#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 SGESVXX computes the solution to system of linear equations A * X = B for GE matrices */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SGESVXX + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, */ /* EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, */ /* BERR, N_ERR_BNDS, ERR_BNDS_NORM, */ /* ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, */ /* INFO ) */ /* CHARACTER EQUED, FACT, TRANS */ /* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, */ /* $ N_ERR_BNDS */ /* REAL RCOND, RPVGRW */ /* INTEGER IPIV( * ), IWORK( * ) */ /* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), */ /* $ X( LDX , * ),WORK( * ) */ /* REAL R( * ), C( * ), PARAMS( * ), BERR( * ), */ /* $ ERR_BNDS_NORM( NRHS, * ), */ /* $ ERR_BNDS_COMP( NRHS, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SGESVXX uses the LU factorization to compute the solution to a */ /* > real system of linear equations A * X = B, where A is an */ /* > N-by-N matrix and X and B are N-by-NRHS matrices. */ /* > */ /* > If requested, both normwise and maximum componentwise error bounds */ /* > are returned. SGESVXX will return a solution with a tiny */ /* > guaranteed error (O(eps) where eps is the working machine */ /* > precision) unless the matrix is very ill-conditioned, in which */ /* > case a warning is returned. Relevant condition numbers also are */ /* > calculated and returned. */ /* > */ /* > SGESVXX accepts user-provided factorizations and equilibration */ /* > factors; see the definitions of the FACT and EQUED options. */ /* > Solving with refinement and using a factorization from a previous */ /* > SGESVXX call will also produce a solution with either O(eps) */ /* > errors or warnings, but we cannot make that claim for general */ /* > user-provided factorizations and equilibration factors if they */ /* > differ from what SGESVXX would itself produce. */ /* > \endverbatim */ /* > \par Description: */ /* ================= */ /* > */ /* > \verbatim */ /* > */ /* > The following steps are performed: */ /* > */ /* > 1. If FACT = 'E', real scaling factors are computed to equilibrate */ /* > the system: */ /* > */ /* > TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B */ /* > TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B */ /* > TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B */ /* > */ /* > Whether or not the system will be equilibrated depends on the */ /* > scaling of the matrix A, but if equilibration is used, A is */ /* > overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') */ /* > or diag(C)*B (if TRANS = 'T' or 'C'). */ /* > */ /* > 2. If FACT = 'N' or 'E', the LU decomposition is used to factor */ /* > the matrix A (after equilibration if FACT = 'E') as */ /* > */ /* > A = P * L * U, */ /* > */ /* > where P is a permutation matrix, L is a unit lower triangular */ /* > matrix, and U is upper triangular. */ /* > */ /* > 3. If some U(i,i)=0, so that U is exactly singular, then the */ /* > routine returns with INFO = i. Otherwise, the factored form of A */ /* > is used to estimate the condition number of the matrix A (see */ /* > argument RCOND). If the reciprocal of the condition number is less */ /* > than machine precision, the routine still goes on to solve for X */ /* > and compute error bounds as described below. */ /* > */ /* > 4. The system of equations is solved for X using the factored form */ /* > of A. */ /* > */ /* > 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), */ /* > the routine will use iterative refinement to try to get a small */ /* > error and error bounds. Refinement calculates the residual to at */ /* > least twice the working precision. */ /* > */ /* > 6. If equilibration was used, the matrix X is premultiplied by */ /* > diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so */ /* > that it solves the original system before equilibration. