integer i, mi;
377 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
378 if (w[i-1]>m) mi=i ,m=w[i-1];
381 static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
382 integer n = *n_, incx = *incx_, incy = *incy_, i;
384 _Fcomplex zdotc = {0.0, 0.0};
385 if (incx == 1 && incy == 1) {
386 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
387 zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
388 zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
391 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
392 zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
393 zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
399 _Complex float zdotc = 0.0;
400 if (incx == 1 && incy == 1) {
401 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
402 zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
405 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
406 zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
412 static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
413 integer n = *n_, incx = *incx_, incy = *incy_, i;
415 _Dcomplex zdotc = {0.0, 0.0};
416 if (incx == 1 && incy == 1) {
417 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
418 zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
419 zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
422 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
423 zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
424 zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
430 _Complex double zdotc = 0.0;
431 if (incx == 1 && incy == 1) {
432 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
433 zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
436 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
437 zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
443 static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
444 integer n = *n_, incx = *incx_, incy = *incy_, i;
446 _Fcomplex zdotc = {0.0, 0.0};
447 if (incx == 1 && incy == 1) {
448 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
449 zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
450 zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
453 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
454 zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
455 zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
461 _Complex float zdotc = 0.0;
462 if (incx == 1 && incy == 1) {
463 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
464 zdotc += Cf(&x[i]) * Cf(&y[i]);
467 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
468 zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
474 static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
475 integer n = *n_, incx = *incx_, incy = *incy_, i;
477 _Dcomplex zdotc = {0.0, 0.0};
478 if (incx == 1 && incy == 1) {
479 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
480 zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
481 zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
484 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
485 zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
486 zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
492 _Complex double zdotc = 0.0;
493 if (incx == 1 && incy == 1) {
494 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
495 zdotc += Cd(&x[i]) * Cd(&y[i]);
498 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
499 zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
505 /* -- translated by f2c (version 20000121).
506 You must link the resulting object file with the libraries:
507 -lf2c -lm (in that order)
513 /* Table of constant values */
515 static integer c__10 = 10;
516 static integer c__1 = 1;
517 static integer c__2 = 2;
518 static integer c__3 = 3;
519 static integer c__4 = 4;
520 static integer c_n1 = -1;
522 /* > \brief <b> SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY mat
525 /* =========== DOCUMENTATION =========== */
527 /* Online html documentation available at */
528 /* http://www.netlib.org/lapack/explore-html/ */
531 /* > Download SSYEVR + dependencies */
532 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssyevr.
