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__1 = 1;
516 static doublecomplex c_b14 = {-1.,0.};
517 static doublecomplex c_b15 = {1.,0.};
518 static doublereal c_b37 = 1.;
520 /* > \brief \b ZLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetri
521 c indefinite matrices by performing extra-precise iterative refinement and provides error bounds and b
522 ackward error estimates for the solution. */
524 /* =========== DOCUMENTATION =========== */
526 /* Online html documentation available at */
527 /* http://www.netlib.org/lapack/explore-html/ */
530 /* > Download ZLA_SYRFSX_EXTENDED + dependencies */
531 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_syr
534 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_syr
537 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_syr
545 /* SUBROUTINE ZLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, */
546 /* AF, LDAF, IPIV, COLEQU, C, B, LDB, */
547 /* Y, LDY, BERR_OUT, N_NORMS, */
548 /* ERR_BNDS_NORM, ERR_BNDS_COMP, RES, */
549 /* AYB, DY, Y_TAIL, RCOND, ITHRESH, */
550 /* RTHRESH, DZ_UB, IGNORE_CWISE, */
553 /* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, */
554 /* $ N_NORMS, ITHRESH */
556 /* LOGICAL COLEQU, IGNORE_CWISE */
557 /* DOUBLE PRECISION RTHRESH, DZ_UB */
558 /* INTEGER IPIV( * ) */
559 /* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), */
560 /* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) */
561 /* DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ), */
562 /* $ ERR_BNDS_NORM( NRHS, * ), */
563 /* $ ERR_BNDS_COMP( NRHS, * ) */
566 /* > \par Purpose: */
571 /* > ZLA_SYRFSX_EXTENDED improves the computed solution to a system of */
572 /* > linear equations by performing extra-precise iterative refinement */
573 /* > and provides error bounds and backward error estimates for the solution. */
574 /* > This subroutine is called by ZSYRFSX to perform iterative refinement. */
575 /* > In addition to normwise error bound, the code provides maximum */
576 /* > componentwise error bound if possible. See comments for ERR_BNDS_NORM */
577 /* > and ERR_BNDS_COMP for details of the error bounds. Note that this */
578 /* > subroutine is only resonsible for setting the second fields of */
579 /* > ERR_BNDS_NORM and ERR_BNDS_COMP. */
585 /* > \param[in] PREC_TYPE */
587 /* > PREC_TYPE is INTEGER */
588 /* > Specifies the intermediate precision to be used in refinement. */
589 /* > The value is defined by ILAPREC(P) where P is a CHARACTER and P */
590 /* > = 'S': Single */
591 /* > = 'D': Double */
592 /* > = 'I': Indigenous */
593 /* > = 'X' or 'E': Extra */
596 /* > \param[in] UPLO */
598 /* > UPLO is CHARACTER*1 */
599 /* > = 'U': Upper triangle of A is stored; */
600 /* > = 'L': Lower triangle of A is stored. */
606 /* > The number of linear equations, i.e., the order of the */
607 /* > matrix A. N >= 0. */
610 /* > \param[in] NRHS */
612 /* > NRHS is INTEGER */
613 /* > The number of right-hand-sides, i.e., the number of columns of the */
619 /* > A is COMPLEX*16 array, dimension (LDA,N) */
620 /* > On entry, the N-by-N matrix A. */
623 /* > \param[in] LDA */
625 /* > LDA is INTEGER */
626 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
629 /* > \param[in] AF */
631 /* > AF is COMPLEX*16 array, dimension (LDAF,N) */
632 /* > The block diagonal matrix D and the multipliers used to */
633 /* > obtain the factor U or L as computed by ZSYTRF. */
636 /* > \param[in] LDAF */
638 /* > LDAF is INTEGER */
639 /* > The leading dimension of the array AF. LDAF >= f2cmax(1,N). */
642 /* > \param[in] IPIV */
644 /* > IPIV is INTEGER array, dimension (N) */
645 /* > Details of the interchanges and the block structure of D */