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \verbatim */ /* > Some optional parameters are bundled in the PARAMS array. These */ /* > settings determine how refinement is performed, but often the */ /* > defaults are acceptable. If the defaults are acceptable, users */ /* > can pass NPARAMS = 0 which prevents the source code from accessing */ /* > the PARAMS argument. */ /* > \endverbatim */ /* > */ /* > \param[in] FACT */ /* > \verbatim */ /* > FACT is CHARACTER*1 */ /* > Specifies whether or not the factored form of the matrix A is */ /* > supplied on entry, and if not, whether the matrix A should be */ /* > equilibrated before it is factored. */ /* > = 'F': On entry, AF and IPIV contain the factored form of A. */ /* > If EQUED is not 'N', the matrix A has been */ /* > equilibrated with scaling factors given by R and C. */ /* > A, AF, and IPIV are not modified. */ /* > = 'N': The matrix A will be copied to AF and factored. */ /* > = 'E': The matrix A will be equilibrated if necessary, then */ /* > copied to AF and factored. */ /* > \endverbatim */ /* > */ /* > \param[in] TRANS */ /* > \verbatim */ /* > TRANS is CHARACTER*1 */ /* > Specifies the form of the system of equations: */ /* > = 'N': A * X = B (No transpose) */ /* > = 'T': A**T * X = B (Transpose) */ /* > = 'C': A**H * X = B (Conjugate Transpose = 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 matrices B and X. NRHS >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is REAL array, dimension (LDA,N) */ /* > On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is */ /* > not 'N', then A must have been equilibrated by the scaling */ /* > factors in R and/or C. A is not modified if FACT = 'F' or */ /* > 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ /* > */ /* > On exit, if EQUED .ne. 'N', A is scaled as follows: */ /* > EQUED = 'R': A := diag(R) * A */ /* > EQUED = 'C': A := A * diag(C) */ /* > EQUED = 'B': A := diag(R) * A * diag(C). */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in,out] AF */ /* > \verbatim */ /* > AF is REAL array, dimension (LDAF,N) */ /* > If FACT = 'F', then AF is an input argument and on entry */ /* > contains the factors L and U from the factorization */ /* > A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then */ /* > AF is the factored form of the equilibrated matrix A. */ /* > */ /* > If FACT = 'N', then AF is an output argument and on exit */ /* > returns the factors L and U from the factorization A = P*L*U */ /* > of the original matrix A. */ /* > */ /* > If FACT = 'E', then AF is an output argument and on exit */ /* > returns the factors L and U from the factorization A = P*L*U */ /* > of the equilibrated matrix A (see the description of A for */ /* > the form of the equilibrated matrix). */ /* > \endverbatim */ /* > */ /* > \param[in] LDAF */ /* > \verbatim */ /* > LDAF is INTEGER */ /* > The leading dimension of the array AF. LDAF >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in,out] IPIV */ /* > \verbatim */ /* > IPIV is INTEGER array, dimension (N) */ /* > If FACT = 'F', then IPIV is an input argument and on entry */ /* > contains 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). */ /* > */ /* > If FACT = 'N', then IPIV is an output argument and on exit */ /* > contains the pivot indices from the factorization A = P*L*U */ /* > of the original matrix A. */ /* > */ /* > If FACT = 'E', then IPIV is an output argument and on exit */ /* > contains the pivot indices from the factorization A = P*L*U */ /* > of the equilibrated matrix A. */ /* > \endverbatim */ /* > */ /* > \param[in,out] EQUED */ /* > \verbatim */ /* > EQUED is CHARACTER*1 */ /* > Specifies the form of equilibration that was done. */ /* > = 'N': No equilibration (always true if FACT = 'N'). */ /* > = 'R': Row equilibration, i.e., A has been premultiplied by */ /* > diag(R). */ /* > = 'C': Column equilibration, i.e., A has been postmultiplied */ /* > by diag(C). */ /* > = 'B': Both row and column equilibration, i.e., A has been */ /* > replaced by diag(R) * A * diag(C). */ /* > EQUED is an input argument if FACT = 'F'; otherwise, it is an */ /* > output argument. */ /* > \endverbatim */ /* > */ /* > \param[in,out] R */ /* > \verbatim */ /* > R is REAL array, dimension (N) */ /* > The row scale factors for A. If EQUED = 'R' or 'B', A is */ /* > multiplied on the left by diag(R); if EQUED = 'N' or 'C', R */ /* > is not accessed. R is an input argument if FACT = 'F'; */ /* > otherwise, R is an output argument. If FACT = 'F' and */ /* > EQUED = 'R' or 'B', each element of R must be positive. */ /* > If R is output, each element of R is a power of the radix. */ /* > If R is input, each element of R 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,out] C */ /* > \verbatim */ /* > C is REAL array, dimension (N) */ /* > The column scale factors for A. If EQUED = 'C' or 'B', A is */ /* > multiplied on the right by diag(C); if EQUED = 'N' or 'R', C */ /* > is not accessed. C is an input argument