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 real c_b6 = -1.f;
517 static real c_b8 = 1.f;
519 /* > \brief \b SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general
520 matrices by performing extra-precise iterative refinement and provides error bounds and backward error
521 estimates for the solution. */
523 /* =========== DOCUMENTATION =========== */
525 /* Online html documentation available at */
526 /* http://www.netlib.org/lapack/explore-html/ */
529 /* > Download SLA_GERFSX_EXTENDED + dependencies */
530 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_ger
533 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_ger
536 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_ger
544 /* SUBROUTINE SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A, */
545 /* LDA, AF, LDAF, IPIV, COLEQU, C, B, */
546 /* LDB, Y, LDY, BERR_OUT, N_NORMS, */
547 /* ERRS_N, ERRS_C, RES, */
548 /* AYB, DY, Y_TAIL, RCOND, ITHRESH, */
549 /* RTHRESH, DZ_UB, IGNORE_CWISE, */
552 /* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, */
553 /* $ TRANS_TYPE, N_NORMS, ITHRESH */
554 /* LOGICAL COLEQU, IGNORE_CWISE */
555 /* REAL RTHRESH, DZ_UB */
556 /* INTEGER IPIV( * ) */
557 /* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), */
558 /* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) */
559 /* REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), */
560 /* $ ERRS_N( NRHS, * ), */
561 /* $ ERRS_C( NRHS, * ) */
564 /* > \par Purpose: */
569 /* > SLA_GERFSX_EXTENDED improves the computed solution to a system of */
570 /* > linear equations by performing extra-precise iterative refinement */
571 /* > and provides error bounds and backward error estimates for the solution. */
572 /* > This subroutine is called by SGERFSX to perform iterative refinement. */
573 /* > In addition to normwise error bound, the code provides maximum */
574 /* > componentwise error bound if possible. See comments for ERRS_N */
575 /* > and ERRS_C for details of the error bounds. Note that this */
576 /* > subroutine is only resonsible for setting the second fields of */
577 /* > ERRS_N and ERRS_C. */
583 /* > \param[in] PREC_TYPE */
585 /* > PREC_TYPE is INTEGER */
586 /* > Specifies the intermediate precision to be used in refinement. */
587 /* > The value is defined by ILAPREC(P) where P is a CHARACTER and P */
588 /* > = 'S': Single */
589 /* > = 'D': Double */
590 /* > = 'I': Indigenous */
591 /* > = 'X' or 'E': Extra */
594 /* > \param[in] TRANS_TYPE */
596 /* > TRANS_TYPE is INTEGER */
597 /* > Specifies the transposition operation on A. */
598 /* > The value is defined by ILATRANS(T) where T is a CHARACTER and T */
599 /* > = 'N': No transpose */
600 /* > = 'T': Transpose */
601 /* > = 'C': Conjugate transpose */
607 /* > The number of linear equations, i.e., the order of the */
608 /* > matrix A. N >= 0. */
611 /* > \param[in] NRHS */
613 /* > NRHS is INTEGER */
614 /* > The number of right-hand-sides, i.e., the number of columns of the */
620 /* > A is REAL array, dimension (LDA,N) */
621 /* > On entry, the N-by-N matrix A. */
624 /* > \param[in] LDA */
626 /* > LDA is INTEGER */
627 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
630 /* > \param[in] AF */
632 /* > AF is REAL array, dimension (LDAF,N) */
633 /* > The factors L and U from the factorization */
634 /* > A = P*L*U as computed by SGETRF. */
637 /* > \param[in] LDAF */
639 /* > LDAF is INTEGER */
640 /* > The leading dimension of the array AF. LDAF >= f2cmax(1,N). */
643 /* > \param[in] IPIV */
645 /* > IPIV is INTEGER array, dimension (N) */
646 /* > The pivot indices from the factorization A = P*L*U */
647 /* > as computed by SGETRF; row i of the matrix was interchanged */
648 /* > with row IPIV(i). */
651 /* > \param[in] COLEQU */
653 /* > COLEQU is LOGICAL */
654 /* > If .TRUE. then column equilibration was done to A before calling */
655 /* > this routine. This is needed to compute the solution and error */
656 /* > bounds correctly. */
661 /* > C is REAL array, dimension (N) */
662 /* > The column scale factors for A. If COLEQU = .FALSE., C */
