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__1 = 1;
516 static integer c_n1 = -1;
517 static doublereal c_b27 = 1.;
518 static doublereal c_b38 = 0.;
520 /* > \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE mat
523 /* =========== DOCUMENTATION =========== */
525 /* Online html documentation available at */
526 /* http://www.netlib.org/lapack/explore-html/ */
529 /* > Download DGEGV + dependencies */
530 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f
533 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f
536 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f
544 /* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, */
545 /* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) */
547 /* CHARACTER JOBVL, JOBVR */
548 /* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N */
549 /* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), */
550 /* $ B( LDB, * ), BETA( * ), VL( LDVL, * ), */
551 /* $ VR( LDVR, * ), WORK( * ) */
554 /* > \par Purpose: */
559 /* > This routine is deprecated and has been replaced by routine DGGEV. */
561 /* > DGEGV computes the eigenvalues and, optionally, the left and/or right */
562 /* > eigenvectors of a real matrix pair (A,B). */
563 /* > Given two square matrices A and B, */
564 /* > the generalized nonsymmetric eigenvalue problem (GNEP) is to find the */
565 /* > eigenvalues lambda and corresponding (non-zero) eigenvectors x such */
568 /* > A*x = lambda*B*x. */
570 /* > An alternate form is to find the eigenvalues mu and corresponding */
571 /* > eigenvectors y such that */
573 /* > mu*A*y = B*y. */
575 /* > These two forms are equivalent with mu = 1/lambda and x = y if */
576 /* > neither lambda nor mu is zero. In order to deal with the case that */
577 /* > lambda or mu is zero or small, two values alpha and beta are returned */
578 /* > for each eigenvalue, such that lambda = alpha/beta and */
579 /* > mu = beta/alpha. */
581 /* > The vectors x and y in the above equations are right eigenvectors of */
582 /* > the matrix pair (A,B). Vectors u and v satisfying */
584 /* > u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B */
586 /* > are left eigenvectors of (A,B). */
588 /* > Note: this routine performs "full balancing" on A and B */
594 /* > \param[in] JOBVL */
596 /* > JOBVL is CHARACTER*1 */
597 /* > = 'N': do not compute the left generalized eigenvectors; */
598 /* > = 'V': compute the left generalized eigenvectors (returned */
602 /* > \param[in] JOBVR */
604 /* > JOBVR is CHARACTER*1 */
605 /* > = 'N': do not compute the right generalized eigenvectors; */
606 /* > = 'V': compute the right generalized eigenvectors (returned */
613 /* > The order of the matrices A, B, VL, and VR. N >= 0. */
616 /* > \param[in,out] A */
618 /* > A is DOUBLE PRECISION array, dimension (LDA, N) */
619 /* > On entry, the matrix A. */
620 /* > If JOBVL = 'V' or JOBVR = 'V', then on exit A */
621 /* > contains the real Schur form of A from the generalized Schur */
622 /* > factorization of the pair (A,B) after balancing. */
623 /* > If no eigenvectors were computed, then only the diagonal */
624 /* > blocks from the Schur form will be correct. See DGGHRD and */
625 /* > DHGEQZ for details. */
628 /* > \param[in] LDA */
630 /* > LDA is INTEGER */
631 /* > The leading dimension of A. LDA >= f2cmax(1,N). */
634 /* > \param[in,out] B */
636 /* > B is DOUBLE PRECISION array, dimension (LDB, N) */
637 /* > On entry, the matrix B. */
638 /* > If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the */
639 /* > upper triangular matrix obtained from B in the generalized */