646 /* > as determined by ZSYTRF. */
649 /* > \param[in] COLEQU */
651 /* > COLEQU is LOGICAL */
652 /* > If .TRUE. then column equilibration was done to A before calling */
653 /* > this routine. This is needed to compute the solution and error */
654 /* > bounds correctly. */
659 /* > C is DOUBLE PRECISION array, dimension (N) */
660 /* > The column scale factors for A. If COLEQU = .FALSE., C */
661 /* > is not accessed. If C is input, each element of C should be a power */
662 /* > of the radix to ensure a reliable solution and error estimates. */
663 /* > Scaling by powers of the radix does not cause rounding errors unless */
664 /* > the result underflows or overflows. Rounding errors during scaling */
665 /* > lead to refining with a matrix that is not equivalent to the */
666 /* > input matrix, producing error estimates that may not be */
672 /* > B is COMPLEX*16 array, dimension (LDB,NRHS) */
673 /* > The right-hand-side matrix B. */
676 /* > \param[in] LDB */
678 /* > LDB is INTEGER */
679 /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */
682 /* > \param[in,out] Y */
684 /* > Y is COMPLEX*16 array, dimension (LDY,NRHS) */
685 /* > On entry, the solution matrix X, as computed by ZSYTRS. */
686 /* > On exit, the improved solution matrix Y. */
689 /* > \param[in] LDY */
691 /* > LDY is INTEGER */
692 /* > The leading dimension of the array Y. LDY >= f2cmax(1,N). */
695 /* > \param[out] BERR_OUT */
697 /* > BERR_OUT is DOUBLE PRECISION array, dimension (NRHS) */
698 /* > On exit, BERR_OUT(j) contains the componentwise relative backward */
699 /* > error for right-hand-side j from the formula */
700 /* > f2cmax(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
701 /* > where abs(Z) is the componentwise absolute value of the matrix */
702 /* > or vector Z. This is computed by ZLA_LIN_BERR. */
705 /* > \param[in] N_NORMS */
707 /* > N_NORMS is INTEGER */
708 /* > Determines which error bounds to return (see ERR_BNDS_NORM */
709 /* > and ERR_BNDS_COMP). */
710 /* > If N_NORMS >= 1 return normwise error bounds. */
711 /* > If N_NORMS >= 2 return componentwise error bounds. */
714 /* > \param[in,out] ERR_BNDS_NORM */
716 /* > ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
717 /* > For each right-hand side, this array contains information about */
718 /* > various error bounds and condition numbers corresponding to the */
719 /* > normwise relative error, which is defined as follows: */
721 /* > Normwise relative error in the ith solution vector: */
722 /* > max_j (abs(XTRUE(j,i) - X(j,i))) */
723 /* > ------------------------------ */
724 /* > max_j abs(X(j,i)) */
726 /* > The array is indexed by the type of error information as described */
727 /* > below. There currently are up to three pieces of information */
730 /* > The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
731 /* > right-hand side. */
733 /* > The second index in ERR_BNDS_NORM(:,err) contains the following */
734 /* > three fields: */
735 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
736 /* > reciprocal condition number is less than the threshold */
737 /* > sqrt(n) * slamch('Epsilon'). */
739 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
740 /* > almost certainly within a factor of 10 of the true error */
741 /* > so long as the next entry is greater than the threshold */
742 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
743 /* > be trusted if the previous boolean is true. */
745 /* > err = 3 Reciprocal condition number: Estimated normwise */
746 /* > reciprocal condition number. Compared with the threshold */
747 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