663 /* > is not accessed. If C is input, each element of C should be a power */
664 /* > of the radix to ensure a reliable solution and error estimates. */
665 /* > Scaling by powers of the radix does not cause rounding errors unless */
666 /* > the result underflows or overflows. Rounding errors during scaling */
667 /* > lead to refining with a matrix that is not equivalent to the */
668 /* > input matrix, producing error estimates that may not be */
674 /* > B is REAL array, dimension (LDB,NRHS) */
675 /* > The right-hand-side matrix B. */
678 /* > \param[in] LDB */
680 /* > LDB is INTEGER */
681 /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */
684 /* > \param[in,out] Y */
686 /* > Y is REAL array, dimension (LDY,NRHS) */
687 /* > On entry, the solution matrix X, as computed by SGETRS. */
688 /* > On exit, the improved solution matrix Y. */
691 /* > \param[in] LDY */
693 /* > LDY is INTEGER */
694 /* > The leading dimension of the array Y. LDY >= f2cmax(1,N). */
697 /* > \param[out] BERR_OUT */
699 /* > BERR_OUT is REAL array, dimension (NRHS) */
700 /* > On exit, BERR_OUT(j) contains the componentwise relative backward */
701 /* > error for right-hand-side j from the formula */
702 /* > f2cmax(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
703 /* > where abs(Z) is the componentwise absolute value of the matrix */
704 /* > or vector Z. This is computed by SLA_LIN_BERR. */
707 /* > \param[in] N_NORMS */
709 /* > N_NORMS is INTEGER */
710 /* > Determines which error bounds to return (see ERRS_N */
712 /* > If N_NORMS >= 1 return normwise error bounds. */
713 /* > If N_NORMS >= 2 return componentwise error bounds. */
716 /* > \param[in,out] ERRS_N */
718 /* > ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS) */
719 /* > For each right-hand side, this array contains information about */
720 /* > various error bounds and condition numbers corresponding to the */
721 /* > normwise relative error, which is defined as follows: */
723 /* > Normwise relative error in the ith solution vector: */
724 /* > max_j (abs(XTRUE(j,i) - X(j,i))) */
725 /* > ------------------------------ */
726 /* > max_j abs(X(j,i)) */
728 /* > The array is indexed by the type of error information as described */
729 /* > below. There currently are up to three pieces of information */
732 /* > The first index in ERRS_N(i,:) corresponds to the ith */
733 /* > right-hand side. */
735 /* > The second index in ERRS_N(:,err) contains the following */
736 /* > three fields: */
737 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
738 /* > reciprocal condition number is less than the threshold */
739 /* > sqrt(n) * slamch('Epsilon'). */
741 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
742 /* > almost certainly within a factor of 10 of the true error */
743 /* > so long as the next entry is greater than the threshold */
744 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
745 /* > be trusted if the previous boolean is true. */
747 /* > err = 3 Reciprocal condition number: Estimated normwise */
748 /* > reciprocal condition number. Compared with the threshold */
749 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
750 /* > estimate is "guaranteed". These reciprocal condition */
751 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
752 /* > appropriately scaled matrix Z. */
753 /* > Let Z = S*A, where S scales each row by a power of the */
754 /* > radix so all absolute row sums of Z are approximately 1. */
756 /* > This subroutine is only responsible for setting the second field */
758 /* > See Lapack Working Note 165 for further details and extra */
762 /* > \param[in,out] ERRS_C */
764 /* > ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS) */
765 /* > For each right-hand side, this array contains information about */
766 /* > various error bounds and condition numbers corresponding to the */
767 /* > componentwise relative error, which is defined as follows: */
769 /* > Componentwise relative error in the ith solution vector: */
770 /* > abs(XTRUE(j,i) - X(j,i)) */
771 /* > max_j ---------------------- */
774 /* > The array is indexed by the right-hand side i (on which the */