640 /* > Schur factorization of the pair (A,B) after balancing. */
641 /* > If no eigenvectors were computed, then only those elements of */
642 /* > B corresponding to the diagonal blocks from the Schur form of */
643 /* > A will be correct. See DGGHRD and DHGEQZ for details. */
646 /* > \param[in] LDB */
648 /* > LDB is INTEGER */
649 /* > The leading dimension of B. LDB >= f2cmax(1,N). */
652 /* > \param[out] ALPHAR */
654 /* > ALPHAR is DOUBLE PRECISION array, dimension (N) */
655 /* > The real parts of each scalar alpha defining an eigenvalue of */
659 /* > \param[out] ALPHAI */
661 /* > ALPHAI is DOUBLE PRECISION array, dimension (N) */
662 /* > The imaginary parts of each scalar alpha defining an */
663 /* > eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th */
664 /* > eigenvalue is real; if positive, then the j-th and */
665 /* > (j+1)-st eigenvalues are a complex conjugate pair, with */
666 /* > ALPHAI(j+1) = -ALPHAI(j). */
669 /* > \param[out] BETA */
671 /* > BETA is DOUBLE PRECISION array, dimension (N) */
672 /* > The scalars beta that define the eigenvalues of GNEP. */
674 /* > Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and */
675 /* > beta = BETA(j) represent the j-th eigenvalue of the matrix */
676 /* > pair (A,B), in one of the forms lambda = alpha/beta or */
677 /* > mu = beta/alpha. Since either lambda or mu may overflow, */
678 /* > they should not, in general, be computed. */
681 /* > \param[out] VL */
683 /* > VL is DOUBLE PRECISION array, dimension (LDVL,N) */
684 /* > If JOBVL = 'V', the left eigenvectors u(j) are stored */
685 /* > in the columns of VL, in the same order as their eigenvalues. */
686 /* > If the j-th eigenvalue is real, then u(j) = VL(:,j). */
687 /* > If the j-th and (j+1)-st eigenvalues form a complex conjugate */
689 /* > u(j) = VL(:,j) + i*VL(:,j+1) */
691 /* > u(j+1) = VL(:,j) - i*VL(:,j+1). */
693 /* > Each eigenvector is scaled so that its largest component has */
694 /* > abs(real part) + abs(imag. part) = 1, except for eigenvectors */
695 /* > corresponding to an eigenvalue with alpha = beta = 0, which */
696 /* > are set to zero. */
697 /* > Not referenced if JOBVL = 'N'. */
700 /* > \param[in] LDVL */
702 /* > LDVL is INTEGER */
703 /* > The leading dimension of the matrix VL. LDVL >= 1, and */
704 /* > if JOBVL = 'V', LDVL >= N. */
707 /* > \param[out] VR */
709 /* > VR is DOUBLE PRECISION array, dimension (LDVR,N) */
710 /* > If JOBVR = 'V', the right eigenvectors x(j) are stored */
711 /* > in the columns of VR, in the same order as their eigenvalues. */
712 /* > If the j-th eigenvalue is real, then x(j) = VR(:,j). */
713 /* > If the j-th and (j+1)-st eigenvalues form a complex conjugate */
715 /* > x(j) = VR(:,j) + i*VR(:,j+1) */
717 /* > x(j+1) = VR(:,j) - i*VR(:,j+1). */
719 /* > Each eigenvector is scaled so that its largest component has */
720 /* > abs(real part) + abs(imag. part) = 1, except for eigenvalues */
721 /* > corresponding to an eigenvalue with alpha = beta = 0, which */
722 /* > are set to zero. */
723 /* > Not referenced if JOBVR = 'N'. */
726 /* > \param[in] LDVR */
728 /* > LDVR is INTEGER */
729 /* > The leading dimension of the matrix VR. LDVR >= 1, and */
730 /* > if JOBVR = 'V', LDVR >= N. */
733 /* > \param[out] WORK */
735 /* > WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
736 /* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
739 /* > \param[in] LWORK */
741 /* > LWORK is INTEGER */
742 /* > The dimension of the array WORK. LWORK >= f2cmax(1,8*N). */
743 /* > For good performance, LWORK must generally be larger. */
744 /* > To compute the optimal value of LWORK, call ILAENV to get */
