#include #include #include #include #include #ifdef complex #undef complex #endif #ifdef I #undef I #endif #if defined(_WIN64) typedef long long BLASLONG; typedef unsigned long long BLASULONG; #else typedef long BLASLONG; typedef unsigned long BLASULONG; #endif #ifdef LAPACK_ILP64 typedef BLASLONG blasint; #if defined(_WIN64) #define blasabs(x) llabs(x) #else #define blasabs(x) labs(x) #endif #else typedef int blasint; #define blasabs(x) abs(x) #endif typedef blasint integer; typedef unsigned int uinteger; typedef char *address; typedef short int shortint; typedef float real; typedef double doublereal; typedef struct { real r, i; } complex; typedef struct { doublereal r, i; } doublecomplex; #ifdef _MSC_VER static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;} static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;} static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;} static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;} #else static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;} static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;} static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;} static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;} #endif #define pCf(z) (*_pCf(z)) #define pCd(z) (*_pCd(z)) typedef int logical; typedef short int shortlogical; typedef char logical1; typedef char integer1; #define TRUE_ (1) #define FALSE_ (0) /* Extern is for use with -E */ #ifndef Extern #define Extern extern #endif /* I/O stuff */ typedef int flag; typedef int ftnlen; typedef int ftnint; /*external read, write*/ typedef struct { flag cierr; ftnint ciunit; flag ciend; char *cifmt; ftnint cirec; } cilist; /*internal read, write*/ typedef struct { flag icierr; char *iciunit; flag iciend; char *icifmt; ftnint icirlen; ftnint icirnum; } icilist; /*open*/ typedef struct { flag oerr; ftnint ounit; char *ofnm; ftnlen ofnmlen; char *osta; char *oacc; char *ofm; ftnint orl; char *oblnk; } olist; /*close*/ typedef struct { flag cerr; ftnint cunit; char *csta; } cllist; /*rewind, backspace, endfile*/ typedef struct { flag aerr; ftnint aunit; } alist; /* inquire */ typedef struct { flag inerr; ftnint inunit; char *infile; ftnlen infilen; ftnint *inex; /*parameters in standard's order*/ ftnint *inopen; ftnint *innum; ftnint *innamed; char *inname; ftnlen innamlen; char *inacc; ftnlen inacclen; char *inseq; ftnlen inseqlen; char *indir; ftnlen indirlen; char *infmt; ftnlen infmtlen; char *inform; ftnint informlen; char *inunf; ftnlen inunflen; ftnint *inrecl; ftnint *innrec; char *inblank; ftnlen inblanklen; } inlist; #define VOID void union Multitype { /* for multiple entry points */ integer1 g; shortint h; integer i; /* longint j; */ real r; doublereal d; complex c; doublecomplex z; }; typedef union Multitype Multitype; struct Vardesc { /* for Namelist */ char *name; char *addr; ftnlen *dims; int type; }; typedef struct Vardesc Vardesc; struct Namelist { char *name; Vardesc **vars; int nvars; }; typedef struct Namelist Namelist; #define abs(x) ((x) >= 0 ? (x) : -(x)) #define dabs(x) (fabs(x)) #define f2cmin(a,b) ((a) <= (b) ? (a) : (b)) #define f2cmax(a,b) ((a) >= (b) ? (a) : (b)) #define dmin(a,b) (f2cmin(a,b)) #define dmax(a,b) (f2cmax(a,b)) #define bit_test(a,b) ((a) >> (b) & 1) #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b))) #define bit_set(a,b) ((a) | ((uinteger)1 << (b))) #define abort_() { sig_die("Fortran abort routine called", 1); } #define c_abs(z) (cabsf(Cf(z))) #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); } #ifdef _MSC_VER #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]);} #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]);} #else #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);} #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);} #endif #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));} #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));} #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));} //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));} #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));} #define d_abs(x) (fabs(*(x))) #define d_acos(x) (acos(*(x))) #define d_asin(x) (asin(*(x))) #define d_atan(x) (atan(*(x))) #define