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]/Cd(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> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for
525 /* @precisions fortran z -> s d c */
527 /* =========== DOCUMENTATION =========== */
529 /* Online html documentation available at */
530 /* http://www.netlib.org/lapack/explore-html/ */
533 /* > Download ZHEEVR_2STAGE + dependencies */
534 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_
537 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevr_
540 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr_
548 /* SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, */
549 /* IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, */
550 /* WORK, LWORK, RWORK, LRWORK, IWORK, */
555 /* CHARACTER JOBZ, RANGE, UPLO */
556 /* INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, */
558 /* DOUBLE PRECISION ABSTOL, VL, VU */
559 /* INTEGER ISUPPZ( * ), IWORK( * ) */
560 /* DOUBLE PRECISION RWORK( * ), W( * ) */
561 /* COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * ) */
564 /* > \par Purpose: */
569 /* > ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors */
570 /* > of a complex Hermitian matrix A using the 2stage technique for */
571 /* > the reduction to tridiagonal. Eigenvalues and eigenvectors can */
572 /* > be selected by specifying either a range of values or a range of */
573 /* > indices for the desired eigenvalues. */
575 /* > ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call */
576 /* > to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute */
577 /* > eigenspectrum using Relatively Robust Representations. ZSTEMR */
578 /* > computes eigenvalues by the dqds algorithm, while orthogonal */
579 /* > eigenvectors are computed from various "good" L D L^T representations */
580 /* > (also known as Relatively Robust Representations). Gram-Schmidt */
581 /* > orthogonalization is avoided as far as possible. More specifically, */
582 /* > the various steps of the algorithm are as follows. */
584 /* > For each unreduced block (submatrix) of T, */
585 /* > (a) Compute T - sigma I = L D L^T, so that L and D */
586 /* > define all the wanted eigenvalues to high relative accuracy. */
587 /* > This means that small relative changes in the entries of D and L */
588 /* > cause only small relative changes in the eigenvalues and */
589 /* > eigenvectors. The standard (unfactored) representation of the */
590 /* > tridiagonal matrix T does not have this property in general. */
591 /* > (b) Compute the eigenvalues to suitable accuracy. */
592 /* > If the eigenvectors are desired, the algorithm attains full */
593 /* > accuracy of the computed eigenvalues only right before */
594 /* > the corresponding vectors have to be computed, see steps c) and d). */
595 /* > (c) For each cluster of close eigenvalues, select a new */
596 /* > shift close to the cluster, find a new factorization, and refine */
597 /* > the shifted eigenvalues to suitable accuracy. */
598 /* > (d) For each eigenvalue with a large enough relative separation compute */
599 /* > the corresponding eigenvector by forming a rank revealing twisted */
600 /* > factorization. Go back to (c) for any clusters that remain. */
602 /* > The desired accuracy of the output can be specified by the input */
603 /* > parameter ABSTOL. */
605 /* > For more details, see DSTEMR's documentation and: */
606 /* > - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */
607 /* > to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */
608 /* > Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */
609 /* > - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */
610 /* > Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */
611 /* > 2004. Also LAPACK Working Note 154. */
612 /* > - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */
613 /* > tridiagonal eigenvalue/eigenvector problem", */
614 /* > Computer Science Division Technical Report No. UCB/CSD-97-971, */
615 /* > UC Berkeley, May 1997. */
618 /* > Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested */
619 /* > on machines which conform to the ieee-754 floating point standard. */
620 /* > ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN on non-ieee machines and */