745 /* > blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: */
746 /* > NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; */
747 /* > The optimal LWORK is: */
748 /* > 2*N + MAX( 6*N, N*(NB+1) ). */
750 /* > If LWORK = -1, then a workspace query is assumed; the routine */
751 /* > only calculates the optimal size of the WORK array, returns */
752 /* > this value as the first entry of the WORK array, and no error */
753 /* > message related to LWORK is issued by XERBLA. */
756 /* > \param[out] INFO */
758 /* > INFO is INTEGER */
759 /* > = 0: successful exit */
760 /* > < 0: if INFO = -i, the i-th argument had an illegal value. */
762 /* > The QZ iteration failed. No eigenvectors have been */
763 /* > calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) */
764 /* > should be correct for j=INFO+1,...,N. */
765 /* > > N: errors that usually indicate LAPACK problems: */
766 /* > =N+1: error return from DGGBAL */
767 /* > =N+2: error return from DGEQRF */
768 /* > =N+3: error return from DORMQR */
769 /* > =N+4: error return from DORGQR */
770 /* > =N+5: error return from DGGHRD */
771 /* > =N+6: error return from DHGEQZ (other than failed */
773 /* > =N+7: error return from DTGEVC */
774 /* > =N+8: error return from DGGBAK (computing VL) */
775 /* > =N+9: error return from DGGBAK (computing VR) */
776 /* > =N+10: error return from DLASCL (various calls) */
782 /* > \author Univ. of Tennessee */
783 /* > \author Univ. of California Berkeley */
784 /* > \author Univ. of Colorado Denver */
785 /* > \author NAG Ltd. */
787 /* > \date December 2016 */
789 /* > \ingroup doubleGEeigen */
791 /* > \par Further Details: */
792 /* ===================== */
799 /* > This driver calls DGGBAL to both permute and scale rows and columns */
800 /* > of A and B. The permutations PL and PR are chosen so that PL*A*PR */
801 /* > and PL*B*R will be upper triangular except for the diagonal blocks */
802 /* > A(i:j,i:j) and B(i:j,i:j), with i and j as close together as */
803 /* > possible. The diagonal scaling matrices DL and DR are chosen so */
804 /* > that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to */
805 /* > one (except for the elements that start out zero.) */
807 /* > After the eigenvalues and eigenvectors of the balanced matrices */
808 /* > have been computed, DGGBAK transforms the eigenvectors back to what */
809 /* > they would have been (in perfect arithmetic) if they had not been */
812 /* > Contents of A and B on Exit */
813 /* > -------- -- - --- - -- ---- */
815 /* > If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or */
816 /* > both), then on exit the arrays A and B will contain the real Schur */
817 /* > form[*] of the "balanced" versions of A and B. If no eigenvectors */
818 /* > are computed, then only the diagonal blocks will be correct. */
820 /* > [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", */
821 /* > by Golub & van Loan, pub. by Johns Hopkins U. Press. */
824 /* ===================================================================== */
825 /* Subroutine */ int dgegv_(char *jobvl, char *jobvr, integer *n, doublereal *
826 a, integer *lda, doublereal *b, integer *ldb, doublereal *alphar,
827 doublereal *alphai, doublereal *beta, doublereal *vl, integer *ldvl,
828 doublereal *vr, integer *ldvr, doublereal *work, integer *lwork,
831 /* System generated locals */
832 integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
833 vr_offset, i__1, i__2;
834 doublereal d__1, d__2, d__3, d__4;
836 /* Local variables */
837 doublereal absb, anrm, bnrm;
842 doublereal anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta;