775 /* > componentwise relative error depends), and the type of error */
776 /* > information as described below. There currently are up to three */
777 /* > pieces of information returned for each right-hand side. If */
778 /* > componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
779 /* > ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most */
780 /* > the first (:,N_ERR_BNDS) entries are returned. */
782 /* > The first index in ERRS_C(i,:) corresponds to the ith */
783 /* > right-hand side. */
785 /* > The second index in ERRS_C(:,err) contains the following */
786 /* > three fields: */
787 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
788 /* > reciprocal condition number is less than the threshold */
789 /* > sqrt(n) * slamch('Epsilon'). */
791 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
792 /* > almost certainly within a factor of 10 of the true error */
793 /* > so long as the next entry is greater than the threshold */
794 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
795 /* > be trusted if the previous boolean is true. */
797 /* > err = 3 Reciprocal condition number: Estimated componentwise */
798 /* > reciprocal condition number. Compared with the threshold */
799 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
800 /* > estimate is "guaranteed". These reciprocal condition */
801 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
802 /* > appropriately scaled matrix Z. */
803 /* > Let Z = S*(A*diag(x)), where x is the solution for the */
804 /* > current right-hand side and S scales each row of */
805 /* > A*diag(x) by a power of the radix so all absolute row */
806 /* > sums of Z are approximately 1. */
808 /* > This subroutine is only responsible for setting the second field */
810 /* > See Lapack Working Note 165 for further details and extra */
814 /* > \param[in] RES */
816 /* > RES is REAL array, dimension (N) */
817 /* > Workspace to hold the intermediate residual. */
820 /* > \param[in] AYB */
822 /* > AYB is REAL array, dimension (N) */
823 /* > Workspace. This can be the same workspace passed for Y_TAIL. */
826 /* > \param[in] DY */
828 /* > DY is REAL array, dimension (N) */
829 /* > Workspace to hold the intermediate solution. */
832 /* > \param[in] Y_TAIL */
834 /* > Y_TAIL is REAL array, dimension (N) */
835 /* > Workspace to hold the trailing bits of the intermediate solution. */
838 /* > \param[in] RCOND */
840 /* > RCOND is REAL */
841 /* > Reciprocal scaled condition number. This is an estimate of the */
842 /* > reciprocal Skeel condition number of the matrix A after */
843 /* > equilibration (if done). If this is less than the machine */
844 /* > precision (in particular, if it is zero), the matrix is singular */
845 /* > to working precision. Note that the error may still be small even */
846 /* > if this number is very small and the matrix appears ill- */
850 /* > \param[in] ITHRESH */
852 /* > ITHRESH is INTEGER */
853 /* > The maximum number of residual computations allowed for */
854 /* > refinement. The default is 10. For 'aggressive' set to 100 to */
855 /* > permit convergence using approximate factorizations or */
856 /* > factorizations other than LU. If the factorization uses a */
857 /* > technique other than Gaussian elimination, the guarantees in */
858 /* > ERRS_N and ERRS_C may no longer be trustworthy. */
861 /* > \param[in] RTHRESH */
863 /* > RTHRESH is REAL */
864 /* > Determines when to stop refinement if the error estimate stops */
865 /* > decreasing. Refinement will stop when the next solution no longer */
866 /* > satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */
867 /* > the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */
868 /* > default value is 0.5. For 'aggressive' set to 0.9 to permit */
869 /* > convergence on extremely ill-conditioned matrices. See LAWN 165 */
870 /* > for more details. */
873 /* > \param[in] DZ_UB */
875 /* > DZ_UB is REAL */
876 /* > Determines when to start considering componentwise convergence. */
877 /* > Componentwise convergence is only considered after each component */
878 /* > of the solution Y is stable, which we definte as the relative */