535 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssyevr.
538 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssyevr.
546 /* SUBROUTINE SSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, */
547 /* ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, */
548 /* IWORK, LIWORK, INFO ) */
550 /* CHARACTER JOBZ, RANGE, UPLO */
551 /* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N */
552 /* REAL ABSTOL, VL, VU */
553 /* INTEGER ISUPPZ( * ), IWORK( * ) */
554 /* REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) */
557 /* > \par Purpose: */
562 /* > SSYEVR computes selected eigenvalues and, optionally, eigenvectors */
563 /* > of a real symmetric matrix A. Eigenvalues and eigenvectors can be */
564 /* > selected by specifying either a range of values or a range of */
565 /* > indices for the desired eigenvalues. */
567 /* > SSYEVR first reduces the matrix A to tridiagonal form T with a call */
568 /* > to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute */
569 /* > the eigenspectrum using Relatively Robust Representations. SSTEMR */
570 /* > computes eigenvalues by the dqds algorithm, while orthogonal */
571 /* > eigenvectors are computed from various "good" L D L^T representations */
572 /* > (also known as Relatively Robust Representations). Gram-Schmidt */
573 /* > orthogonalization is avoided as far as possible. More specifically, */
574 /* > the various steps of the algorithm are as follows. */
576 /* > For each unreduced block (submatrix) of T, */
577 /* > (a) Compute T - sigma I = L D L^T, so that L and D */
578 /* > define all the wanted eigenvalues to high relative accuracy. */
579 /* > This means that small relative changes in the entries of D and L */
580 /* > cause only small relative changes in the eigenvalues and */
581 /* > eigenvectors. The standard (unfactored) representation of the */
582 /* > tridiagonal matrix T does not have this property in general. */
583 /* > (b) Compute the eigenvalues to suitable accuracy. */
584 /* > If the eigenvectors are desired, the algorithm attains full */
585 /* > accuracy of the computed eigenvalues only right before */
586 /* > the corresponding vectors have to be computed, see steps c) and d). */
587 /* > (c) For each cluster of close eigenvalues, select a new */
588 /* > shift close to the cluster, find a new factorization, and refine */
589 /* > the shifted eigenvalues to suitable accuracy. */
590 /* > (d) For each eigenvalue with a large enough relative separation compute */
591 /* > the corresponding eigenvector by forming a rank revealing twisted */
592 /* > factorization. Go back to (c) for any clusters that remain. */
594 /* > The desired accuracy of the output can be specified by the input */
595 /* > parameter ABSTOL. */
597 /* > For more details, see SSTEMR's documentation and: */
598 /* > - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */
599 /* > to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */
600 /* > Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */
601 /* > - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */
602 /* > Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */
603 /* > 2004. Also LAPACK Working Note 154. */
604 /* > - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */
605 /* > tridiagonal eigenvalue/eigenvector problem", */
606 /* > Computer Science Division Technical Report No. UCB/CSD-97-971, */
607 /* > UC Berkeley, May 1997. */
610 /* > Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested */
611 /* > on machines which conform to the ieee-754 floating point standard. */
612 /* > SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and */
613 /* > when partial spectrum requests are made. */
615 /* > Normal execution of SSTEMR may create NaNs and infinities and */
616 /* > hence may abort due to a floating point exception in environments */
617 /* > which do not handle NaNs and infinities in the ieee standard default */
624 /* > \param[in] JOBZ */
626 /* > JOBZ is CHARACTER*1 */
627 /* > = 'N': Compute eigenvalues only; */
628 /* > = 'V': Compute eigenvalues and eigenvectors. */