if FACT = 'F'; */ /* > otherwise, C is an output argument. If FACT = 'F' and */ /* > EQUED = 'C' or 'B', each element of C must be positive. */ /* > If C is output, each element of C is a power of the radix. */ /* > 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,out] B */ /* > \verbatim */ /* > B is REAL array, dimension (LDB,NRHS) */ /* > On entry, the N-by-NRHS right hand side matrix B. */ /* > On exit, */ /* > if EQUED = 'N', B is not modified; */ /* > if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by */ /* > diag(R)*B; */ /* > if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is */ /* > overwritten by diag(C)*B. */ /* > \endverbatim */ /* > */ /* > \param[in] LDB */ /* > \verbatim */ /* > LDB is INTEGER */ /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] X */ /* > \verbatim */ /* > X is REAL array, dimension (LDX,NRHS) */ /* > If INFO = 0, the N-by-NRHS solution matrix X to the original */ /* > system of equations. Note that A and B are modified on exit */ /* > if EQUED .ne. 'N', and the solution to the equilibrated system is */ /* > inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or */ /* > inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDX */ /* > \verbatim */ /* > LDX is INTEGER */ /* > The leading dimension of the array X. LDX >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] 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[out] RPVGRW */ /* > \verbatim */ /* > RPVGRW is REAL */ /* > Reciprocal pivot growth. On exit, this contains the reciprocal */ /* > pivot growth factor norm(A)/norm(U). The "f2cmax absolute element" */ /* > norm is used. If this is much less than 1, then the stability of */ /* > the LU factorization of the (equilibrated) matrix A could be poor. */ /* > This also means that the solution X, estimated condition numbers, */ /* > and error bounds could be unreliable. If factorization fails with */ /* > 0 for the leading INFO columns of A. In SGESVX, this quantity is */ /* > returned in WORK(1). */ /* > \endverbatim */ /* > */ /* > \param[out] BERR */ /* > \verbatim */ /* > BERR is REAL array, dimension (NRHS) */ /* > Componentwise relative backward error. This is the */ /* > componentwise relative backward error of each solution vector X(j) */ /* > (i.e., the smallest relative change in any element of A or B that */ /* > makes X(j) an exact solution). */ /* > \endverbatim */ /* > */ /* > \param[in] N_ERR_BNDS */ /* > \verbatim */ /* > N_ERR_BNDS is INTEGER */ /* > Number of error bounds to return for each right hand side */ /* > and each type (normwise or componentwise). See ERR_BNDS_NORM and */ /* > ERR_BNDS_COMP below. */ /* > \endverbatim */ /* > */ /* > \param[out] ERR_BNDS_NORM */ /* > \verbatim */ /* > ERR_BNDS_NORM 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 ERR_BNDS_NORM(i,:) corresponds to the ith */ /* > right-hand side. */ /* > */ /* > The second index in ERR_BNDS_NORM(:,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. */ /* > */ /* > See Lapack Working Note 165 for further details and extra */ /* > cautions. */ /* > \endverbatim */ /* > */ /* > \param[out] ERR_BNDS_COMP */ /* > \verbatim */ /* > ERR_BNDS_COMP 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 */ /* > ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most */ /* > the first (:,N_ERR_BNDS) entries are returned. */ /* > */ /* > The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */ /* > right-hand side. */ /* > */ /* > The second index in ERR_BNDS_COMP(:,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. */ /* > */ /* > See Lapack Working Note 165 for further details and extra */ /* > cautions. */ /* > \endverbatim */ /* > */ /* > \param[in] NPARAMS */ /* > \verbatim */ /* > NPARAMS is INTEGER */ /* > Specifies the number of parameters set in PARAMS. If <= 0, the */ /* > PARAMS array is never referenced and default values are used. */ /* > \endverbatim */ /* > */ /* > \param[in,out] PARAMS */ /* > \verbatim */ /* > PARAMS is REAL array, dimension NPARAMS */ /* > Specifies algorithm parameters. If an entry is < 0.0, then */ /* > that entry will be filled with default value used for that */ /* > parameter. Only positions up to NPARAMS are accessed; defaults */ /* > are used for higher-numbered parameters. */ /* > */ /* > PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative */ /* > refinement or not. */ /* > Default: 1.0 */ /* > = 0.0: No refinement is performed, and no error bounds are */ /* > computed. */ /* > = 1.0: Use the double-precision refinement algorithm, */ /* > possibly with doubled-single computations if the */ /* > compilation environment does not support DOUBLE */ /* > PRECISION. */ /* > (other values are reserved for future use) */ /* > */ /* > PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual */ /* > computations allowed for refinement. */ /* > Default: 10 */ /* > 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 */ /* > err_bnds_norm and err_bnds_comp may no longer be */ /* > trustworthy. */ /* > */ /* > PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code */ /* > will attempt to find a solution with small componentwise */ /* > relative error in the double-precision algorithm. Positive */ /* > is true, 0.0 is false. */ /* > Default: 1.0 (attempt componentwise convergence) */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension (4*N) */ /* > \endverbatim */ /* > */ /* > \param[out] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (N) */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: Successful exit. The solution to every right-hand side is */ /* > guaranteed. */ /* > < 0: If INFO = -i, the i-th argument had an illegal value */ /* > > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization */ /* > has been completed, but the factor U is exactly singular, so */ /* > the solution and error bounds could not be computed. RCOND = 0 */ /* > is returned. */ /* > = N+J: The solution corresponding to the Jth right-hand side is */ /* > not guaranteed. The solutions corresponding to other right- */ /* > hand sides K with K > J may not be guaranteed as well, but */ /* > only the first such right-hand side is reported. If a small */ /* > componentwise error is not requested (PARAMS(3) = 0.0) then */ /* > the Jth right-hand side is the first with a normwise error */ /* > bound that is not guaranteed (the smallest J such */ /* > that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) */ /* > the Jth right-hand side is the first with either a normwise or */ /* > componentwise error bound that is not guaranteed (the smallest */ /* > J such that either ERR_BNDS_NORM(J,1) = 0.0 or */ /* > ERR_BNDS_COMP(J,1) = 0.0). See the definition of */ /* > ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information */ /* > about all of the right-hand sides check ERR_BNDS_NORM or */ /* > ERR_BNDS_COMP. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date April 2012 */ /* > \ingroup realGEsolve */ /* ===================================================================== */ /* Subroutine */ int sgesvxx_(char *fact, char *trans, integer *n, integer * nrhs, real *a, integer *lda, real *af, integer *ldaf, integer *ipiv, char *equed, real *r__, real *c__, real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *rpvgrw, real *berr, integer * n_err_bnds__, real *err_bnds_norm__, real *err_bnds_comp__, integer * nparams, real *params, real *work, integer *iwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, err_bnds_norm_dim1, err_bnds_norm_offset, err_bnds_comp_dim1, err_bnds_comp_offset, i__1; real r__1, r__2; /* Local variables */ real amax; extern real sla_gerpvgrw_(integer *, integer *, real *, integer *, real * , integer *); integer j; extern logical lsame_(char *, char *); real rcmin, rcmax; logical equil; real colcnd; extern real slamch_(char *); logical nofact; extern /* Subroutine */ int slaqge_(integer *, integer *, real *, integer *, real *, real *, real *, real *, real *, char *), xerbla_(char *, integer *, ftnlen); real bignum; integer infequ; logical colequ; extern /* Subroutine */ int sgetrf_(integer *, integer *, real *, integer *, integer *, integer *); real rowcnd; extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *); logical notran; extern /* Subroutine */ int sgetrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *); real smlnum; logical rowequ; extern /* Subroutine */ int slascl2_(integer *, integer *, real *, real *, integer *), sgeequb_(integer *, integer *, real *, integer *, real *, real *, real *, real *, real *, integer *), sgerfsx_(char *, char *, integer *, integer *, real *, integer *, real *, integer *, integer *, real *, real *, real *, integer *, real *, integer *, real *, real *, integer *, real *, real *, integer *, real *, real *, integer *, integer *); /* -- LAPACK driver 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..