621 /* > when partial spectrum requests are made. */
623 /* > Normal execution of ZSTEMR may create NaNs and infinities and */
624 /* > hence may abort due to a floating point exception in environments */
625 /* > which do not handle NaNs and infinities in the ieee standard default */
632 /* > \param[in] JOBZ */
634 /* > JOBZ is CHARACTER*1 */
635 /* > = 'N': Compute eigenvalues only; */
636 /* > = 'V': Compute eigenvalues and eigenvectors. */
637 /* > Not available in this release. */
640 /* > \param[in] RANGE */
642 /* > RANGE is CHARACTER*1 */
643 /* > = 'A': all eigenvalues will be found. */
644 /* > = 'V': all eigenvalues in the half-open interval (VL,VU] */
645 /* > will be found. */
646 /* > = 'I': the IL-th through IU-th eigenvalues will be found. */
647 /* > For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and */
648 /* > ZSTEIN are called */
651 /* > \param[in] UPLO */
653 /* > UPLO is CHARACTER*1 */
654 /* > = 'U': Upper triangle of A is stored; */
655 /* > = 'L': Lower triangle of A is stored. */
661 /* > The order of the matrix A. N >= 0. */
664 /* > \param[in,out] A */
666 /* > A is COMPLEX*16 array, dimension (LDA, N) */
667 /* > On entry, the Hermitian matrix A. If UPLO = 'U', the */
668 /* > leading N-by-N upper triangular part of A contains the */
669 /* > upper triangular part of the matrix A. If UPLO = 'L', */
670 /* > the leading N-by-N lower triangular part of A contains */
671 /* > the lower triangular part of the matrix A. */
672 /* > On exit, the lower triangle (if UPLO='L') or the upper */
673 /* > triangle (if UPLO='U') of A, including the diagonal, is */
677 /* > \param[in] LDA */
679 /* > LDA is INTEGER */
680 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
683 /* > \param[in] VL */
685 /* > VL is DOUBLE PRECISION */
686 /* > If RANGE='V', the lower bound of the interval to */
687 /* > be searched for eigenvalues. VL < VU. */
688 /* > Not referenced if RANGE = 'A' or 'I'. */
691 /* > \param[in] VU */
693 /* > VU is DOUBLE PRECISION */
694 /* > If RANGE='V', the upper bound of the interval to */
695 /* > be searched for eigenvalues. VL < VU. */
696 /* > Not referenced if RANGE = 'A' or 'I'. */
699 /* > \param[in] IL */
701 /* > IL is INTEGER */
702 /* > If RANGE='I', the index of the */
703 /* > smallest 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] IU */
710 /* > IU is INTEGER */
711 /* > If RANGE='I', the index of the */
712 /* > largest eigenvalue to be returned. */
713 /* > 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. */
714 /* > Not referenced if RANGE = 'A' or 'V'. */
717 /* > \param[in] ABSTOL */
719 /* > ABSTOL is DOUBLE PRECISION */
720 /* > The absolute error tolerance for the eigenvalues. */
721 /* > An approximate eigenvalue is accepted as converged */
722 /* > when it is determined to lie in an interval [a,b] */
723 /* > of width less than or equal to */
725 /* > ABSTOL + EPS * f2cmax( |a|,|b| ) , */
727 /* > where EPS is the machine precision. If ABSTOL is less than */
728 /* > or equal to zero, then EPS*|T| will be used in its place, */
729 /* > where |T| is the 1-norm of the tridiagonal matrix obtained */
730 /* > by reducing A to tridiagonal form. */
732 /* > See "Computing Small Singular Values of Bidiagonal Matrices */
733 /* > with Guaranteed High Relative Accuracy," by Demmel and */
734 /* > Kahan, LAPACK Working Note #3. */
736 /* > If high relative accuracy is important, set ABSTOL to */
737 /* > DLAMCH( 'Safe minimum' ). Doing so will guarantee that */
738 /* > eigenvalues are computed to high relative accuracy when */
739 /* > possible in future releases. The current code does not */
740 /* > make any guarantees about high relative accuracy, but */
741 /* > future releases will. See J. Barlow and J. Demmel, */
742 /* > "Computing Accurate Eigensystems of Scaled Diagonally */
743 /* > Dominant Matrices", LAPACK Working Note #7, for a discussion */
744 /* > of which matrices define their eigenvalues to high relative */
748 /* > \param[out] M */
751 /* > The total number of eigenvalues found. 0 <= M <= N. */
752 /* > If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */
755 /* > \param[out] W */
757 /* > W is DOUBLE PRECISION array, dimension (N) */