748 /* > estimate is "guaranteed". These reciprocal condition */
749 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
750 /* > appropriately scaled matrix Z. */
751 /* > Let Z = S*A, where S scales each row by a power of the */
752 /* > radix so all absolute row sums of Z are approximately 1. */
754 /* > This subroutine is only responsible for setting the second field */
756 /* > See Lapack Working Note 165 for further details and extra */
760 /* > \param[in,out] ERR_BNDS_COMP */
762 /* > ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
763 /* > For each right-hand side, this array contains information about */
764 /* > various error bounds and condition numbers corresponding to the */
765 /* > componentwise relative error, which is defined as follows: */
767 /* > Componentwise relative error in the ith solution vector: */
768 /* > abs(XTRUE(j,i) - X(j,i)) */
769 /* > max_j ---------------------- */
772 /* > The array is indexed by the right-hand side i (on which the */
773 /* > componentwise relative error depends), and the type of error */
774 /* > information as described below. There currently are up to three */
775 /* > pieces of information returned for each right-hand side. If */
776 /* > componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
777 /* > ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most */
778 /* > the first (:,N_ERR_BNDS) entries are returned. */
780 /* > The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
781 /* > right-hand side. */
783 /* > The second index in ERR_BNDS_COMP(:,err) contains the following */
784 /* > three fields: */
785 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
786 /* > reciprocal condition number is less than the threshold */
787 /* > sqrt(n) * slamch('Epsilon'). */
789 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
790 /* > almost certainly within a factor of 10 of the true error */
791 /* > so long as the next entry is greater than the threshold */
792 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
793 /* > be trusted if the previous boolean is true. */
795 /* > err = 3 Reciprocal condition number: Estimated componentwise */
796 /* > reciprocal condition number. Compared with the threshold */
797 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
798 /* > estimate is "guaranteed". These reciprocal condition */
799 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
800 /* > appropriately scaled matrix Z. */
801 /* > Let Z = S*(A*diag(x)), where x is the solution for the */
802 /* > current right-hand side and S scales each row of */
803 /* > A*diag(x) by a power of the radix so all absolute row */
804 /* > sums of Z are approximately 1. */
806 /* > This subroutine is only responsible for setting the second field */
808 /* > See Lapack Working Note 165 for further details and extra */
812 /* > \param[in] RES */
814 /* > RES is COMPLEX*16 array, dimension (N) */
815 /* > Workspace to hold the intermediate residual. */
818 /* > \param[in] AYB */
820 /* > AYB is DOUBLE PRECISION array, dimension (N) */
824 /* > \param[in] DY */
826 /* > DY is COMPLEX*16 array, dimension (N) */
827 /* > Workspace to hold the intermediate solution. */
830 /* > \param[in] Y_TAIL */
832 /* > Y_TAIL is COMPLEX*16 array, dimension (N) */
833 /* > Workspace to hold the trailing bits of the intermediate solution. */
836 /* > \param[in] RCOND */
838 /* > RCOND is DOUBLE PRECISION */
839 /* > Reciprocal scaled condition number. This is an estimate of the */
840 /* > reciprocal Skeel condition number of the matrix A after */
841 /* > equilibration (if done). If this is less than the machine */