-- */ /* April 2012 */ /* ================================================================== */ /* Parameter adjustments */ err_bnds_comp_dim1 = *nrhs; err_bnds_comp_offset = 1 + err_bnds_comp_dim1 * 1; err_bnds_comp__ -= err_bnds_comp_offset; err_bnds_norm_dim1 = *nrhs; err_bnds_norm_offset = 1 + err_bnds_norm_dim1 * 1; err_bnds_norm__ -= err_bnds_norm_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; --r__; --c__; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; x_dim1 = *ldx; x_offset = 1 + x_dim1 * 1; x -= x_offset; --berr; --params; --work; --iwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); notran = lsame_(trans, "N"); smlnum = slamch_("Safe minimum"); bignum = 1.f / smlnum; if (nofact || equil) { *(unsigned char *)equed = 'N'; rowequ = FALSE_; colequ = FALSE_; } else { rowequ = lsame_(equed, "R") || lsame_(equed, "B"); colequ = lsame_(equed, "C") || lsame_(equed, "B"); } /* Default is failure. If an input parameter is wrong or */ /* factorization fails, make everything look horrible. Only the */ /* pivot growth is set here, the rest is initialized in SGERFSX. */ *rpvgrw = 0.f; /* Test the input parameters. PARAMS is not tested until SGERFSX. */ if (! nofact && ! equil && ! lsame_(fact, "F")) { *info = -1; } else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*nrhs < 0) { *info = -4; } else if (*lda < f2cmax(1,*n)) { *info = -6; } else if (*ldaf < f2cmax(1,*n)) { *info = -8; } else if (lsame_(fact, "F") && ! (rowequ || colequ || lsame_(equed, "N"))) { *info = -10; } else { if (rowequ) { rcmin = bignum; rcmax = 0.f; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ r__1 = rcmin, r__2 = r__[j]; rcmin = f2cmin(r__1,r__2); /* Computing MAX */ r__1 = rcmax, r__2 = r__[j]; rcmax = f2cmax(r__1,r__2); /* L10: */ } if (rcmin <= 0.f) { *info = -11; } else if (*n > 0) { rowcnd = f2cmax(rcmin,smlnum) / f2cmin(rcmax,bignum); } else { rowcnd = 1.f; } } if (colequ && *info == 0) { rcmin = bignum; rcmax = 0.f; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ r__1 = rcmin, r__2 = c__[j]; rcmin = f2cmin(r__1,r__2); /* Computing MAX */ r__1 = rcmax, r__2 = c__[j]; rcmax = f2cmax(r__1,r__2); /* L20: */ } if (rcmin <= 0.f) { *info = -12; } else if (*n > 0) { colcnd = f2cmax(rcmin,smlnum) / f2cmin(rcmax,bignum); } else { colcnd = 1.f; } } if (*info == 0) { if (*ldb < f2cmax(1,*n)) { *info = -14; } else if (*ldx < f2cmax(1,*n)) { *info = -16; } } } if (*info != 0) { i__1 = -(*info); xerbla_("SGESVXX", &i__1, (ftnlen)7); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ sgeequb_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &colcnd, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ slaqge_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, & colcnd, &amax, equed); rowequ = lsame_(equed, "R") || lsame_(equed, "B"); colequ = lsame_(equed, "C") || lsame_(equed, "B"); } /* If the scaling factors are not applied, set them to 1.0. */ if (! rowequ) { i__1 = *n; for (j = 1; j <= i__1; ++j) { r__[j] = 1.f; } } if (! colequ) { i__1 = *n; for (j = 1; j <= i__1; ++j) { c__[j] = 1.f; } } } /* Scale the right-hand side. */ if (notran) { if (rowequ) { slascl2_(n, nrhs, &r__[1], &b[b_offset], ldb); } } else { if (colequ) { slascl2_(n, nrhs, &c__[1], &b[b_offset], ldb); } } if (nofact || equil) { /* Compute the LU factorization of A. */ slacpy_("Full", n, n, &a[a_offset], lda, &af[af_offset], ldaf); sgetrf_(n, n, &af[af_offset], ldaf, &ipiv[1], info); /* Return if INFO is non-zero. */ if (*info > 0) { /* Pivot in column INFO is exactly 0 */ /* Compute the reciprocal pivot growth factor of the */ /* leading rank-deficient INFO columns of A. */ *rpvgrw = sla_gerpvgrw_(n, info, &a[a_offset], lda, &af[ af_offset], ldaf); return 0; } } /* Compute the reciprocal pivot growth factor RPVGRW. */ *rpvgrw = sla_gerpvgrw_(n, n, &a[a_offset], lda, &af[af_offset], ldaf); /* Compute the solution matrix X. */ slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); sgetrs_(trans, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ sgerfsx_(trans, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & ipiv[1], &r__[1], &c__[1], &b[b_offset], ldb, &x[x_offset], ldx, rcond, &berr[1], n_err_bnds__, &err_bnds_norm__[ err_bnds_norm_offset], &err_bnds_comp__[err_bnds_comp_offset], nparams, ¶ms[1], &work[1], &iwork[1], info); /* Scale solutions. */ if (colequ && notran) { slascl2_(n, nrhs, &c__[1], &x[x_offset], ldx); } else if (rowequ && ! notran) { slascl2_(n, nrhs, &r__[1], &x[x_offset], ldx); } return 0; /* End of SGESVXX */ } /* sgesvxx_ */