631 /* > \param[in] RANGE */
633 /* > RANGE is CHARACTER*1 */
634 /* > = 'A': all eigenvalues will be found. */
635 /* > = 'V': all eigenvalues in the half-open interval (VL,VU] */
636 /* > will be found. */
637 /* > = 'I': the IL-th through IU-th eigenvalues will be found. */
638 /* > For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and */
639 /* > SSTEIN are called */
642 /* > \param[in] UPLO */
644 /* > UPLO is CHARACTER*1 */
645 /* > = 'U': Upper triangle of A is stored; */
646 /* > = 'L': Lower triangle of A is stored. */
652 /* > The order of the matrix A. N >= 0. */
655 /* > \param[in,out] A */
657 /* > A is REAL array, dimension (LDA, N) */
658 /* > On entry, the symmetric matrix A. If UPLO = 'U', the */
659 /* > leading N-by-N upper triangular part of A contains the */
660 /* > upper triangular part of the matrix A. If UPLO = 'L', */
661 /* > the leading N-by-N lower triangular part of A contains */
662 /* > the lower triangular part of the matrix A. */
663 /* > On exit, the lower triangle (if UPLO='L') or the upper */
664 /* > triangle (if UPLO='U') of A, including the diagonal, is */
668 /* > \param[in] LDA */
670 /* > LDA is INTEGER */
671 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
674 /* > \param[in] VL */
677 /* > If RANGE='V', the lower bound of the interval to */
678 /* > be searched for eigenvalues. VL < VU. */
679 /* > Not referenced if RANGE = 'A' or 'I'. */
682 /* > \param[in] VU */
685 /* > If RANGE='V', the upper bound of the interval to */
686 /* > be searched for eigenvalues. VL < VU. */
687 /* > Not referenced if RANGE = 'A' or 'I'. */
690 /* > \param[in] IL */
692 /* > IL is INTEGER */
693 /* > If RANGE='I', the index of the */
694 /* > smallest eigenvalue to be returned. */
695 /* > 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. */
696 /* > Not referenced if RANGE = 'A' or 'V'. */
699 /* > \param[in] IU */
701 /* > IU is INTEGER */
702 /* > If RANGE='I', the index of the */
703 /* > largest eigenvalue to be returned. */
704 /* > 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. */
705 /* > Not referenced if RANGE = 'A' or 'V'. */
708 /* > \param[in] ABSTOL */
710 /* > ABSTOL is REAL */
711 /* > The absolute error tolerance for the eigenvalues. */
712 /* > An approximate eigenvalue is accepted as converged */
713 /* > when it is determined to lie in an interval [a,b] */
714 /* > of width less than or equal to */
716 /* > ABSTOL + EPS * f2cmax( |a|,|b| ) , */
718 /* > where EPS is the machine precision. If ABSTOL is less than */
719 /* > or equal to zero, then EPS*|T| will be used in its place, */
720 /* > where |T| is the 1-norm of the tridiagonal matrix obtained */
721 /* > by reducing A to tridiagonal form. */
723 /* > See "Computing Small Singular Values of Bidiagonal Matrices */
724 /* > with Guaranteed High Relative Accuracy," by Demmel and */
725 /* > Kahan, LAPACK Working Note #3. */
727 /* > If high relative accuracy is important, set ABSTOL to */
728 /* > SLAMCH( 'Safe minimum' ). Doing so will guarantee that */
729 /* > eigenvalues are computed to high relative accuracy when */
730 /* > possible in future releases. The current code does not */
731 /* > make any guarantees about high relative accuracy, but */
732 /* > future releases will. See J. Barlow and J. Demmel, */
733 /* > "Computing Accurate Eigensystems of Scaled Diagonally */
734 /* > Dominant Matrices", LAPACK Working Note #7, for a discussion */
735 /* > of which matrices define their eigenvalues to high relative */
739 /* > \param[out] M */
742 /* > The total number of eigenvalues found. 0 <= M <= N. */
743 /* > If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */
746 /* > \param[out] W */
748 /* > W is REAL array, dimension (N) */
749 /* > The first M elements contain the selected eigenvalues in */
750 /* > ascending order. */
753 /* > \param[out] Z */
755 /* > Z is REAL array, dimension (LDZ, f2cmax(1,M)) */
756 /* > If JOBZ = 'V', then if INFO = 0, the first M columns of Z */
757 /* > contain the orthonormal eigenvectors of the matrix A */
758 /* > corresponding to the selected eigenvalues, with the i-th */