843 extern logical lsame_(char *, char *);
844 integer ileft, iinfo, icols, iwork, irows, jc;
845 extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
846 integer *, doublereal *, doublereal *, integer *, doublereal *,
847 integer *, integer *);
849 extern /* Subroutine */ int dggbal_(char *, integer *, doublereal *,
850 integer *, doublereal *, integer *, integer *, integer *,
851 doublereal *, doublereal *, doublereal *, integer *);
853 extern doublereal dlamch_(char *), dlange_(char *, integer *,
854 integer *, doublereal *, integer *, doublereal *);
857 extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
858 integer *, doublereal *, integer *, doublereal *, integer *,
859 doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal
860 *, doublereal *, integer *, integer *, doublereal *, integer *,
863 extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
864 integer *, doublereal *, doublereal *, integer *, integer *),
865 dlacpy_(char *, integer *, integer *, doublereal *, integer *,
866 doublereal *, integer *);
868 extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
869 doublereal *, doublereal *, doublereal *, integer *);
873 extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
874 integer *, integer *, doublereal *, integer *, doublereal *,
875 integer *, doublereal *, doublereal *, doublereal *, doublereal *,
876 integer *, doublereal *, integer *, doublereal *, integer *,
877 integer *), dtgevc_(char *, char *,
878 logical *, integer *, doublereal *, integer *, doublereal *,
879 integer *, doublereal *, integer *, doublereal *, integer *,
880 integer *, integer *, doublereal *, integer *),
881 xerbla_(char *, integer *);
882 integer ijobvl, iright;
884 extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
885 integer *, integer *, ftnlen, ftnlen);
887 extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
888 doublereal *, integer *, doublereal *, doublereal *, integer *,
891 integer lwkmin, nb1, nb2, nb3;
892 extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
893 integer *, doublereal *, integer *, doublereal *, doublereal *,
894 integer *, doublereal *, integer *, integer *);
902 /* -- LAPACK driver routine (version 3.7.0) -- */
903 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
904 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
908 /* ===================================================================== */
911 /* Decode the input arguments */
913 /* Parameter adjustments */
915 a_offset = 1 + a_dim1 * 1;
918 b_offset = 1 + b_dim1 * 1;
924 vl_offset = 1 + vl_dim1 * 1;
927 vr_offset = 1 + vr_dim1 * 1;
932 if (lsame_(jobvl, "N")) {
935 } else if (lsame_(jobvl, "V")) {
943 if (lsame_(jobvr, "N")) {
946 } else if (lsame_(jobvr, "V")) {
955 /* Test the input arguments */
959 lwkmin = f2cmax(i__1,1);
961 work[1] = (doublereal) lwkopt;
962 lquery = *lwork == -1;
966 } else if (ijobvr <= 0) {
970 } else if (*lda < f2cmax(1,*n)) {
972 } else if (*ldb < f2cmax(1,*n)) {
974 } else if (*ldvl < 1 || ilvl && *ldvl < *n) {
976 } else if (*ldvr < 1 || ilvr && *ldvr < *n) {
978 } else if (*lwork < lwkmin && ! lquery) {
983 nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, (
985 nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1, (ftnlen)6, (
987 nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1, (ftnlen)6, (
990 i__1 = f2cmax(nb1,nb2);
991 nb = f2cmax(i__1,nb3);
993 i__1 = *n * 6, i__2 = *n * (nb + 1);
994 lopt = (*n << 1) + f2cmax(i__1,i__2);
995 work[1] = (doublereal) lopt;
1000 xerbla_("DGEGV ", &i__1);
1002 } else if (lquery) {
1006 /* Quick return if possible */