879 /* > change in each component being less than DZ_UB. The default value */
880 /* > is 0.25, requiring the first bit to be stable. See LAWN 165 for */
881 /* > more details. */
884 /* > \param[in] IGNORE_CWISE */
886 /* > IGNORE_CWISE is LOGICAL */
887 /* > If .TRUE. then ignore componentwise convergence. Default value */
891 /* > \param[out] INFO */
893 /* > INFO is INTEGER */
894 /* > = 0: Successful exit. */
895 /* > < 0: if INFO = -i, the ith argument to SGETRS had an illegal */
902 /* > \author Univ. of Tennessee */
903 /* > \author Univ. of California Berkeley */
904 /* > \author Univ. of Colorado Denver */
905 /* > \author NAG Ltd. */
907 /* > \date December 2016 */
909 /* > \ingroup realGEcomputational */
911 /* ===================================================================== */
912 /* Subroutine */ int sla_gerfsx_extended_(integer *prec_type__, integer *
913 trans_type__, integer *n, integer *nrhs, real *a, integer *lda, real *
914 af, integer *ldaf, integer *ipiv, logical *colequ, real *c__, real *b,
915 integer *ldb, real *y, integer *ldy, real *berr_out__, integer *
916 n_norms__, real *errs_n__, real *errs_c__, real *res, real *ayb, real
917 *dy, real *y_tail__, real *rcond, integer *ithresh, real *rthresh,
918 real *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, errs_n_dim1, errs_n_offset, errs_c_dim1, errs_c_offset,
927 /* Local variables */
928 real dx_x__, dz_z__, ymin;
929 extern /* Subroutine */ int sla_lin_berr_(integer *, integer *, integer *
930 , real *, real *, real *);
932 extern /* Subroutine */ int blas_sgemv_x_(integer *, integer *, integer *
933 , real *, real *, integer *, real *, integer *, real *, real *,
934 integer *, integer *);
936 integer y_prec_state__, i__, j;
937 extern /* Subroutine */ int blas_sgemv2_x_(integer *, integer *, integer
938 *, real *, real *, integer *, real *, real *, integer *, real *,
939 real *, integer *, integer *), sla_geamv_(integer *, integer *,
940 integer *, real *, real *, integer *, real *, integer *, real *,
941 real *, integer *), sgemv_(char *, integer *, integer *, real *,
942 real *, integer *, real *, integer *, real *, real *, integer *);
947 extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
950 extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
952 real myhugeval, prev_dz_z__, yk;
953 extern real slamch_(char *);
954 real final_dx_x__, final_dz_z__, normdx;
955 extern /* Subroutine */ int sgetrs_(char *, integer *, integer *, real *,
956 integer *, integer *, real *, integer *, integer *),
957 sla_wwaddw_(integer *, real *, real *, real *);
958 extern /* Character */ VOID chla_transtype_(char *, integer *);
962 integer x_state__, z_state__;
966 /* -- LAPACK computational routine (version 3.7.0) -- */
967 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
968 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
972 /* ===================================================================== */
975 /* Parameter adjustments */
977 errs_c_offset = 1 + errs_c_dim1 * 1;
978 errs_c__ -= errs_c_offset;
980 errs_n_offset = 1 + errs_n_dim1 * 1;
981 errs_n__ -= errs_n_offset;
983 a_offset = 1 + a_dim1 * 1;
986 af_offset = 1 + af_dim1 * 1;
991 b_offset = 1 + b_dim1 * 1;
994 y_offset = 1 + y_dim1 * 1;
1006 chla_transtype_(ch__1, trans_type__);
1007 *(unsigned char *)trans = *(unsigned char *)&ch__1[0];
1008 eps = slamch_("Epsilon");
1009 myhugeval = slamch_("Overflow");
1010 /* Force MYHUGEVAL to Inf */
1011 myhugeval *= myhugeval;
1012 /* Using MYHUGEVAL may lead to spurious underflows. */
1013 incr_thresh__ = (real) (*n) * eps;
1016 for (j = 1; j <= i__1; ++j) {
1018 if (y_prec_state__ == 2) {
1020 for (i__ = 1; i__ <= i__2; ++i__) {
1021 y_tail__[i__] = 0.f;
1028 final_dx_x__ = myhugeval;
1029 final_dz_z__ = myhugeval;
1030 prevnormdx = myhugeval;
1031 prev_dz_z__ = myhugeval;
1036 incr_prec__ = FALSE_;
1038 for (cnt = 1; cnt <= i__2; ++cnt) {
1040 /* Compute residual RES = B_s - op(A_s) * Y, */
1041 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1043 scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1044 if (y_prec_state__ == 0) {
