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_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general
520 banded matrices by performing extra-precise iterative refinement and provides error bounds and backwar
521 d error estimates for the solution. */
523 /* =========== DOCUMENTATION =========== */
525 /* Online html documentation available at */
526 /* http://www.netlib.org/lapack/explore-html/ */
529 /* > Download SLA_GBRFSX_EXTENDED + dependencies */
530 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_gbr
533 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_gbr
536 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_gbr
544 /* SUBROUTINE SLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU, */
545 /* NRHS, AB, LDAB, AFB, LDAFB, IPIV, */
546 /* COLEQU, C, B, LDB, Y, LDY, */
547 /* BERR_OUT, N_NORMS, ERR_BNDS_NORM, */
548 /* ERR_BNDS_COMP, RES, AYB, DY, */
549 /* Y_TAIL, RCOND, ITHRESH, RTHRESH, */
550 /* DZ_UB, IGNORE_CWISE, INFO ) */
552 /* INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS, */
553 /* $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH */
554 /* LOGICAL COLEQU, IGNORE_CWISE */
555 /* REAL RTHRESH, DZ_UB */
556 /* INTEGER IPIV( * ) */
557 /* REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), */
558 /* $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*) */
559 /* REAL C( * ), AYB(*), RCOND, BERR_OUT(*), */
560 /* $ ERR_BNDS_NORM( NRHS, * ), */
561 /* $ ERR_BNDS_COMP( NRHS, * ) */
564 /* > \par Purpose: */
569 /* > SLA_GBRFSX_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 SGBRFSX to perform iterative refinement. */
573 /* > In addition to normwise error bound, the code provides maximum */
574 /* > componentwise error bound if possible. See comments for ERR_BNDS_NORM */
575 /* > and ERR_BNDS_COMP for details of the error bounds. Note that this */
576 /* > subroutine is only resonsible for setting the second fields of */
577 /* > ERR_BNDS_NORM and ERR_BNDS_COMP. */
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] KL */
613 /* > KL is INTEGER */
614 /* > The number of subdiagonals within the band of A. KL >= 0. */
617 /* > \param[in] KU */
619 /* > KU is INTEGER */
620 /* > The number of superdiagonals within the band of A. KU >= 0 */
623 /* > \param[in] NRHS */
625 /* > NRHS is INTEGER */
626 /* > The number of right-hand-sides, i.e., the number of columns of the */
630 /* > \param[in] AB */
632 /* > AB is REAL array, dimension (LDAB,N) */
633 /* > On entry, the N-by-N matrix AB. */
636 /* > \param[in] LDAB */
638 /* > LDAB is INTEGER */
639 /* > The leading dimension of the array AB. LDAB >= f2cmax(1,N). */
642 /* > \param[in] AFB */
644 /* > AFB is REAL array, dimension (LDAFB,N) */
645 /* > The factors L and U from the factorization */
646 /* > A = P*L*U as computed by SGBTRF. */
649 /* > \param[in] LDAFB */
651 /* > LDAFB is INTEGER */
652 /* > The leading dimension of the array AF. LDAFB >= f2cmax(1,N). */
655 /* > \param[in] IPIV */
657 /* > IPIV is INTEGER array, dimension (N) */
658 /* > The pivot indices from the factorization A = P*L*U */
659 /* > as computed by SGBTRF; row i of the matrix was interchanged */
660 /* > with row IPIV(i). */
663 /* > \param[in] COLEQU */
665 /* > COLEQU is LOGICAL */
666 /* > If .TRUE. then column equilibration was done to A before calling */
667 /* > this routine. This is needed to compute the solution and error */
668 /* > bounds correctly. */
673 /* > C is REAL array, dimension (N) */
674 /* > The column scale factors for A. If COLEQU = .FALSE., C */
675 /* > is not accessed. If C is input, each element of C should be a power */
676 /* > of the radix to ensure a reliable solution and error estimates. */
677 /* > Scaling by powers of the radix does not cause rounding errors unless */
678 /* > the result underflows or overflows. Rounding errors during scaling */
679 /* > lead to refining with a matrix that is not equivalent to the */
680 /* > input matrix, producing error estimates that may not be */
686 /* > B is REAL array, dimension (LDB,NRHS) */
687 /* > The right-hand-side matrix B. */
690 /* > \param[in] LDB */
692 /* > LDB is INTEGER */