758 /* > The first M elements contain the selected eigenvalues in */
759 /* > ascending order. */
762 /* > \param[out] Z */
764 /* > Z is COMPLEX*16 array, dimension (LDZ, f2cmax(1,M)) */
765 /* > If JOBZ = 'V', then if INFO = 0, the first M columns of Z */
766 /* > contain the orthonormal eigenvectors of the matrix A */
767 /* > corresponding to the selected eigenvalues, with the i-th */
768 /* > column of Z holding the eigenvector associated with W(i). */
769 /* > If JOBZ = 'N', then Z is not referenced. */
770 /* > Note: the user must ensure that at least f2cmax(1,M) columns are */
771 /* > supplied in the array Z; if RANGE = 'V', the exact value of M */
772 /* > is not known in advance and an upper bound must be used. */
775 /* > \param[in] LDZ */
777 /* > LDZ is INTEGER */
778 /* > The leading dimension of the array Z. LDZ >= 1, and if */
779 /* > JOBZ = 'V', LDZ >= f2cmax(1,N). */
782 /* > \param[out] ISUPPZ */
784 /* > ISUPPZ is INTEGER array, dimension ( 2*f2cmax(1,M) ) */
785 /* > The support of the eigenvectors in Z, i.e., the indices */
786 /* > indicating the nonzero elements in Z. The i-th eigenvector */
787 /* > is nonzero only in elements ISUPPZ( 2*i-1 ) through */
788 /* > ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal */
789 /* > matrix). The support of the eigenvectors of A is typically */
790 /* > 1:N because of the unitary transformations applied by ZUNMTR. */
791 /* > Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 */
794 /* > \param[out] WORK */
796 /* > WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) */
797 /* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
800 /* > \param[in] LWORK */
802 /* > LWORK is INTEGER */
803 /* > The dimension of the array WORK. */
804 /* > If JOBZ = 'N' and N > 1, LWORK must be queried. */
805 /* > LWORK = MAX(1, 26*N, dimension) where */
806 /* > dimension = f2cmax(stage1,stage2) + (KD+1)*N + N */
807 /* > = N*KD + N*f2cmax(KD+1,FACTOPTNB) */
808 /* > + f2cmax(2*KD*KD, KD*NTHREADS) */
809 /* > + (KD+1)*N + N */
810 /* > where KD is the blocking size of the reduction, */
811 /* > FACTOPTNB is the blocking used by the QR or LQ */
812 /* > algorithm, usually FACTOPTNB=128 is a good choice */
813 /* > NTHREADS is the number of threads used when */
814 /* > openMP compilation is enabled, otherwise =1. */
815 /* > If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available */
817 /* > If LWORK = -1, then a workspace query is assumed; the routine */
818 /* > only calculates the optimal sizes of the WORK, RWORK and */
819 /* > IWORK arrays, returns these values as the first entries of */
820 /* > the WORK, RWORK and IWORK arrays, and no error message */
821 /* > related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
824 /* > \param[out] RWORK */
826 /* > RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) */
827 /* > On exit, if INFO = 0, RWORK(1) returns the optimal */
828 /* > (and minimal) LRWORK. */
831 /* > \param[in] LRWORK */
833 /* > LRWORK is INTEGER */
834 /* > The length of the array RWORK. LRWORK >= f2cmax(1,24*N). */
836 /* > If LRWORK = -1, then a workspace query is assumed; the */
837 /* > routine only calculates the optimal sizes of the WORK, RWORK */
838 /* > and IWORK arrays, returns these values as the first entries */
839 /* > of the WORK, RWORK and IWORK arrays, and no error message */
840 /* > related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
843 /* > \param[out] IWORK */
845 /* > IWORK is INTEGER array, dimension (MAX(1,LIWORK)) */
846 /* > On exit, if INFO = 0, IWORK(1) returns the optimal */
847 /* > (and minimal) LIWORK. */
850 /* > \param[in] LIWORK */
852 /* > LIWORK is INTEGER */
853 /* > The dimension of the array IWORK. LIWORK >= f2cmax(1,10*N). */
855 /* > If LIWORK = -1, then a workspace query is assumed; the */
856 /* > routine only calculates the optimal sizes of the WORK, RWORK */
857 /* > and IWORK arrays, returns these values as the first entries */
858 /* > of the WORK, RWORK and IWORK arrays, and no error message */
859 /* > related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
862 /* > \param[out] INFO */
864 /* > INFO is INTEGER */
865 /* > = 0: successful exit */