d_atn2(x, y) (atan2(*(x),*(y))) #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); } #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); } #define d_cos(x) (cos(*(x))) #define d_cosh(x) (cosh(*(x))) #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 ) #define d_exp(x) (exp(*(x))) #define d_imag(z) (cimag(Cd(z))) #define r_imag(z) (cimagf(Cf(z))) #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define d_log(x) (log(*(x))) #define d_mod(x, y) (fmod(*(x), *(y))) #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x))) #define d_nint(x) u_nint(*(x)) #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a))) #define d_sign(a,b) u_sign(*(a),*(b)) #define r_sign(a,b) u_sign(*(a),*(b)) #define d_sin(x) (sin(*(x))) #define d_sinh(x) (sinh(*(x))) #define d_sqrt(x) (sqrt(*(x))) #define d_tan(x) (tan(*(x))) #define d_tanh(x) (tanh(*(x))) #define i_abs(x) abs(*(x)) #define i_dnnt(x) ((integer)u_nint(*(x))) #define i_len(s, n) (n) #define i_nint(x) ((integer)u_nint(*(x))) #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b))) #define pow_dd(ap, bp) ( pow(*(ap), *(bp))) #define pow_si(B,E) spow_ui(*(B),*(E)) #define pow_ri(B,E) spow_ui(*(B),*(E)) #define pow_di(B,E) dpow_ui(*(B),*(E)) #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));} #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));} #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));} #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++ = ' '; } #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d)))) #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]; } #define sig_die(s, kill) { exit(1); } #define s_stop(s, n) {exit(0);} static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n"; #define z_abs(z) (cabs(Cd(z))) #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));} #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));} #define myexit_() break; #define mycycle() continue; #define myceiling(w) {ceil(w)} #define myhuge(w) {HUGE_VAL} //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);} #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)} /* procedure parameter types for -A and -C++ */ #define F2C_proc_par_types 1 #ifdef __cplusplus typedef logical (*L_fp)(...); #else typedef logical (*L_fp)(); #endif static float spow_ui(float x, integer n) { float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static double dpow_ui(double x, integer n) { double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #ifdef _MSC_VER static _Fcomplex cpow_ui(complex x, integer n) { complex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i; for(u = n; ; ) { if(u & 01) pow.r *= x.r, pow.i *= x.i; if(u >>= 1) x.r *= x.r, x.i *= x.i; else break; } } _Fcomplex p={pow.r, pow.i}; return p; } #else static _Complex float cpow_ui(_Complex float x, integer n) { _Complex float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif #ifdef _MSC_VER static _Dcomplex zpow_ui(_Dcomplex x, integer n) { _Dcomplex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1]; for(u = n; ; ) { if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1]; if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1]; else break; } } _Dcomplex p = {pow._Val[0], pow._Val[1]}; return p; } #else static _Complex double zpow_ui(_Complex double x, integer n) { _Complex double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif static integer pow_ii(integer x, integer n) { integer pow; unsigned long int u; if (n <= 0) { if (n == 0 || x == 1) pow = 1; else if (x != -1) pow = x == 0 ? 1/x : 0; else n = -n; } if ((n > 0) || !(n == 0 || x == 1 || x != -1)) { u = n; for(pow = 1; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static integer dmaxloc_(double *w, integer s, integer e, integer *n) { double m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static integer smaxloc_(float *w, integer s, integer e, integer *n) { float m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) { integer n = *n_, incx = *incx_, incy = *incy_, i; #ifdef _MSC_VER _Fcomplex zdotc = {0.0, 0.0}; if (incx == 1 && incy == 1) { for (i=0;i \brief \b SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SLA_GERFSX_EXTENDED + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A, */ /* LDA, AF, LDAF, IPIV, COLEQU, C, B, */ /* LDB, Y, LDY, BERR_OUT, N_NORMS, */ /* ERRS_N, ERRS_C, RES, */ /* AYB, DY, Y_TAIL, RCOND, ITHRESH, */ /* RTHRESH, DZ_UB, IGNORE_CWISE, */ /* INFO ) */ /* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, */ /* $ TRANS_TYPE, N_NORMS, ITHRESH */ /* LOGICAL COLEQU, IGNORE_CWISE */ /* REAL RTHRESH, DZ_UB */ /* INTEGER IPIV( * ) */ /* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), */ /* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) */ /* REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), */ /* $ ERRS_N( NRHS, * ), */ /* $ ERRS_C( NRHS, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SLA_GERFSX_EXTENDED improves the computed solution to a system of */ /* > linear equations by performing extra-precise iterative refinement */ /* > and provides error bounds and backward error estimates for the solution. */ /* > This subroutine is called by SGERFSX to perform iterative refinement. */ /* > In addition to normwise error bound, the code provides maximum */ /* > componentwise error bound if possible. See comments for ERRS_N */ /* > and ERRS_C for details of the error bounds. Note that this */ /* > subroutine is only resonsible for setting the second fields of */ /* > ERRS_N and ERRS_C. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] PREC_TYPE */ /* > \verbatim */ /* > PREC_TYPE is INTEGER */ /* > Specifies the intermediate precision to be used in refinement. */ /* > The value is defined by ILAPREC(P) where P is a CHARACTER and P */ /* > = 'S': Single */ /* > = 'D': Double */ /* > = 'I': Indigenous */ /* > = 'X' or 'E': Extra */ /* > \endverbatim */ /* > */ /* > \param[in] TRANS_TYPE */ /* > \verbatim */ /* > TRANS_TYPE is INTEGER */ /* > Specifies the transposition operation on A. */ /* > The value is defined by ILATRANS(T) where T is a CHARACTER and T */ /* > = 'N': No transpose */ /* > = 'T': Transpose */ /* > = 'C': Conjugate transpose */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The number of linear equations, i.e., the order of the */ /* > matrix A. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] NRHS */ /* > \verbatim */ /* > NRHS is INTEGER */ /* > The number of right-hand-sides, i.e., the number of columns of the */ /* > matrix B. */ /* > \endverbatim */ /* > */ /* > \param[in] A */ /* > \verbatim */ /* > A is REAL array, dimension (LDA,N) */ /* > On entry, the N-by-N matrix A. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in] AF */ /* > \verbatim */ /* > AF is REAL array, dimension (LDAF,N) */ /* > The factors L and U from the factorization */ /* > A = P*L*U as computed by SGETRF. */ /* > \endverbatim */ /* > */ /* > \param[in] LDAF */ /* > \verbatim */ /* > LDAF is INTEGER */ /* > The leading dimension of the array AF. LDAF >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in] IPIV */ /* > \verbatim */ /* > IPIV is INTEGER array, dimension (N) */ /* > The pivot indices from the factorization A = P*L*U */ /* > as computed by SGETRF; row i of the matrix was interchanged */ /* > with row IPIV(i). */ /* > \endverbatim */ /* > */ /* > \param[in] COLEQU */ /* > \verbatim */ /* > COLEQU is LOGICAL */ /* > If .TRUE. then column equilibration was done to A before calling */ /* > this routine. This is needed to compute the solution and error */ /* > bounds correctly. */ /* > \endverbatim */ /* > */ /* > \param[in] C */ /* > \verbatim */ /* > C is REAL array, dimension (N) */ /* > The column scale factors for A. If COLEQU = .FALSE., C */ /* > is not accessed. If C is input, each element of C should be a power */ /* > of the radix to ensure a reliable solution and error estimates. */ /* > Scaling by powers of the radix does not cause rounding errors unless */ /* > the result underflows or overflows. Rounding errors during scaling */ /* > lead to refining with a matrix that is not equivalent to the */ /* > input matrix, producing error estimates that may not be */ /* > reliable. */ /* > \endverbatim */ /* > */ /* > \param[in] B */ /* > \verbatim */ /* > B is REAL array, dimension (LDB,NRHS) */ /* > The right-hand-side matrix B. */ /* > \endverbatim */ /* > */ /* > \param[in] LDB */ /* > \verbatim */ /* > LDB is INTEGER */ /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in,out] Y */ /* > \verbatim */ /* > Y is REAL array, dimension (LDY,NRHS) */ /* > On entry, the solution matrix X, as computed by SGETRS. */ /* > On exit, the improved solution matrix Y. */ /* > \endverbatim */ /* > */ /* > \param[in] LDY */ /* > \verbatim */ /* > LDY is INTEGER */ /* > The leading dimension of the array Y. LDY >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] BERR_OUT */ /* > \verbatim */ /* > BERR_OUT is REAL array, dimension (NRHS) */ /* > On exit, BERR_OUT(j) contains the componentwise relative backward */ /* > error for right-hand-side j from the formula */ /* > f2cmax(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */ /* > where abs(Z) is the componentwise absolute value of the matrix */ /* > or vector Z. This is computed by SLA_LIN_BERR. */ /* > \endverbatim */ /* > */ /* > \param[in] N_NORMS */ /* > \verbatim */ /* > N_NORMS is INTEGER */ /* > Determines which error bounds to return (see ERRS_N */ /* > and ERRS_C). */ /* > If N_NORMS >= 1 return normwise error bounds. */ /* > If N_NORMS >= 2 return componentwise error bounds. */ /* > \endverbatim */ /* > */ /* > \param[in,out] ERRS_N */ /* > \verbatim */ /* > ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS) */ /* > For each right-hand side, this array contains information about */ /* > various error bounds and condition numbers corresponding to the */ /* > normwise relative error, which is defined as follows: */ /* > */ /* > Normwise relative error in the ith solution vector: */ /* > max_j (abs(XTRUE(j,i) - X(j,i))) */ /* > ------------------------------ */ /* > max_j abs(X(j,i)) */ /* > */ /* > The array is indexed by the type of error information as described */ /* > below. There currently are up to three pieces of information */ /* > returned. */ /* > */ /* > The first index in ERRS_N(i,:) corresponds to the ith */ /* > right-hand side. */ /* > */ /* > The second index in ERRS_N(:,err) contains the following */ /* > three fields: */ /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */ /* > reciprocal condition number is less than the threshold */ /* > sqrt(n) * slamch('Epsilon'). */ /* > */ /* > err = 2 "Guaranteed" error bound: The estimated forward error, */ /* > almost certainly within a factor of 10 of the true error */ /* > so long as the next entry is greater than the threshold */ /* > sqrt(n) * slamch('Epsilon'). This error bound should only */ /* > be trusted if the previous boolean is true. */ /* > */ /* > err = 3 Reciprocal condition number: Estimated normwise */ /* > reciprocal condition number. Compared with the threshold */ /* > sqrt(n) * slamch('Epsilon') to determine if the error */ /* > estimate is "guaranteed". These reciprocal condition */ /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */ /* > appropriately scaled matrix Z. */ /* > Let Z = S*A, where S scales each row by a power of the */ /* > radix so all absolute row sums of Z are approximately 1. */ /* > */ /* > This subroutine is only responsible for setting the second field */ /* > above. */ /* > See Lapack Working Note 165 for further details and extra */ /* > cautions. */ /* > \endverbatim */ /* > */ /* > \param[in,out] ERRS_C */ /* > \verbatim */ /* > ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS) */ /* > For each right-hand side, this array contains information about */ /* > various error bounds and condition numbers corresponding to the */ /* > componentwise relative error, which is defined as follows: */ /* > */ /* > Componentwise relative error in the ith solution vector: */ /* > abs(XTRUE(j,i) - X(j,i)) */ /* > max_j ---------------------- */ /* > abs(X(j,i)) */ /* > */ /* > The array is indexed by the right-hand side i (on which the */ /* > componentwise relative error depends), and the type of error */ /* > information as described below. There currently are up to three */ /* > pieces of information returned for each right-hand side. If */ /* > componentwise accuracy is not requested (PARAMS(3) = 0.0), then */ /* > ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most */ /* > the first (:,N_ERR_BNDS) entries are returned. */ /* > */ /* > The first index in ERRS_C(i,:) corresponds to the ith */ /* > right-hand side. */ /* > */ /* > The