693 /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */
696 /* > \param[in,out] Y */
698 /* > Y is REAL array, dimension (LDY,NRHS) */
699 /* > On entry, the solution matrix X, as computed by SGBTRS. */
700 /* > On exit, the improved solution matrix Y. */
703 /* > \param[in] LDY */
705 /* > LDY is INTEGER */
706 /* > The leading dimension of the array Y. LDY >= f2cmax(1,N). */
709 /* > \param[out] BERR_OUT */
711 /* > BERR_OUT is REAL array, dimension (NRHS) */
712 /* > On exit, BERR_OUT(j) contains the componentwise relative backward */
713 /* > error for right-hand-side j from the formula */
714 /* > f2cmax(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
715 /* > where abs(Z) is the componentwise absolute value of the matrix */
716 /* > or vector Z. This is computed by SLA_LIN_BERR. */
719 /* > \param[in] N_NORMS */
721 /* > N_NORMS is INTEGER */
722 /* > Determines which error bounds to return (see ERR_BNDS_NORM */
723 /* > and ERR_BNDS_COMP). */
724 /* > If N_NORMS >= 1 return normwise error bounds. */
725 /* > If N_NORMS >= 2 return componentwise error bounds. */
728 /* > \param[in,out] ERR_BNDS_NORM */
730 /* > ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) */
731 /* > For each right-hand side, this array contains information about */
732 /* > various error bounds and condition numbers corresponding to the */
733 /* > normwise relative error, which is defined as follows: */
735 /* > Normwise relative error in the ith solution vector: */
736 /* > max_j (abs(XTRUE(j,i) - X(j,i))) */
737 /* > ------------------------------ */
738 /* > max_j abs(X(j,i)) */
740 /* > The array is indexed by the type of error information as described */
741 /* > below. There currently are up to three pieces of information */
744 /* > The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
745 /* > right-hand side. */
747 /* > The second index in ERR_BNDS_NORM(:,err) contains the following */
748 /* > three fields: */
749 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
750 /* > reciprocal condition number is less than the threshold */
751 /* > sqrt(n) * slamch('Epsilon'). */
753 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
754 /* > almost certainly within a factor of 10 of the true error */
755 /* > so long as the next entry is greater than the threshold */
756 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
757 /* > be trusted if the previous boolean is true. */
759 /* > err = 3 Reciprocal condition number: Estimated normwise */
760 /* > reciprocal condition number. Compared with the threshold */
761 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
762 /* > estimate is "guaranteed". These reciprocal condition */
763 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
764 /* > appropriately scaled matrix Z. */
765 /* > Let Z = S*A, where S scales each row by a power of the */
766 /* > radix so all absolute row sums of Z are approximately 1. */
768 /* > This subroutine is only responsible for setting the second field */
770 /* > See Lapack Working Note 165 for further details and extra */
774 /* > \param[in,out] ERR_BNDS_COMP */
776 /* > ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) */
777 /* > For each right-hand side, this array contains information about */
778 /* > various error bounds and condition numbers corresponding to the */
779 /* > componentwise relative error, which is defined as follows: */
781 /* > Componentwise relative error in the ith solution vector: */
782 /* > abs(XTRUE(j,i) - X(j,i)) */
783 /* > max_j ---------------------- */
786 /* > The array is indexed by the right-hand side i (on which the */
787 /* > componentwise relative error depends), and the type of error */
788 /* > information as described below. There currently are up to three */
789 /* > pieces of information returned for each right-hand side. If */
790 /* > componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
791 /* > ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most */
792 /* > the first (:,N_ERR_BNDS) entries are returned. */
794 /* > The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
795 /* > right-hand side. */