842 /* > precision (in particular, if it is zero), the matrix is singular */
843 /* > to working precision. Note that the error may still be small even */
844 /* > if this number is very small and the matrix appears ill- */
848 /* > \param[in] ITHRESH */
850 /* > ITHRESH is INTEGER */
851 /* > The maximum number of residual computations allowed for */
852 /* > refinement. The default is 10. For 'aggressive' set to 100 to */
853 /* > permit convergence using approximate factorizations or */
854 /* > factorizations other than LU. If the factorization uses a */
855 /* > technique other than Gaussian elimination, the guarantees in */
856 /* > ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. */
859 /* > \param[in] RTHRESH */
861 /* > RTHRESH is DOUBLE PRECISION */
862 /* > Determines when to stop refinement if the error estimate stops */
863 /* > decreasing. Refinement will stop when the next solution no longer */
864 /* > satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */
865 /* > the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */
866 /* > default value is 0.5. For 'aggressive' set to 0.9 to permit */
867 /* > convergence on extremely ill-conditioned matrices. See LAWN 165 */
868 /* > for more details. */
871 /* > \param[in] DZ_UB */
873 /* > DZ_UB is DOUBLE PRECISION */
874 /* > Determines when to start considering componentwise convergence. */
875 /* > Componentwise convergence is only considered after each component */
876 /* > of the solution Y is stable, which we definte as the relative */
877 /* > change in each component being less than DZ_UB. The default value */
878 /* > is 0.25, requiring the first bit to be stable. See LAWN 165 for */
879 /* > more details. */
882 /* > \param[in] IGNORE_CWISE */
884 /* > IGNORE_CWISE is LOGICAL */
885 /* > If .TRUE. then ignore componentwise convergence. Default value */
889 /* > \param[out] INFO */
891 /* > INFO is INTEGER */
892 /* > = 0: Successful exit. */
893 /* > < 0: if INFO = -i, the ith argument to ZLA_HERFSX_EXTENDED had an illegal */
900 /* > \author Univ. of Tennessee */
901 /* > \author Univ. of California Berkeley */
902 /* > \author Univ. of Colorado Denver */
903 /* > \author NAG Ltd. */
905 /* > \date June 2017 */
907 /* > \ingroup complex16SYcomputational */
909 /* ===================================================================== */
910 /* Subroutine */ int zla_syrfsx_extended_(integer *prec_type__, char *uplo,
911 integer *n, integer *nrhs, doublecomplex *a, integer *lda,
912 doublecomplex *af, integer *ldaf, integer *ipiv, logical *colequ,
913 doublereal *c__, doublecomplex *b, integer *ldb, doublecomplex *y,
914 integer *ldy, doublereal *berr_out__, integer *n_norms__, doublereal *
915 err_bnds_norm__, doublereal *err_bnds_comp__, doublecomplex *res,
916 doublereal *ayb, doublecomplex *dy, doublecomplex *y_tail__,
917 doublereal *rcond, integer *ithresh, doublereal *rthresh, doublereal *
918 dz_ub__, logical *ignore_cwise__, integer *info)
920 /* System generated locals */
921 integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, y_dim1,
922 y_offset, err_bnds_norm_dim1, err_bnds_norm_offset,
923 err_bnds_comp_dim1, err_bnds_comp_offset, i__1, i__2, i__3, i__4;
924 doublereal d__1, d__2;
926 /* Local variables */
927 doublereal dx_x__, dz_z__, ymin;
928 extern /* Subroutine */ int zla_lin_berr_(integer *, integer *, integer *
929 , doublecomplex *, doublereal *, doublereal *);
930 doublereal dxratmax, dzratmax;
931 integer y_prec_state__, uplo2, i__, j;
932 extern /* Subroutine */ int blas_zsymv_x_(integer *, integer *,
933 doublecomplex *, doublecomplex *, integer *, doublecomplex *,
934 integer *, doublecomplex *, doublecomplex *, integer *, integer *)