866 /* > < 0: if INFO = -i, the i-th argument had an illegal value */
867 /* > > 0: Internal error */
873 /* > \author Univ. of Tennessee */
874 /* > \author Univ. of California Berkeley */
875 /* > \author Univ. of Colorado Denver */
876 /* > \author NAG Ltd. */
878 /* > \date June 2016 */
880 /* > \ingroup complex16HEeigen */
882 /* > \par Contributors: */
883 /* ================== */
885 /* > Inderjit Dhillon, IBM Almaden, USA \n */
886 /* > Osni Marques, LBNL/NERSC, USA \n */
887 /* > Ken Stanley, Computer Science Division, University of */
888 /* > California at Berkeley, USA \n */
889 /* > Jason Riedy, Computer Science Division, University of */
890 /* > California at Berkeley, USA \n */
892 /* > \par Further Details: */
893 /* ===================== */
897 /* > All details about the 2stage techniques are available in: */
899 /* > Azzam Haidar, Hatem Ltaief, and Jack Dongarra. */
900 /* > Parallel reduction to condensed forms for symmetric eigenvalue problems */
901 /* > using aggregated fine-grained and memory-aware kernels. In Proceedings */
902 /* > of 2011 International Conference for High Performance Computing, */
903 /* > Networking, Storage and Analysis (SC '11), New York, NY, USA, */
904 /* > Article 8 , 11 pages. */
905 /* > http://doi.acm.org/10.1145/2063384.2063394 */
907 /* > A. Haidar, J. Kurzak, P. Luszczek, 2013. */
908 /* > An improved parallel singular value algorithm and its implementation */
909 /* > for multicore hardware, In Proceedings of 2013 International Conference */
910 /* > for High Performance Computing, Networking, Storage and Analysis (SC '13). */
911 /* > Denver, Colorado, USA, 2013. */
912 /* > Article 90, 12 pages. */
913 /* > http://doi.acm.org/10.1145/2503210.2503292 */
915 /* > A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. */
916 /* > A novel hybrid CPU-GPU generalized eigensolver for electronic structure */
917 /* > calculations based on fine-grained memory aware tasks. */
918 /* > International Journal of High Performance Computing Applications. */
919 /* > Volume 28 Issue 2, Pages 196-209, May 2014. */
920 /* > http://hpc.sagepub.com/content/28/2/196 */
924 /* ===================================================================== */
925 /* Subroutine */ int zheevr_2stage_(char *jobz, char *range, char *uplo,
926 integer *n, doublecomplex *a, integer *lda, doublereal *vl,
927 doublereal *vu, integer *il, integer *iu, doublereal *abstol, integer
928 *m, doublereal *w, doublecomplex *z__, integer *ldz, integer *isuppz,
929 doublecomplex *work, integer *lwork, doublereal *rwork, integer *
930 lrwork, integer *iwork, integer *liwork, integer *info)
932 /* System generated locals */
933 integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2;
934 doublereal d__1, d__2;
936 /* Local variables */
937 extern integer ilaenv2stage_(integer *, char *, char *, integer *,
938 integer *, integer *, integer *);
941 doublereal rmin, rmax;
943 integer itmp1, i__, j;
944 extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
946 integer indrd, indre;
948 extern logical lsame_(char *, char *);
950 extern /* Subroutine */ int zhetrd_2stage_(char *, char *, integer *,
951 doublecomplex *, integer *, doublereal *, doublereal *,
952 doublecomplex *, doublecomplex *, integer *, doublecomplex *,
953 integer *, integer *);
955 integer indwk, lhtrd;
956 extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
957 doublereal *, integer *);
962 extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *,
963 doublecomplex *, integer *);
965 extern doublereal dlamch_(char *);
966 logical alleig, indeig;
967 integer iscale, ieeeok, indibl, indrdd, indifl, indree;
970 extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
971 integer *, integer *, ftnlen, ftnlen);
972 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen), zdscal_(
973 integer *, doublereal *, doublecomplex *, integer *);
974 doublereal abstll, bignum;
975 integer indtau, indisp;
976 extern /* Subroutine */ int dsterf_(integer *, doublereal *, doublereal *,
978 integer indiwo, indwkn;
979 extern /* Subroutine */ int dstebz_(char *, char *, integer *, doublereal
980 *, doublereal *, integer *, integer *, doublereal *, doublereal *,