797 /* > The second index in ERR_BNDS_COMP(:,err) contains the following */
798 /* > three fields: */
799 /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */
800 /* > reciprocal condition number is less than the threshold */
801 /* > sqrt(n) * slamch('Epsilon'). */
803 /* > err = 2 "Guaranteed" error bound: The estimated forward error, */
804 /* > almost certainly within a factor of 10 of the true error */
805 /* > so long as the next entry is greater than the threshold */
806 /* > sqrt(n) * slamch('Epsilon'). This error bound should only */
807 /* > be trusted if the previous boolean is true. */
809 /* > err = 3 Reciprocal condition number: Estimated componentwise */
810 /* > reciprocal condition number. Compared with the threshold */
811 /* > sqrt(n) * slamch('Epsilon') to determine if the error */
812 /* > estimate is "guaranteed". These reciprocal condition */
813 /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
814 /* > appropriately scaled matrix Z. */
815 /* > Let Z = S*(A*diag(x)), where x is the solution for the */
816 /* > current right-hand side and S scales each row of */
817 /* > A*diag(x) by a power of the radix so all absolute row */
818 /* > sums of Z are approximately 1. */
820 /* > This subroutine is only responsible for setting the second field */
822 /* > See Lapack Working Note 165 for further details and extra */
826 /* > \param[in] RES */
828 /* > RES is REAL array, dimension (N) */
829 /* > Workspace to hold the intermediate residual. */
832 /* > \param[in] AYB */
834 /* > AYB is REAL array, dimension (N) */
835 /* > Workspace. This can be the same workspace passed for Y_TAIL. */
838 /* > \param[in] DY */
840 /* > DY is REAL array, dimension (N) */
841 /* > Workspace to hold the intermediate solution. */
844 /* > \param[in] Y_TAIL */
846 /* > Y_TAIL is REAL array, dimension (N) */
847 /* > Workspace to hold the trailing bits of the intermediate solution. */
850 /* > \param[in] RCOND */
852 /* > RCOND is REAL */
853 /* > Reciprocal scaled condition number. This is an estimate of the */
854 /* > reciprocal Skeel condition number of the matrix A after */
855 /* > equilibration (if done). If this is less than the machine */
856 /* > precision (in particular, if it is zero), the matrix is singular */
857 /* > to working precision. Note that the error may still be small even */
858 /* > if this number is very small and the matrix appears ill- */
862 /* > \param[in] ITHRESH */
864 /* > ITHRESH is INTEGER */
865 /* > The maximum number of residual computations allowed for */
866 /* > refinement. The default is 10. For 'aggressive' set to 100 to */
867 /* > permit convergence using approximate factorizations or */
868 /* > factorizations other than LU. If the factorization uses a */
869 /* > technique other than Gaussian elimination, the guarantees in */
870 /* > ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. */
873 /* > \param[in] RTHRESH */
875 /* > RTHRESH is REAL */
876 /* > Determines when to stop refinement if the error estimate stops */
877 /* > decreasing. Refinement will stop when the next solution no longer */
878 /* > satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */
879 /* > the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */
880 /* > default value is 0.5. For 'aggressive' set to 0.9 to permit */
881 /* > convergence on extremely ill-conditioned matrices. See LAWN 165 */
882 /* > for more details. */
885 /* > \param[in] DZ_UB */
887 /* > DZ_UB is REAL */
888 /* > Determines when to start considering componentwise convergence. */
889 /* > Componentwise convergence is only considered after each component */
890 /* > of the solution Y is stable, which we definte as the relative */
891 /* > change in each component being less than DZ_UB. The default value */
892 /* > is 0.25, requiring the first bit to be stable. See LAWN 165 for */
893 /* > more details. */
896 /* > \param[in] IGNORE_CWISE */
898 /* > IGNORE_CWISE is LOGICAL */
899 /* > If .TRUE. then ignore componentwise convergence. Default value */
903 /* > \param[out] INFO */
905 /* > INFO is INTEGER */
906 /* > = 0: Successful exit. */
907 /* > < 0: if INFO = -i, the ith argument to SGBTRS had an illegal */