second index in ERRS_C(:,err) contains the following */ /* > three fields: */ /* > err = 1 "Trust/don't trust" boolean. Trust the answer if the */ /* > reciprocal condition number is less than the threshold */ /* > sqrt(n) * slamch('Epsilon'). */ /* > */ /* > err = 2 "Guaranteed" error bound: The estimated forward error, */ /* > almost certainly within a factor of 10 of the true error */ /* > so long as the next entry is greater than the threshold */ /* > sqrt(n) * slamch('Epsilon'). This error bound should only */ /* > be trusted if the previous boolean is true. */ /* > */ /* > err = 3 Reciprocal condition number: Estimated componentwise */ /* > reciprocal condition number. Compared with the threshold */ /* > sqrt(n) * slamch('Epsilon') to determine if the error */ /* > estimate is "guaranteed". These reciprocal condition */ /* > numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */ /* > appropriately scaled matrix Z. */ /* > Let Z = S*(A*diag(x)), where x is the solution for the */ /* > current right-hand side and S scales each row of */ /* > A*diag(x) by a power of the radix so all absolute row */ /* > sums of Z are approximately 1. */ /* > */ /* > This subroutine is only responsible for setting the second field */ /* > above. */ /* > See Lapack Working Note 165 for further details and extra */ /* > cautions. */ /* > \endverbatim */ /* > */ /* > \param[in] RES */ /* > \verbatim */ /* > RES is REAL array, dimension (N) */ /* > Workspace to hold the intermediate residual. */ /* > \endverbatim */ /* > */ /* > \param[in] AYB */ /* > \verbatim */ /* > AYB is REAL array, dimension (N) */ /* > Workspace. This can be the same workspace passed for Y_TAIL. */ /* > \endverbatim */ /* > */ /* > \param[in] DY */ /* > \verbatim */ /* > DY is REAL array, dimension (N) */ /* > Workspace to hold the intermediate solution. */ /* > \endverbatim */ /* > */ /* > \param[in] Y_TAIL */ /* > \verbatim */ /* > Y_TAIL is REAL array, dimension (N) */ /* > Workspace to hold the trailing bits of the intermediate solution. */ /* > \endverbatim */ /* > */ /* > \param[in] RCOND */ /* > \verbatim */ /* > RCOND is REAL */ /* > Reciprocal scaled condition number. This is an estimate of the */ /* > reciprocal Skeel condition number of the matrix A after */ /* > equilibration (if done). If this is less than the machine */ /* > precision (in particular, if it is zero), the matrix is singular */ /* > to working precision. Note that the error may still be small even */ /* > if this number is very small and the matrix appears ill- */ /* > conditioned. */ /* > \endverbatim */ /* > */ /* > \param[in] ITHRESH */ /* > \verbatim */ /* > ITHRESH is INTEGER */ /* > The maximum number of residual computations allowed for */ /* > refinement. The default is 10. For 'aggressive' set to 100 to */ /* > permit convergence using approximate factorizations or */ /* > factorizations other than LU. If the factorization uses a */ /* > technique other than Gaussian elimination, the guarantees in */ /* > ERRS_N and ERRS_C may no longer be trustworthy. */ /* > \endverbatim */ /* > */ /* > \param[in] RTHRESH */ /* > \verbatim */ /* > RTHRESH is REAL */ /* > Determines when to stop refinement if the error estimate stops */ /* > decreasing. Refinement will stop when the next solution no longer */ /* > satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */ /* > the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */ /* > default value is 0.5. For 'aggressive' set to 0.9 to permit */ /* > convergence on extremely ill-conditioned matrices. See LAWN 165 */ /* > for more details. */ /* > \endverbatim */ /* > */ /* > \param[in] DZ_UB */ /* > \verbatim */ /* > DZ_UB is REAL */ /* > Determines when to start considering componentwise convergence. */ /* > Componentwise convergence is only considered after each component */ /* > of the solution Y is stable, which we definte as the relative */ /* > change in each component being less than DZ_UB. The default value */ /* > is 0.25, requiring the first bit to be stable. See LAWN 165 for */ /* > more details. */ /* > \endverbatim */ /* > */ /* > \param[in] IGNORE_CWISE */ /* > \verbatim */ /* > IGNORE_CWISE is LOGICAL */ /* > If .TRUE. then ignore componentwise convergence. Default value */ /* > is .FALSE.. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: Successful exit. */ /* > < 0: if INFO = -i, the ith argument to SGETRS had an illegal */ /* > value */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup realGEcomputational */ /* ===================================================================== */ /* Subroutine */ int sla_gerfsx_extended_(integer *prec_type__, integer * trans_type__, integer *n, integer *nrhs, real *a, integer *lda, real * af, integer *ldaf, integer *ipiv, logical *colequ, real *c__, real *b, integer *ldb, real *y, integer *ldy, real *berr_out__, integer * n_norms__, real *errs_n__, real *errs_c__, real *res, real *ayb, real *dy, real *y_tail__, real *rcond, integer *ithresh, real *rthresh, real *dz_ub__, logical *ignore_cwise__, integer *info) { /* System generated locals */ integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, y_dim1, y_offset, errs_n_dim1, errs_n_offset, errs_c_dim1, errs_c_offset, i__1, i__2, i__3; real r__1, r__2; char ch__1[1]; /* Local variables */ real dx_x__, dz_z__, ymin; extern /* Subroutine */ int sla_lin_berr_(integer *, integer *, integer * , real *, real *, real *); real dxratmax; extern /* Subroutine */ int blas_sgemv_x_(integer *, integer *, integer * , real *, real *, integer *, real *, integer *, real *, real *, integer *, integer *); real dzratmax; integer y_prec_state__, i__, j; extern /* Subroutine */ int blas_sgemv2_x_(integer *, integer *, integer *, real *, real *, integer *, real *, real *, integer *, real *, real *, integer *, integer *), sla_geamv_(integer *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *), sgemv_(char *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); real dxrat; logical incr_prec__; real dzrat; char trans[1]; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); real normx, normy; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *); real myhugeval, prev_dz_z__, yk; extern real slamch_(char *); real final_dx_x__, final_dz_z__, normdx; extern /* Subroutine */ int sgetrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *), sla_wwaddw_(integer *, real *, real *, real *); extern /* Character */ VOID chla_transtype_(char *, integer *); real prevnormdx; integer cnt; real dyk, eps; integer x_state__, z_state__; real incr_thresh__; /* -- LAPACK computational routine (version 3.7.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* December 2016 */ /* ===================================================================== */ /* Parameter adjustments */ errs_c_dim1 = *nrhs; errs_c_offset = 1 + errs_c_dim1 * 1; errs_c__ -= errs_c_offset; errs_n_dim1 = *nrhs; errs_n_offset = 1 + errs_n_dim1 * 1; errs_n__ -= errs_n_offset; a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; af_dim1 = *ldaf; af_offset = 1 + af_dim1 * 1; af -= af_offset; --ipiv; --c__; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; y_dim1 = *ldy; y_offset = 1 + y_dim1 * 1; y -= y_offset; --berr_out__; --res; --ayb; --dy; --y_tail__; /* Function Body */ if (*info != 0) { return 0; } chla_transtype_(ch__1, trans_type__); *(unsigned char *)trans = *(unsigned char *)&ch__1[0]; eps = slamch_("Epsilon"); myhugeval = slamch_("Overflow"); /* Force MYHUGEVAL to Inf */ myhugeval *= myhugeval; /* Using MYHUGEVAL may lead to spurious underflows. */ incr_thresh__ = (real) (*n) * eps; i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { y_prec_state__ = 1; if (y_prec_state__ == 2) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { y_tail__[i__] = 0.f; } } dxrat = 0.f; dxratmax = 0.f; dzrat = 0.f; dzratmax = 0.f; final_dx_x__ = myhugeval; final_dz_z__ = myhugeval; prevnormdx = myhugeval; prev_dz_z__ = myhugeval; dz_z__ = myhugeval; dx_x__ = myhugeval; x_state__ = 1; z_state__ = 0; incr_prec__ = FALSE_; i__2 = *ithresh; for (cnt = 1; cnt <= i__2; ++cnt) { /* Compute residual RES = B_s - op(A_s) * Y, */ /* op(A) = A, A**T, or A**H depending on TRANS (and type). */ scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1); if (y_prec_state__ == 0) { sgemv_(trans, n, n, &c_b6, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1); } else if (y_prec_state__ == 1) { blas_sgemv_x__(trans_type__, n, n, &c_b6, &a[a_offset], lda, & y[j * y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1, prec_type__); } else { blas_sgemv2_x__(trans_type__, n, n, &c_b6, &a[a_offset], lda, &y[j * y_dim1 + 1], &y_tail__[1], &c__1, &c_b8, &res[ 1], &c__1, prec_type__); } /* XXX: RES is no longer needed. */ scopy_(n, &res[1], &c__1, &dy[1], &c__1); sgetrs_(trans, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &dy[1], n, info); /* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */ normx = 0.f; normy = 0.f; normdx = 0.f; dz_z__ = 0.f; ymin = myhugeval; i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { yk = (r__1 = y[i__ + j * y_dim1], abs(r__1)); dyk = (r__1 = dy[i__], abs(r__1)); if (yk != 0.f) { /* Computing MAX */ r__1 = dz_z__, r__2 = dyk / yk; dz_z__ = f2cmax(r__1,r__2); } else if (dyk != 0.f) { dz_z__ = myhugeval; } ymin = f2cmin(ymin,yk); normy = f2cmax(normy,yk); if (*colequ) { /* Computing MAX */ r__1 = normx, r__2 = yk * c__[i__]; normx = f2cmax(r__1,r__2); /* Computing MAX */ r__1 = normdx, r__2 = dyk * c__[i__]; normdx = f2cmax(r__1,r__2); } else { normx = normy; normdx = f2cmax(normdx,dyk); } } if (normx != 0.f) { dx_x__ = normdx / normx; } else if (normdx == 0.f) { dx_x__ = 0.f; } else { dx_x__ = myhugeval; } dxrat = normdx / prevnormdx; dzrat = dz_z__ / prev_dz_z__; /* Check termination criteria */ if (! (*ignore_cwise__) && ymin * *rcond < incr_thresh__ * normy && y_prec_state__ < 2) { incr_prec__ = TRUE_; } if (x_state__ == 3 && dxrat <= *rthresh) { x_state__ = 1; } if (x_state__ == 1) { if (dx_x__ <= eps) { x_state__ = 2; } else if (dxrat > *rthresh) { if (y_prec_state__ != 2) { incr_prec__ = TRUE_; } else { x_state__ = 3; } } else { if (dxrat > dxratmax) { dxratmax = dxrat; } } if (x_state__ > 1) { final_dx_x__ = dx_x__; } } if (z_state__ == 0 && dz_z__ <= *dz_ub__) { z_state__ = 1; } if (z_state__ == 3 && dzrat <= *rthresh) { z_state__ = 1; } if (z_state__ == 1) { if (dz_z__ <= eps) { z_state__ = 2; } else if (dz_z__ > *dz_ub__) { z_state__ = 0; dzratmax = 0.f; final_dz_z__ = myhugeval; } else if (dzrat > *rthresh) { if (y_prec_state__ != 2) { incr_prec__ = TRUE_; } else { z_state__ = 3; } } else { if (dzrat > dzratmax) { dzratmax = dzrat; } } if (z_state__ > 1) { final_dz_z__ = dz_z__; } } /* Exit if both normwise and componentwise stopped working, */ /* but if componentwise is unstable, let it go at least two */ /* iterations. */ if (x_state__ != 1) { if (*ignore_cwise__) { goto L666; } if (z_state__ == 3 || z_state__ == 2) { goto L666; } if (z_state__ == 0 && cnt > 1) { goto L666; } } if (incr_prec__) { incr_prec__ = FALSE_; ++y_prec_state__; i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { y_tail__[i__] = 0.f; } } prevnormdx = normdx; prev_dz_z__ = dz_z__; /* Update soluton. */ if (y_prec_state__ < 2) { saxpy_(n, &c_b8, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1); } else { sla_wwaddw_(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]); } } /* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't CALL MYEXIT. */ L666: /* Set final_* when cnt hits ithresh. */ if (x_state__ == 1) { final_dx_x__ = dx_x__; } if (z_state__ == 1) { final_dz_z__ = dz_z__; } /* Compute error bounds */ if (*n_norms__ >= 1) { errs_n__[j + (errs_n_dim1 << 1)] = final_dx_x__ / (1 - dxratmax); } if (*n_norms__ >= 2) { errs_c__[j + (errs_c_dim1 << 1)] = final_dz_z__ / (1 - dzratmax); } /* Compute componentwise relative backward error from formula */ /* f2cmax(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */ /* where abs(Z) is the componentwise absolute value of the matrix */ /* or vector Z. */ /* Compute residual RES = B_s - op(A_s) * Y, */ /* op(A) = A, A**T, or A**H depending on TRANS (and type). */ scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1); sgemv_(trans, n, n, &c_b6, &a[a_offset], lda, &y[j * y_dim1 + 1], & c__1, &c_b8, &res[1], &c__1); i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { ayb[i__] = (r__1 = b[i__ + j * b_dim1], abs(r__1)); } /* Compute abs(op(A_s))*abs(Y) + abs(B_s). */ sla_geamv_(trans_type__, n, n, &c_b8, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &c_b8, &ayb[1], &c__1); sla_lin_berr_(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]); /* End of loop for each RHS. */ } return 0; } /* sla_gerfsx_extended__ */