981 doublereal *, integer *, integer *, doublereal *, integer *,
982 integer *, doublereal *, integer *, integer *);
983 integer indrwk, liwmin;
985 integer lrwmin, llwrkn, llwork, nsplit;
987 extern /* Subroutine */ int zstein_(integer *, doublereal *, doublereal *,
988 integer *, doublereal *, integer *, integer *, doublecomplex *,
989 integer *, doublereal *, integer *, integer *, integer *);
991 extern doublereal zlansy_(char *, char *, integer *, doublecomplex *,
992 integer *, doublereal *);
993 extern /* Subroutine */ int zstemr_(char *, char *, integer *, doublereal
994 *, doublereal *, doublereal *, doublereal *, integer *, integer *,
995 integer *, doublereal *, doublecomplex *, integer *, integer *,
996 integer *, logical *, doublereal *, integer *, integer *, integer
997 *, integer *), zunmtr_(char *, char *, char *,
998 integer *, integer *, doublecomplex *, integer *, doublecomplex *,
999 doublecomplex *, integer *, doublecomplex *, integer *, integer *
1001 doublereal eps, vll, vuu;
1002 integer indhous, llrwork;
1007 /* -- LAPACK driver routine (version 3.8.0) -- */
1008 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
1009 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
1013 /* ===================================================================== */
1016 /* Test the input parameters. */
1018 /* Parameter adjustments */
1020 a_offset = 1 + a_dim1 * 1;
1024 z_offset = 1 + z_dim1 * 1;
1032 ieeeok = ilaenv_(&c__10, "ZHEEVR", "N", &c__1, &c__2, &c__3, &c__4, (
1033 ftnlen)6, (ftnlen)1);
1035 lower = lsame_(uplo, "L");
1036 wantz = lsame_(jobz, "V");
1037 alleig = lsame_(range, "A");
1038 valeig = lsame_(range, "V");
1039 indeig = lsame_(range, "I");
1041 lquery = *lwork == -1 || *lrwork == -1 || *liwork == -1;
1043 kd = ilaenv2stage_(&c__1, "ZHETRD_2STAGE", jobz, n, &c_n1, &c_n1, &c_n1);
1044 ib = ilaenv2stage_(&c__2, "ZHETRD_2STAGE", jobz, n, &kd, &c_n1, &c_n1);
1045 lhtrd = ilaenv2stage_(&c__3, "ZHETRD_2STAGE", jobz, n, &kd, &ib, &c_n1);
1046 lwtrd = ilaenv2stage_(&c__4, "ZHETRD_2STAGE", jobz, n, &kd, &ib, &c_n1);
1047 lwmin = *n + lhtrd + lwtrd;
1049 i__1 = 1, i__2 = *n * 24;
1050 lrwmin = f2cmax(i__1,i__2);
1052 i__1 = 1, i__2 = *n * 10;
1053 liwmin = f2cmax(i__1,i__2);
1056 if (! lsame_(jobz, "N")) {
1058 } else if (! (alleig || valeig || indeig)) {
1060 } else if (! (lower || lsame_(uplo, "U"))) {
1062 } else if (*n < 0) {
1064 } else if (*lda < f2cmax(1,*n)) {
1068 if (*n > 0 && *vu <= *vl) {
1071 } else if (indeig) {
1072 if (*il < 1 || *il > f2cmax(1,*n)) {
1074 } else if (*iu < f2cmin(*n,*il) || *iu > *n) {
1080 if (*ldz < 1 || wantz && *ldz < *n) {
1086 work[1].r = (doublereal) lwmin, work[1].i = 0.;
1087 rwork[1] = (doublereal) lrwmin;
1090 if (*lwork < lwmin && ! lquery) {
1092 } else if (*lrwork < lrwmin && ! lquery) {
1094 } else if (*liwork < liwmin && ! lquery) {
1101 xerbla_("ZHEEVR_2STAGE", &i__1, (ftnlen)13);
1103 } else if (lquery) {
1107 /* Quick return if possible */
1111 work[1].r = 1., work[1].i = 0.;
1116 work[1].r = 2., work[1].i = 0.;
1117 if (alleig || indeig) {
1124 if (*vl < a[i__1].r && *vu >= a[i__2].r) {
1132 z__[i__1].r = 1., z__[i__1].i = 0.;
1139 /* Get machine constants. */
1141 safmin = dlamch_("Safe minimum");
1142 eps = dlamch_("Precision");
1143 smlnum = safmin / eps;
1144 bignum = 1. / smlnum;
1145 rmin = sqrt(smlnum);
1147 d__1 = sqrt(bignum), d__2 = 1. / sqrt(sqrt(safmin));
1148 rmax = f2cmin(d__1,d__2);
1150 /* Scale matrix to allowable range, if necessary. */
1158 anrm = zlansy_("M", uplo, n, &a[a_offset], lda, &rwork[1]);
1159 if (anrm > 0. && anrm < rmin) {
1161 sigma = rmin / anrm;
1162 } else if (anrm > rmax) {
1164 sigma = rmax / anrm;
1169 for (j = 1; j <= i__1; ++j) {
1171 zdscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1);
1176 for (j = 1; j <= i__1; ++j) {
1177 zdscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1);
1182 abstll = *abstol * sigma;
1189 /* Initialize indices into workspaces. Note: The IWORK indices are */
1190 /* used only if DSTERF or ZSTEMR fail. */
1191 /* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the */
1192 /* elementary reflectors used in ZHETRD. */
1194 /* INDWK is the starting offset of the remaining complex workspace, */