914 /* > \author Univ. of Tennessee */
915 /* > \author Univ. of California Berkeley */
916 /* > \author Univ. of Colorado Denver */
917 /* > \author NAG Ltd. */
919 /* > \date June 2017 */
921 /* > \ingroup realGBcomputational */
923 /* ===================================================================== */
924 /* Subroutine */ int sla_gbrfsx_extended_(integer *prec_type__, integer *
925 trans_type__, integer *n, integer *kl, integer *ku, integer *nrhs,
926 real *ab, integer *ldab, real *afb, integer *ldafb, integer *ipiv,
927 logical *colequ, real *c__, real *b, integer *ldb, real *y, integer *
928 ldy, real *berr_out__, integer *n_norms__, real *err_bnds_norm__,
929 real *err_bnds_comp__, real *res, real *ayb, real *dy, real *y_tail__,
930 real *rcond, integer *ithresh, real *rthresh, real *dz_ub__, logical
931 *ignore_cwise__, integer *info)
933 /* System generated locals */
934 integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
935 y_dim1, y_offset, err_bnds_norm_dim1, err_bnds_norm_offset,
936 err_bnds_comp_dim1, err_bnds_comp_offset, i__1, i__2, i__3;
940 /* Local variables */
941 real dx_x__, dz_z__, ymin;
942 extern /* Subroutine */ int sla_lin_berr_(integer *, integer *, integer *
943 , real *, real *, real *), blas_sgbmv_x__(integer *, integer *,
944 integer *, integer *, integer *, real *, real *, integer *, real *
945 , integer *, real *, real *, integer *, integer *);
946 real dxratmax, dzratmax;
947 integer y_prec_state__, i__, j, m;
948 extern /* Subroutine */ int blas_sgbmv2_x_(integer *, integer *, integer
949 *, integer *, integer *, real *, real *, integer *, real *, real *
950 , integer *, real *, real *, integer *, integer *), sla_gbamv__(
951 integer *, integer *, integer *, integer *, integer *, real *,
952 real *, integer *, real *, integer *, real *, real *, integer *),
953 sgbmv_(char *, integer *, integer *, integer *, integer *, real *,
954 real *, integer *, real *, integer *, real *, real *, integer *);
959 extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
962 extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
964 real myhugeval, prev_dz_z__, yk;
965 extern real slamch_(char *);
966 real final_dx_x__, final_dz_z__;
967 extern /* Subroutine */ int sgbtrs_(char *, integer *, integer *, integer
968 *, integer *, real *, integer *, integer *, real *, integer *,
971 extern /* Subroutine */ int sla_wwaddw_(integer *, real *, real *, real *
973 extern /* Character */ VOID chla_transtype_(char *, integer *);
977 integer x_state__, z_state__;
981 /* -- LAPACK computational routine (version 3.7.1) -- */
982 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
983 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
987 /* ===================================================================== */
990 /* Parameter adjustments */
991 err_bnds_comp_dim1 = *nrhs;
992 err_bnds_comp_offset = 1 + err_bnds_comp_dim1 * 1;
993 err_bnds_comp__ -= err_bnds_comp_offset;
994 err_bnds_norm_dim1 = *nrhs;
995 err_bnds_norm_offset = 1 + err_bnds_norm_dim1 * 1;
996 err_bnds_norm__ -= err_bnds_norm_offset;
998 ab_offset = 1 + ab_dim1 * 1;
1001 afb_offset = 1 + afb_dim1 * 1;
1006 b_offset = 1 + b_dim1 * 1;
1009 y_offset = 1 + y_dim1 * 1;
1021 chla_transtype_(ch__1, trans_type__);
1022 *(unsigned char *)trans = *(unsigned char *)&ch__1[0];
1023 eps = slamch_("Epsilon");
1024 myhugeval = slamch_("Overflow");
1025 /* Force MYHUGEVAL to Inf */
1026 myhugeval *= myhugeval;
1027 /* Using MYHUGEVAL may lead to spurious underflows. */
1028 incr_thresh__ = (real) (*n) * eps;
1031 for (j = 1; j <= i__1; ++j) {
1033 if (y_prec_state__ == 2) {
1035 for (i__ = 1; i__ <= i__2; ++i__) {
1036 y_tail__[i__] = 0.f;
1043 final_dx_x__ = myhugeval;
1044 final_dz_z__ = myhugeval;
1045 prevnormdx = myhugeval;
1046 prev_dz_z__ = myhugeval;
1051 incr_prec__ = FALSE_;
1053 for (cnt = 1; cnt <= i__2; ++cnt) {
1055 /* Compute residual RES = B_s - op(A_s) * Y, */
1056 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1058 scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1059 if (y_prec_state__ == 0) {
1060 sgbmv_(trans, &m, n, kl, ku, &c_b6, &ab[ab_offset], ldab, &y[