1012 /* Get machine constants */
1014 eps = dlamch_("E") * dlamch_("B");
1015 safmin = dlamch_("S");
1017 safmax = 1. / safmin;
1018 onepls = eps * 4 + 1.;
1022 anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
1026 if (safmax * anrm < 1.) {
1028 anrm2 = safmax * anrm;
1033 dlascl_("G", &c_n1, &c_n1, &anrm, &c_b27, n, n, &a[a_offset], lda, &
1043 bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
1047 if (safmax * bnrm < 1.) {
1049 bnrm2 = safmax * bnrm;
1054 dlascl_("G", &c_n1, &c_n1, &bnrm, &c_b27, n, n, &b[b_offset], ldb, &
1062 /* Permute the matrix to make it more nearly triangular */
1063 /* Workspace layout: (8*N words -- "work" requires 6*N words) */
1064 /* left_permutation, right_permutation, work... */
1068 iwork = iright + *n;
1069 dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
1070 ileft], &work[iright], &work[iwork], &iinfo);
1076 /* Reduce B to triangular form, and initialize VL and/or VR */
1077 /* Workspace layout: ("work..." must have at least N words) */
1078 /* left_permutation, right_permutation, tau, work... */
1080 irows = ihi + 1 - ilo;
1082 icols = *n + 1 - ilo;
1087 iwork = itau + irows;
1088 i__1 = *lwork + 1 - iwork;
1089 dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
1090 iwork], &i__1, &iinfo);
1093 i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
1094 lwkopt = f2cmax(i__1,i__2);
1101 i__1 = *lwork + 1 - iwork;
1102 dormqr_("L", "T", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
1103 work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwork], &i__1, &
1107 i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
1108 lwkopt = f2cmax(i__1,i__2);
1116 dlaset_("Full", n, n, &c_b38, &c_b27, &vl[vl_offset], ldvl)
1120 dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo +
1121 1 + ilo * vl_dim1], ldvl);
1122 i__1 = *lwork + 1 - iwork;
1123 dorgqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[
1124 itau], &work[iwork], &i__1, &iinfo);
1127 i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
1128 lwkopt = f2cmax(i__1,i__2);
1137 dlaset_("Full", n, n, &c_b38, &c_b27, &vr[vr_offset], ldvr)
1141 /* Reduce to generalized Hessenberg form */
1145 /* Eigenvectors requested -- work on whole matrix. */
1147 dgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
1148 ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo);
1150 dgghrd_("N", "N", &irows, &c__1, &irows, &a[ilo + ilo * a_dim1], lda,
1151 &b[ilo + ilo * b_dim1], ldb, &vl[vl_offset], ldvl, &vr[
1152 vr_offset], ldvr, &iinfo);
1159 /* Perform QZ algorithm */
1160 /* Workspace layout: ("work..." must have at least 1 word) */
1161 /* left_permutation, right_permutation, work... */
1165 *(unsigned char *)chtemp = 'S';
1167 *(unsigned char *)chtemp = 'E';
1169 i__1 = *lwork + 1 - iwork;
1170 dhgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
1171 b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset],
1172 ldvl, &vr[vr_offset], ldvr, &work[iwork], &i__1, &iinfo);
1175 i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
1176 lwkopt = f2cmax(i__1,i__2);
1179 if (iinfo > 0 && iinfo <= *n) {
1181 } else if (iinfo > *n && iinfo <= *n << 1) {
1191 /* Compute Eigenvectors (DTGEVC requires 6*N words of workspace) */
1195 *(unsigned char *)chtemp = 'B';
1197 *(unsigned char *)chtemp = 'L';
1200 *(unsigned char *)chtemp = 'R';
1203 dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
1204 &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
1211 /* Undo balancing on VL and VR, rescale */
1214 dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
1215 vl[vl_offset], ldvl, &iinfo);
1221 for (jc = 1; jc <= i__1; ++jc) {