759 /* > column of Z holding the eigenvector associated with W(i). */
760 /* > If JOBZ = 'N', then Z is not referenced. */
761 /* > Note: the user must ensure that at least f2cmax(1,M) columns are */
762 /* > supplied in the array Z; if RANGE = 'V', the exact value of M */
763 /* > is not known in advance and an upper bound must be used. */
764 /* > Supplying N columns is always safe. */
767 /* > \param[in] LDZ */
769 /* > LDZ is INTEGER */
770 /* > The leading dimension of the array Z. LDZ >= 1, and if */
771 /* > JOBZ = 'V', LDZ >= f2cmax(1,N). */
774 /* > \param[out] ISUPPZ */
776 /* > ISUPPZ is INTEGER array, dimension ( 2*f2cmax(1,M) ) */
777 /* > The support of the eigenvectors in Z, i.e., the indices */
778 /* > indicating the nonzero elements in Z. The i-th eigenvector */
779 /* > is nonzero only in elements ISUPPZ( 2*i-1 ) through */
780 /* > ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal */
781 /* > matrix). The support of the eigenvectors of A is typically */
782 /* > 1:N because of the orthogonal transformations applied by SORMTR. */
783 /* > Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 */
786 /* > \param[out] WORK */
788 /* > WORK is REAL array, dimension (MAX(1,LWORK)) */
789 /* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
792 /* > \param[in] LWORK */
794 /* > LWORK is INTEGER */
795 /* > The dimension of the array WORK. LWORK >= f2cmax(1,26*N). */
796 /* > For optimal efficiency, LWORK >= (NB+6)*N, */
797 /* > where NB is the f2cmax of the blocksize for SSYTRD and SORMTR */
798 /* > returned by ILAENV. */
800 /* > If LWORK = -1, then a workspace query is assumed; the routine */
801 /* > only calculates the optimal sizes of the WORK and IWORK */
802 /* > arrays, returns these values as the first entries of the WORK */
803 /* > and IWORK arrays, and no error message related to LWORK or */
804 /* > LIWORK is issued by XERBLA. */
807 /* > \param[out] IWORK */
809 /* > IWORK is INTEGER array, dimension (MAX(1,LIWORK)) */
810 /* > On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. */
813 /* > \param[in] LIWORK */
815 /* > LIWORK is INTEGER */
816 /* > The dimension of the array IWORK. LIWORK >= f2cmax(1,10*N). */
818 /* > If LIWORK = -1, then a workspace query is assumed; the */
819 /* > routine only calculates the optimal sizes of the WORK and */
820 /* > IWORK arrays, returns these values as the first entries of */
821 /* > the WORK and IWORK arrays, and no error message related to */
822 /* > LWORK or LIWORK is issued by XERBLA. */
825 /* > \param[out] INFO */
827 /* > INFO is INTEGER */
828 /* > = 0: successful exit */
829 /* > < 0: if INFO = -i, the i-th argument had an illegal value */
830 /* > > 0: Internal error */
836 /* > \author Univ. of Tennessee */
837 /* > \author Univ. of California Berkeley */
838 /* > \author Univ. of Colorado Denver */
839 /* > \author NAG Ltd. */
841 /* > \date June 2016 */
843 /* > \ingroup realSYeigen */
845 /* > \par Contributors: */
846 /* ================== */
848 /* > Inderjit Dhillon, IBM Almaden, USA \n */
849 /* > Osni Marques, LBNL/NERSC, USA \n */
850 /* > Ken Stanley, Computer Science Division, University of */
851 /* > California at Berkeley, USA \n */
852 /* > Jason Riedy, Computer Science Division, University of */
853 /* > California at Berkeley, USA \n */
855 /* ===================================================================== */
856 /* Subroutine */ int ssyevr_(char *jobz, char *range, char *uplo, integer *n,
857 real *a, integer *lda, real *vl, real *vu, integer *il, integer *iu,
858 real *abstol, integer *m, real *w, real *z__, integer *ldz, integer *
859 isuppz, real *work, integer *lwork, integer *iwork, integer *liwork,
862 /* System generated locals */
863 integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2;
866 /* Local variables */
872 integer i__, j, inddd, indee;
874 extern logical lsame_(char *, char *);
876 extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
878 integer indwk, lwmin;
880 extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
881 integer *), sswap_(integer *, real *, integer *, real *, integer *