1045 sgemv_(trans, n, n, &c_b6, &a[a_offset], lda, &y[j * y_dim1 +
1046 1], &c__1, &c_b8, &res[1], &c__1);
1047 } else if (y_prec_state__ == 1) {
1048 blas_sgemv_x__(trans_type__, n, n, &c_b6, &a[a_offset], lda, &
1049 y[j * y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1,
1052 blas_sgemv2_x__(trans_type__, n, n, &c_b6, &a[a_offset], lda,
1053 &y[j * y_dim1 + 1], &y_tail__[1], &c__1, &c_b8, &res[
1054 1], &c__1, prec_type__);
1056 /* XXX: RES is no longer needed. */
1057 scopy_(n, &res[1], &c__1, &dy[1], &c__1);
1058 sgetrs_(trans, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &dy[1],
1061 /* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */
1070 for (i__ = 1; i__ <= i__3; ++i__) {
1071 yk = (r__1 = y[i__ + j * y_dim1], abs(r__1));
1072 dyk = (r__1 = dy[i__], abs(r__1));
1075 r__1 = dz_z__, r__2 = dyk / yk;
1076 dz_z__ = f2cmax(r__1,r__2);
1077 } else if (dyk != 0.f) {
1080 ymin = f2cmin(ymin,yk);
1081 normy = f2cmax(normy,yk);
1084 r__1 = normx, r__2 = yk * c__[i__];
1085 normx = f2cmax(r__1,r__2);
1087 r__1 = normdx, r__2 = dyk * c__[i__];
1088 normdx = f2cmax(r__1,r__2);
1091 normdx = f2cmax(normdx,dyk);
1095 dx_x__ = normdx / normx;
1096 } else if (normdx == 0.f) {
1101 dxrat = normdx / prevnormdx;
1102 dzrat = dz_z__ / prev_dz_z__;
1104 /* Check termination criteria */
1106 if (! (*ignore_cwise__) && ymin * *rcond < incr_thresh__ * normy
1107 && y_prec_state__ < 2) {
1108 incr_prec__ = TRUE_;
1110 if (x_state__ == 3 && dxrat <= *rthresh) {
1113 if (x_state__ == 1) {
1114 if (dx_x__ <= eps) {
1116 } else if (dxrat > *rthresh) {
1117 if (y_prec_state__ != 2) {
1118 incr_prec__ = TRUE_;
1123 if (dxrat > dxratmax) {
1127 if (x_state__ > 1) {
1128 final_dx_x__ = dx_x__;
1131 if (z_state__ == 0 && dz_z__ <= *dz_ub__) {
1134 if (z_state__ == 3 && dzrat <= *rthresh) {
1137 if (z_state__ == 1) {
1138 if (dz_z__ <= eps) {
1140 } else if (dz_z__ > *dz_ub__) {
1143 final_dz_z__ = myhugeval;
1144 } else if (dzrat > *rthresh) {
1145 if (y_prec_state__ != 2) {
1146 incr_prec__ = TRUE_;
1151 if (dzrat > dzratmax) {
1155 if (z_state__ > 1) {
1156 final_dz_z__ = dz_z__;
1160 /* Exit if both normwise and componentwise stopped working, */
1161 /* but if componentwise is unstable, let it go at least two */
1164 if (x_state__ != 1) {
1165 if (*ignore_cwise__) {
1168 if (z_state__ == 3 || z_state__ == 2) {
1171 if (z_state__ == 0 && cnt > 1) {
1176 incr_prec__ = FALSE_;
1179 for (i__ = 1; i__ <= i__3; ++i__) {
1180 y_tail__[i__] = 0.f;
1183 prevnormdx = normdx;
1184 prev_dz_z__ = dz_z__;
1186 /* Update soluton. */
1188 if (y_prec_state__ < 2) {
1189 saxpy_(n, &c_b8, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1);
1191 sla_wwaddw_(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]);
1194 /* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't CALL MYEXIT. */
1197 /* Set final_* when cnt hits ithresh. */
1199 if (x_state__ == 1) {
1200 final_dx_x__ = dx_x__;
1202 if (z_state__ == 1) {
1203 final_dz_z__ = dz_z__;
1206 /* Compute error bounds */
1208 if (*n_norms__ >= 1) {
1209 errs_n__[j + (errs_n_dim1 << 1)] = final_dx_x__ / (1 - dxratmax);
1211 if (*n_norms__ >= 2) {
1212 errs_c__[j + (errs_c_dim1 << 1)] = final_dz_z__ / (1 - dzratmax);
1215 /* Compute componentwise relative backward error from formula */
1216 /* f2cmax(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
1217 /* where abs(Z) is the componentwise absolute value of the matrix */
1220 /* Compute residual RES = B_s - op(A_s) * Y, */
1221 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1223 scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1224 sgemv_(trans, n, n, &c_b6, &a[a_offset], lda, &y[j * y_dim1 + 1], &
1225 c__1, &c_b8, &res[1], &c__1);
1227 for (i__ = 1; i__ <= i__2; ++i__) {
1228 ayb[i__] = (r__1 = b[i__ + j * b_dim1], abs(r__1));
1231 /* Compute abs(op(A_s))*abs(Y) + abs(B_s). */
1233 sla_geamv_(trans_type__, n, n, &c_b8, &a[a_offset], lda, &y[j *
1234 y_dim1 + 1], &c__1, &c_b8, &ayb[1], &c__1);
1235 sla_lin_berr_(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]);
1237 /* End of loop for each RHS. */
1242 } /* sla_gerfsx_extended__ */