1061 j * y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1);
1062 } else if (y_prec_state__ == 1) {
1063 blas_sgbmv_x__(trans_type__, n, n, kl, ku, &c_b6, &ab[
1064 ab_offset], ldab, &y[j * y_dim1 + 1], &c__1, &c_b8, &
1065 res[1], &c__1, prec_type__);
1067 blas_sgbmv2_x__(trans_type__, n, n, kl, ku, &c_b6, &ab[
1068 ab_offset], ldab, &y[j * y_dim1 + 1], &y_tail__[1], &
1069 c__1, &c_b8, &res[1], &c__1, prec_type__);
1071 /* XXX: RES is no longer needed. */
1072 scopy_(n, &res[1], &c__1, &dy[1], &c__1);
1073 sgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1]
1076 /* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */
1084 for (i__ = 1; i__ <= i__3; ++i__) {
1085 yk = (r__1 = y[i__ + j * y_dim1], abs(r__1));
1086 dyk = (r__1 = dy[i__], abs(r__1));
1089 r__1 = dz_z__, r__2 = dyk / yk;
1090 dz_z__ = f2cmax(r__1,r__2);
1091 } else if (dyk != 0.f) {
1094 ymin = f2cmin(ymin,yk);
1095 normy = f2cmax(normy,yk);
1098 r__1 = normx, r__2 = yk * c__[i__];
1099 normx = f2cmax(r__1,r__2);
1101 r__1 = normdx, r__2 = dyk * c__[i__];
1102 normdx = f2cmax(r__1,r__2);
1105 normdx = f2cmax(normdx,dyk);
1109 dx_x__ = normdx / normx;
1110 } else if (normdx == 0.f) {
1115 dxrat = normdx / prevnormdx;
1116 dzrat = dz_z__ / prev_dz_z__;
1118 /* Check termination criteria. */
1120 if (! (*ignore_cwise__) && ymin * *rcond < incr_thresh__ * normy
1121 && y_prec_state__ < 2) {
1122 incr_prec__ = TRUE_;
1124 if (x_state__ == 3 && dxrat <= *rthresh) {
1127 if (x_state__ == 1) {
1128 if (dx_x__ <= eps) {
1130 } else if (dxrat > *rthresh) {
1131 if (y_prec_state__ != 2) {
1132 incr_prec__ = TRUE_;
1137 if (dxrat > dxratmax) {
1141 if (x_state__ > 1) {
1142 final_dx_x__ = dx_x__;
1145 if (z_state__ == 0 && dz_z__ <= *dz_ub__) {
1148 if (z_state__ == 3 && dzrat <= *rthresh) {
1151 if (z_state__ == 1) {
1152 if (dz_z__ <= eps) {
1154 } else if (dz_z__ > *dz_ub__) {
1157 final_dz_z__ = myhugeval;
1158 } else if (dzrat > *rthresh) {
1159 if (y_prec_state__ != 2) {
1160 incr_prec__ = TRUE_;
1165 if (dzrat > dzratmax) {
1169 if (z_state__ > 1) {
1170 final_dz_z__ = dz_z__;
1174 /* Exit if both normwise and componentwise stopped working, */
1175 /* but if componentwise is unstable, let it go at least two */
1178 if (x_state__ != 1) {
1179 if (*ignore_cwise__) {
1182 if (z_state__ == 3 || z_state__ == 2) {
1185 if (z_state__ == 0 && cnt > 1) {
1190 incr_prec__ = FALSE_;
1193 for (i__ = 1; i__ <= i__3; ++i__) {
1194 y_tail__[i__] = 0.f;
1197 prevnormdx = normdx;
1198 prev_dz_z__ = dz_z__;
1200 /* Update soluton. */
1202 if (y_prec_state__ < 2) {
1203 saxpy_(n, &c_b8, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1);
1205 sla_wwaddw_(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]);
1208 /* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't CALL MYEXIT. */
1211 /* Set final_* when cnt hits ithresh. */
1213 if (x_state__ == 1) {
1214 final_dx_x__ = dx_x__;
1216 if (z_state__ == 1) {
1217 final_dz_z__ = dz_z__;
1220 /* Compute error bounds. */
1222 if (*n_norms__ >= 1) {
1223 err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = final_dx_x__ / (
1226 if (*n_norms__ >= 2) {
1227 err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = final_dz_z__ / (
1231 /* Compute componentwise relative backward error from formula */
1232 /* f2cmax(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
1233 /* where abs(Z) is the componentwise absolute value of the matrix */
1236 /* Compute residual RES = B_s - op(A_s) * Y, */
1237 /* op(A) = A, A**T, or A**H depending on TRANS (and type). */
1239 scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
1240 sgbmv_(trans, n, n, kl, ku, &c_b6, &ab[ab_offset], ldab, &y[j *
1241 y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1);
1243 for (i__ = 1; i__ <= i__2; ++i__) {
1244 ayb[i__] = (r__1 = b[i__ + j * b_dim1], abs(r__1));
1247 /* Compute abs(op(A_s))*abs(Y) + abs(B_s). */
1249 sla_gbamv_(trans_type__, n, n, kl, ku, &c_b8, &ab[ab_offset], ldab, &
1250 y[j * y_dim1 + 1], &c__1, &c_b8, &ayb[1], &c__1);
1251 sla_lin_berr_(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]);
1253 /* End of loop for each RHS */
1258 } /* sla_gbrfsx_extended__ */