885 logical alleig, indeig;
886 integer iscale, ieeeok, indibl, indifl;
888 extern real slamch_(char *);
890 extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
891 integer *, integer *, ftnlen, ftnlen);
892 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
894 integer indtau, indisp, indiwo, indwkn, liwmin;
896 extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *,
897 real *, integer *, integer *, real *, integer *, real *, integer *
898 , integer *, integer *), ssterf_(integer *, real *, real *,
900 integer llwrkn, llwork, nsplit;
902 extern real slansy_(char *, char *, integer *, real *, integer *, real *);
903 extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
904 real *, integer *, integer *, real *, real *, real *, integer *,
905 integer *, real *, integer *, integer *, real *, integer *,
906 integer *), sstemr_(char *, char *, integer *,
907 real *, real *, real *, real *, integer *, integer *, integer *,
908 real *, real *, integer *, integer *, integer *, logical *, real *
909 , integer *, integer *, integer *, integer *);
912 extern /* Subroutine */ int sormtr_(char *, char *, char *, integer *,
913 integer *, real *, integer *, real *, real *, integer *, real *,
914 integer *, integer *), ssytrd_(char *,
915 integer *, real *, integer *, real *, real *, real *, real *,
916 integer *, integer *);
917 real eps, vll, vuu, tmp1;
920 /* -- LAPACK driver routine (version 3.7.0) -- */
921 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
922 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
926 /* ===================================================================== */
929 /* Test the input parameters. */
931 /* Parameter adjustments */
933 a_offset = 1 + a_dim1 * 1;
937 z_offset = 1 + z_dim1 * 1;
944 ieeeok = ilaenv_(&c__10, "SSYEVR", "N", &c__1, &c__2, &c__3, &c__4, (
945 ftnlen)6, (ftnlen)1);
947 lower = lsame_(uplo, "L");
948 wantz = lsame_(jobz, "V");
949 alleig = lsame_(range, "A");
950 valeig = lsame_(range, "V");
951 indeig = lsame_(range, "I");
953 lquery = *lwork == -1 || *liwork == -1;
956 i__1 = 1, i__2 = *n * 26;
957 lwmin = f2cmax(i__1,i__2);
959 i__1 = 1, i__2 = *n * 10;
960 liwmin = f2cmax(i__1,i__2);
963 if (! (wantz || lsame_(jobz, "N"))) {
965 } else if (! (alleig || valeig || indeig)) {
967 } else if (! (lower || lsame_(uplo, "U"))) {
971 } else if (*lda < f2cmax(1,*n)) {
975 if (*n > 0 && *vu <= *vl) {
979 if (*il < 1 || *il > f2cmax(1,*n)) {
981 } else if (*iu < f2cmin(*n,*il) || *iu > *n) {
987 if (*ldz < 1 || wantz && *ldz < *n) {
993 nb = ilaenv_(&c__1, "SSYTRD", uplo, n, &c_n1, &c_n1, &c_n1, (ftnlen)6,
996 i__1 = nb, i__2 = ilaenv_(&c__1, "SORMTR", uplo, n, &c_n1, &c_n1, &
997 c_n1, (ftnlen)6, (ftnlen)1);
998 nb = f2cmax(i__1,i__2);
1000 i__1 = (nb + 1) * *n;
1001 lwkopt = f2cmax(i__1,lwmin);
1002 work[1] = (real) lwkopt;
1005 if (*lwork < lwmin && ! lquery) {
1007 } else if (*liwork < liwmin && ! lquery) {
1014 xerbla_("SSYEVR", &i__1, (ftnlen)6);
1016 } else if (lquery) {
1020 /* Quick return if possible */
1030 if (alleig || indeig) {
1032 w[1] = a[a_dim1 + 1];
1034 if (*vl < a[a_dim1 + 1] && *vu >= a[a_dim1 + 1]) {
1036 w[1] = a[a_dim1 + 1];
1040 z__[z_dim1 + 1] = 1.f;
1047 /* Get machine constants. */
1049 safmin = slamch_("Safe minimum");
1050 eps = slamch_("Precision");
1051 smlnum = safmin / eps;
1052 bignum = 1.f / smlnum;
1053 rmin = sqrt(smlnum);
1055 r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
1056 rmax = f2cmin(r__1,r__2);
1058 /* Scale matrix to allowable range, if necessary. */
1066 anrm = slansy_("M", uplo, n, &a[a_offset], lda, &work[1]);
1067 if (anrm > 0.f && anrm < rmin) {
1069 sigma = rmin / anrm;
1070 } else if (anrm > rmax) {
1072 sigma = rmax / anrm;
1077 for (j = 1; j <= i__1; ++j) {
1079 sscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1);
1084 for (j = 1; j <= i__1; ++j) {
1085 sscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1);
1089 if (*abstol > 0.f) {
1090 abstll = *abstol * sigma;
1097 /* Initialize indices into workspaces. Note: The IWORK indices are */
1098 /* used only if SSTERF or SSTEMR fail. */