1195 /* and LLWORK is the remaining complex workspace size. */
1196 indhous = indtau + *n;
1197 indwk = indhous + lhtrd;
1198 llwork = *lwork - indwk + 1;
1199 /* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal */
1202 /* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the */
1203 /* tridiagonal matrix from ZHETRD. */
1205 /* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over */
1206 /* -written by ZSTEMR (the DSTERF path copies the diagonal to W). */
1207 indrdd = indre + *n;
1208 /* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over */
1209 /* -written while computing the eigenvalues in DSTERF and ZSTEMR. */
1210 indree = indrdd + *n;
1211 /* INDRWK is the starting offset of the left-over real workspace, and */
1212 /* LLRWORK is the remaining workspace size. */
1213 indrwk = indree + *n;
1214 llrwork = *lrwork - indrwk + 1;
1215 /* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and */
1216 /* stores the block indices of each of the M<=N eigenvalues. */
1218 /* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and */
1219 /* stores the starting and finishing indices of each block. */
1220 indisp = indibl + *n;
1221 /* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors */
1222 /* that corresponding to eigenvectors that fail to converge in */
1223 /* ZSTEIN. This information is discarded; if any fail, the driver */
1224 /* returns INFO > 0. */
1225 indifl = indisp + *n;
1226 /* INDIWO is the offset of the remaining integer workspace. */
1227 indiwo = indifl + *n;
1229 /* Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form. */
1231 zhetrd_2stage_(jobz, uplo, n, &a[a_offset], lda, &rwork[indrd], &rwork[
1232 indre], &work[indtau], &work[indhous], &lhtrd, &work[indwk], &
1235 /* If all eigenvalues are desired */
1236 /* then call DSTERF or ZSTEMR and ZUNMTR. */
1240 if (*il == 1 && *iu == *n) {
1244 if ((alleig || test) && ieeeok == 1) {
1246 dcopy_(n, &rwork[indrd], &c__1, &w[1], &c__1);
1248 dcopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1);
1249 dsterf_(n, &w[1], &rwork[indree], info);
1252 dcopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1);
1253 dcopy_(n, &rwork[indrd], &c__1, &rwork[indrdd], &c__1);
1255 if (*abstol <= *n * 2. * eps) {
1260 zstemr_(jobz, "A", n, &rwork[indrdd], &rwork[indree], vl, vu, il,
1261 iu, m, &w[1], &z__[z_offset], ldz, n, &isuppz[1], &tryrac,
1262 &rwork[indrwk], &llrwork, &iwork[1], liwork, info);
1264 /* Apply unitary matrix used in reduction to tridiagonal */
1265 /* form to eigenvectors returned by ZSTEMR. */
1267 if (wantz && *info == 0) {
1269 llwrkn = *lwork - indwkn + 1;
1270 zunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau]
1271 , &z__[z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
1283 /* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. */
1284 /* Also call DSTEBZ and ZSTEIN if ZSTEMR fails. */
1287 *(unsigned char *)order = 'B';
1289 *(unsigned char *)order = 'E';
1291 dstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &rwork[indrd], &
1292 rwork[indre], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &
1293 rwork[indrwk], &iwork[indiwo], info);
1296 zstein_(n, &rwork[indrd], &rwork[indre], m, &w[1], &iwork[indibl], &
1297 iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[
1298 indiwo], &iwork[indifl], info);
1300 /* Apply unitary matrix used in reduction to tridiagonal */
1301 /* form to eigenvectors returned by ZSTEIN. */
1304 llwrkn = *lwork - indwkn + 1;
1305 zunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[
1306 z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
1309 /* If matrix was scaled, then rescale eigenvalues appropriately. */
1319 dscal_(&imax, &d__1, &w[1], &c__1);
1322 /* If eigenvalues are not in order, then sort them, along with */
1327 for (j = 1; j <= i__1; ++j) {
1331 for (jj = j + 1; jj <= i__2; ++jj) {
1340 itmp1 = iwork[indibl + i__ - 1];
1342 iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
1344 iwork[indibl + j - 1] = itmp1;
1345 zswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
1352 /* Set WORK(1) to optimal workspace size. */
1354 work[1].r = (doublereal) lwmin, work[1].i = 0.;
1355 rwork[1] = (doublereal) lrwmin;
1360 /* End of ZHEEVR_2STAGE */
1362 } /* zheevr_2stage__ */