1222 if (alphai[jc] < 0.) {
1226 if (alphai[jc] == 0.) {
1228 for (jr = 1; jr <= i__2; ++jr) {
1230 d__2 = temp, d__3 = (d__1 = vl[jr + jc * vl_dim1],
1232 temp = f2cmax(d__2,d__3);
1237 for (jr = 1; jr <= i__2; ++jr) {
1239 d__3 = temp, d__4 = (d__1 = vl[jr + jc * vl_dim1],
1240 abs(d__1)) + (d__2 = vl[jr + (jc + 1) *
1241 vl_dim1], abs(d__2));
1242 temp = f2cmax(d__3,d__4);
1246 if (temp < safmin) {
1250 if (alphai[jc] == 0.) {
1252 for (jr = 1; jr <= i__2; ++jr) {
1253 vl[jr + jc * vl_dim1] *= temp;
1258 for (jr = 1; jr <= i__2; ++jr) {
1259 vl[jr + jc * vl_dim1] *= temp;
1260 vl[jr + (jc + 1) * vl_dim1] *= temp;
1269 dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
1270 vr[vr_offset], ldvr, &iinfo);
1276 for (jc = 1; jc <= i__1; ++jc) {
1277 if (alphai[jc] < 0.) {
1281 if (alphai[jc] == 0.) {
1283 for (jr = 1; jr <= i__2; ++jr) {
1285 d__2 = temp, d__3 = (d__1 = vr[jr + jc * vr_dim1],
1287 temp = f2cmax(d__2,d__3);
1292 for (jr = 1; jr <= i__2; ++jr) {
1294 d__3 = temp, d__4 = (d__1 = vr[jr + jc * vr_dim1],
1295 abs(d__1)) + (d__2 = vr[jr + (jc + 1) *
1296 vr_dim1], abs(d__2));
1297 temp = f2cmax(d__3,d__4);
1301 if (temp < safmin) {
1305 if (alphai[jc] == 0.) {
1307 for (jr = 1; jr <= i__2; ++jr) {
1308 vr[jr + jc * vr_dim1] *= temp;
1313 for (jr = 1; jr <= i__2; ++jr) {
1314 vr[jr + jc * vr_dim1] *= temp;
1315 vr[jr + (jc + 1) * vr_dim1] *= temp;
1324 /* End of eigenvector calculation */
1328 /* Undo scaling in alpha, beta */
1330 /* Note: this does not give the alpha and beta for the unscaled */
1333 /* Un-scaling is limited to avoid underflow in alpha and beta */
1334 /* if they are significant. */
1337 for (jc = 1; jc <= i__1; ++jc) {
1338 absar = (d__1 = alphar[jc], abs(d__1));
1339 absai = (d__1 = alphai[jc], abs(d__1));
1340 absb = (d__1 = beta[jc], abs(d__1));
1341 salfar = anrm * alphar[jc];
1342 salfai = anrm * alphai[jc];
1343 sbeta = bnrm * beta[jc];
1347 /* Check for significant underflow in ALPHAI */
1350 d__1 = safmin, d__2 = eps * absar, d__1 = f2cmax(d__1,d__2), d__2 = eps *
1352 if (abs(salfai) < safmin && absai >= f2cmax(d__1,d__2)) {
1355 d__1 = onepls * safmin, d__2 = anrm2 * absai;
1356 scale = onepls * safmin / anrm1 / f2cmax(d__1,d__2);
1358 } else if (salfai == 0.) {
1360 /* If insignificant underflow in ALPHAI, then make the */
1361 /* conjugate eigenvalue real. */
1363 if (alphai[jc] < 0. && jc > 1) {
1364 alphai[jc - 1] = 0.;
1365 } else if (alphai[jc] > 0. && jc < *n) {
1366 alphai[jc + 1] = 0.;
1370 /* Check for significant underflow in ALPHAR */
1373 d__1 = safmin, d__2 = eps * absai, d__1 = f2cmax(d__1,d__2), d__2 = eps *
1375 if (abs(salfar) < safmin && absar >= f2cmax(d__1,d__2)) {
1379 d__3 = onepls * safmin, d__4 = anrm2 * absar;
1380 d__1 = scale, d__2 = onepls * safmin / anrm1 / f2cmax(d__3,d__4);
1381 scale = f2cmax(d__1,d__2);
1384 /* Check for significant underflow in BETA */
1387 d__1 = safmin, d__2 = eps * absar, d__1 = f2cmax(d__1,d__2), d__2 = eps *
1389 if (abs(sbeta) < safmin && absb >= f2cmax(d__1,d__2)) {
1393 d__3 = onepls * safmin, d__4 = bnrm2 * absb;
1394 d__1 = scale, d__2 = onepls * safmin / bnrm1 / f2cmax(d__3,d__4);
1395 scale = f2cmax(d__1,d__2);
1398 /* Check for possible overflow when limiting scaling */
1402 d__1 = abs(salfar), d__2 = abs(salfai), d__1 = f2cmax(d__1,d__2),
1404 temp = scale * safmin * f2cmax(d__1,d__2);
1413 /* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary. */
1416 salfar = scale * alphar[jc] * anrm;
1417 salfai = scale * alphai[jc] * anrm;
1418 sbeta = scale * beta[jc] * bnrm;
1420 alphar[jc] = salfar;
1421 alphai[jc] = salfai;
1427 work[1] = (doublereal) lwkopt;