1099 /* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the */
1100 /* elementary reflectors used in SSYTRD. */
1102 /* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries. */
1104 /* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the */
1105 /* tridiagonal matrix from SSYTRD. */
1107 /* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over */
1108 /* -written by SSTEMR (the SSTERF path copies the diagonal to W). */
1110 /* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over */
1111 /* -written while computing the eigenvalues in SSTERF and SSTEMR. */
1113 /* INDWK is the starting offset of the left-over workspace, and */
1114 /* LLWORK is the remaining workspace size. */
1116 llwork = *lwork - indwk + 1;
1117 /* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and */
1118 /* stores the block indices of each of the M<=N eigenvalues. */
1120 /* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and */
1121 /* stores the starting and finishing indices of each block. */
1122 indisp = indibl + *n;
1123 /* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors */
1124 /* that corresponding to eigenvectors that fail to converge in */
1125 /* SSTEIN. This information is discarded; if any fail, the driver */
1126 /* returns INFO > 0. */
1127 indifl = indisp + *n;
1128 /* INDIWO is the offset of the remaining integer workspace. */
1129 indiwo = indifl + *n;
1131 /* Call SSYTRD to reduce symmetric matrix to tridiagonal form. */
1133 ssytrd_(uplo, n, &a[a_offset], lda, &work[indd], &work[inde], &work[
1134 indtau], &work[indwk], &llwork, &iinfo);
1136 /* If all eigenvalues are desired */
1137 /* then call SSTERF or SSTEMR and SORMTR. */
1141 if (*il == 1 && *iu == *n) {
1145 if ((alleig || test) && ieeeok == 1) {
1147 scopy_(n, &work[indd], &c__1, &w[1], &c__1);
1149 scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
1150 ssterf_(n, &w[1], &work[indee], info);
1153 scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
1154 scopy_(n, &work[indd], &c__1, &work[inddd], &c__1);
1156 if (*abstol <= *n * 2.f * eps) {
1161 sstemr_(jobz, "A", n, &work[inddd], &work[indee], vl, vu, il, iu,
1162 m, &w[1], &z__[z_offset], ldz, n, &isuppz[1], &tryrac, &
1163 work[indwk], lwork, &iwork[1], liwork, info);
1167 /* Apply orthogonal matrix used in reduction to tridiagonal */
1168 /* form to eigenvectors returned by SSTEMR. */
1170 if (wantz && *info == 0) {
1172 llwrkn = *lwork - indwkn + 1;
1173 sormtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau]
1174 , &z__[z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
1180 /* Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are */
1188 /* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
1189 /* Also call SSTEBZ and SSTEIN if SSTEMR fails. */
1192 *(unsigned char *)order = 'B';
1194 *(unsigned char *)order = 'E';
1196 sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &work[indd], &work[
1197 inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &work[
1198 indwk], &iwork[indiwo], info);
1201 sstein_(n, &work[indd], &work[inde], m, &w[1], &iwork[indibl], &iwork[
1202 indisp], &z__[z_offset], ldz, &work[indwk], &iwork[indiwo], &
1203 iwork[indifl], info);
1205 /* Apply orthogonal matrix used in reduction to tridiagonal */
1206 /* form to eigenvectors returned by SSTEIN. */
1209 llwrkn = *lwork - indwkn + 1;
1210 sormtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[
1211 z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
1214 /* If matrix was scaled, then rescale eigenvalues appropriately. */
1216 /* Jump here if SSTEMR/SSTEIN succeeded. */
1225 sscal_(&imax, &r__1, &w[1], &c__1);
1228 /* If eigenvalues are not in order, then sort them, along with */
1229 /* eigenvectors. Note: We do not sort the IFAIL portion of IWORK. */
1230 /* It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do */
1231 /* not return this detailed information to the user. */
1235 for (j = 1; j <= i__1; ++j) {
1239 for (jj = j + 1; jj <= i__2; ++jj) {
1250 sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
1257 /* Set WORK(1) to optimal workspace size. */
1259 work[1] = (real) lwkopt;