936 extern logical lsame_(char *, char *);
941 extern /* Subroutine */ int blas_zsymv2_x_(integer *, integer *,
942 doublecomplex *, doublecomplex *, integer *, doublecomplex *,
943 doublecomplex *, integer *, doublecomplex *, doublecomplex *,
944 integer *, integer *);
945 doublereal normx, normy;
946 extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *,
947 doublecomplex *, integer *), zaxpy_(integer *, doublecomplex *,
948 doublecomplex *, integer *, doublecomplex *, integer *);
949 doublereal myhugeval, prev_dz_z__;
950 extern /* Subroutine */ int zla_syamv_(integer *, integer *, doublereal *
951 , doublecomplex *, integer *, doublecomplex *, integer *,
952 doublereal *, doublereal *, integer *), zsymv_(char *, integer *,
953 doublecomplex *, doublecomplex *, integer *, doublecomplex *,
954 integer *, doublecomplex *, doublecomplex *, integer *);
955 extern doublereal dlamch_(char *);
957 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
958 doublereal final_dx_x__, final_dz_z__, normdx;
959 extern /* Subroutine */ int zla_wwaddw_(integer *, doublecomplex *,
960 doublecomplex *, doublecomplex *), zsytrs_(char *, integer *,
961 integer *, doublecomplex *, integer *, integer *, doublecomplex *,
962 integer *, integer *);
963 doublereal prevnormdx;
966 extern integer ilauplo_(char *);
967 integer x_state__, z_state__;
968 doublereal incr_thresh__;
971 /* -- LAPACK computational routine (version 3.7.1) -- */
972 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
973 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
977 /* ===================================================================== */
980 /* Parameter adjustments */
981 err_bnds_comp_dim1 = *nrhs;
982 err_bnds_comp_offset = 1 + err_bnds_comp_dim1 * 1;
983 err_bnds_comp__ -= err_bnds_comp_offset;
984 err_bnds_norm_dim1 = *nrhs;
985 err_bnds_norm_offset = 1 + err_bnds_norm_dim1 * 1;
986 err_bnds_norm__ -= err_bnds_norm_offset;
988 a_offset = 1 + a_dim1 * 1;
991 af_offset = 1 + af_dim1 * 1;
996 b_offset = 1 + b_dim1 * 1;
999 y_offset = 1 + y_dim1 * 1;
1009 upper = lsame_(uplo, "U");
1010 if (! upper && ! lsame_(uplo, "L")) {
1012 } else if (*n < 0) {
1014 } else if (*nrhs < 0) {
1016 } else if (*lda < f2cmax(1,*n)) {
1018 } else if (*ldaf < f2cmax(1,*n)) {
1020 } else if (*ldb < f2cmax(1,*n)) {
1022 } else if (*ldy < f2cmax(1,*n)) {
1027 xerbla_("ZLA_HERFSX_EXTENDED", &i__1, (ftnlen)19);
1030 eps = dlamch_("Epsilon");
1031 myhugeval = dlamch_("Overflow");
1032 /* Force MYHUGEVAL to Inf */
1033 myhugeval *= myhugeval;
1034 /* Using MYHUGEVAL may lead to spurious underflows. */
1035 incr_thresh__ = (doublereal) (*n) * eps;
1036 if (lsame_(uplo, "L")) {
1037 uplo2 = ilauplo_("L");
1039 uplo2 = ilauplo_("U");
1042 for (j = 1; j <= i__1; ++j) {
1044 if (y_prec_state__ == 2) {
1046 for (i__ = 1; i__ <= i__2; ++i__) {
1048 y_tail__[i__3].r = 0., y_tail__[i__3].i = 0.;
1055 final_dx_x__ = myhugeval;
1056 final_dz_z__ = myhugeval;
1057 prevnormdx = myhugeval;
1058 prev_dz_z__ = myhugeval;
1063 incr_prec__ = FALSE_;
1065 for (cnt = 1; cnt <= i__2; ++cnt) {
1067 /* Compute residual RES = B_s - op(A_s) * Y, */
1068 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1070 zcopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1071 if (y_prec_state__ == 0) {
1072 zsymv_(uplo, n, &c_b14, &a[a_offset], lda, &y[j * y_dim1 + 1],
1073 &c__1, &c_b15, &res[1], &c__1);
1074 } else if (y_prec_state__ == 1) {
1075 blas_zsymv_x__(&uplo2, n, &c_b14, &a[a_offset], lda, &y[j *
1076 y_dim1 + 1], &c__1, &c_b15, &res[1], &c__1,
1079 blas_zsymv2_x__(&uplo2, n, &c_b14, &a[a_offset], lda, &y[j *
1080 y_dim1 + 1], &y_tail__[1], &c__1, &c_b15, &res[1], &
1083 /* XXX: RES is no longer needed. */
1084 zcopy_(n, &res[1], &c__1, &dy[1], &c__1);
1085 zsytrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &dy[1], n,
1088 /* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */
1096 for (i__ = 1; i__ <= i__3; ++i__) {
1097 i__4 = i__ + j * y_dim1;
1098 yk = (d__1 = y[i__4].r, abs(d__1)) + (d__2 = d_imag(&y[i__ +
1099 j * y_dim1]), abs(d__2));
1101 dyk = (d__1 = dy[i__4].r, abs(d__1)) + (d__2 = d_imag(&dy[i__]
1105 d__1 = dz_z__, d__2 = dyk / yk;
1106 dz_z__ = f2cmax(d__1,d__2);
1107 } else if (dyk != 0.) {
1110 ymin = f2cmin(ymin,yk);
1111 normy = f2cmax(normy,yk);
1114 d__1 = normx, d__2 = yk * c__[i__];
1115 normx = f2cmax(d__1,d__2);
1117 d__1 = normdx, d__2 = dyk * c__[i__];
1118 normdx = f2cmax(d__1,d__2);
1121 normdx = f2cmax(normdx,dyk);
1125 dx_x__ = normdx / normx;
1126 } else if (normdx == 0.) {
1131 dxrat = normdx / prevnormdx;
1132 dzrat = dz_z__ / prev_dz_z__;
1134 /* Check termination criteria. */
1136 if (ymin * *rcond < incr_thresh__ * normy && y_prec_state__ < 2) {
1137 incr_prec__ = TRUE_;
1139 if (x_state__ == 3 && dxrat <= *rthresh) {
1142 if (x_state__ == 1) {
1143 if (dx_x__ <= eps) {
1145 } else if (dxrat > *rthresh) {
1146 if (y_prec_state__ != 2) {
1147 incr_prec__ = TRUE_;
1152 if (dxrat > dxratmax) {
1156 if (x_state__ > 1) {
1157 final_dx_x__ = dx_x__;
1160 if (z_state__ == 0 && dz_z__ <= *dz_ub__) {
1163 if (z_state__ == 3 && dzrat <= *rthresh) {
1166 if (z_state__ == 1) {
1167 if (dz_z__ <= eps) {
1169 } else if (dz_z__ > *dz_ub__) {
1172 final_dz_z__ = myhugeval;
1173 } else if (dzrat > *rthresh) {
1174 if (y_prec_state__ != 2) {
1175 incr_prec__ = TRUE_;
1180 if (dzrat > dzratmax) {
1184 if (z_state__ > 1) {
1185 final_dz_z__ = dz_z__;
1188 if (x_state__ != 1 && (*ignore_cwise__ || z_state__ != 1)) {
1192 incr_prec__ = FALSE_;
1195 for (i__ = 1; i__ <= i__3; ++i__) {
1197 y_tail__[i__4].r = 0., y_tail__[i__4].i = 0.;
1200 prevnormdx = normdx;
1201 prev_dz_z__ = dz_z__;
1203 /* Update soluton. */
1205 if (y_prec_state__ < 2) {
1206 zaxpy_(n, &c_b15, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1);
1208 zla_wwaddw_(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]);
1211 /* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't CALL MYEXIT. */
1214 /* Set final_* when cnt hits ithresh. */
1216 if (x_state__ == 1) {
1217 final_dx_x__ = dx_x__;
1219 if (z_state__ == 1) {
1220 final_dz_z__ = dz_z__;
1223 /* Compute error bounds. */
1225 if (*n_norms__ >= 1) {
1226 err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = final_dx_x__ / (
1229 if (*n_norms__ >= 2) {
1230 err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = final_dz_z__ / (
1234 /* Compute componentwise relative backward error from formula */
1235 /* f2cmax(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
1236 /* where abs(Z) is the componentwise absolute value of the matrix */
1239 /* Compute residual RES = B_s - op(A_s) * Y, */
1240 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1242 zcopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1243 zsymv_(uplo, n, &c_b14, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1,
1244 &c_b15, &res[1], &c__1);
1246 for (i__ = 1; i__ <= i__2; ++i__) {
1247 i__3 = i__ + j * b_dim1;
1248 ayb[i__] = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b[i__
1249 + j * b_dim1]), abs(d__2));
1252 /* Compute abs(op(A_s))*abs(Y) + abs(B_s). */
1254 zla_syamv_(&uplo2, n, &c_b37, &a[a_offset], lda, &y[j * y_dim1 + 1],
1255 &c__1, &c_b37, &ayb[1], &c__1);
1256 zla_lin_berr_(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]);
1258 /* End of loop for each RHS. */
1263 } /* zla_syrfsx_extended__ */