#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 SGBSVX computes the solution to system of linear equations A * X = B for GB matrices */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SGBSVX + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, */ /* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, */ /* RCOND, FERR, BERR, WORK, IWORK, INFO ) */ /* CHARACTER EQUED, FACT, TRANS */ /* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS */ /* REAL RCOND */ /* INTEGER IPIV( * ), IWORK( * ) */ /* REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), */ /* $ BERR( * ), C( * ), FERR( * ), R( * ), */ /* $ WORK( * ), X( LDX, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > SGBSVX uses the LU factorization to compute the solution to a real */ /* > system of linear equations A * X = B, A**T * X = B, or A**H * X = B, */ /* > where A is a band matrix of order N with KL subdiagonals and KU */ /* > superdiagonals, and X and B are N-by-NRHS matrices. */ /* > */ /* > Error bounds on the solution and a condition estimate are also */ /* > provided. */ /* > \endverbatim */ /* > \par Description: */ /* ================= */ /* > */ /* > \verbatim */ /* > */ /* > The following steps are performed by this subroutine: */ /* > */ /* > 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 = L * U, */ /* > where L is a product of permutation and unit lower triangular */ /* > matrices with KL subdiagonals, and U is upper triangular with */ /* > KL+KU superdiagonals. */ /* > */ /* > 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. If the */ /* > reciprocal of the condition number is less than machine precision, */ /* > INFO = N+1 is returned as a warning, but 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. Iterative refinement is applied to improve the computed solution */ /* > matrix and calculate error bounds and backward error estimates */ /* > for it. */ /* > */ /* > 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: */ /* ========== */ /* > \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, AFB 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. */ /* > AB, AFB, and IPIV are not modified. */ /* > = 'N': The matrix A will be copied to AFB and factored. */ /* > = 'E': The matrix A will be equilibrated if necessary, then */ /* > copied to AFB 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 (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] KL */ /* > \verbatim */ /* > KL is INTEGER */ /* > The number of subdiagonals within the band of A. KL >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] KU */ /* > \verbatim */ /* > KU is INTEGER */ /* > The number of superdiagonals within the band of A. KU >= 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] AB */ /* > \verbatim */ /* > AB is REAL array, dimension (LDAB,N) */ /* > On entry, the matrix A in band storage, in rows 1 to KL+KU+1. */ /* > The j-th column of A is stored in the j-th column of the */ /* > array AB as follows: */ /* > AB(KU+1+i-j,j) = A(i,j) for f2cmax(1,j-KU)<=i<=f2cmin(N,j+kl) */ /* > */ /* > If FACT = 'F' and EQUED is not 'N', then A must have been */ /* > equilibrated by the scaling factors in R and/or C. AB 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] LDAB */ /* > \verbatim */ /* > LDAB is INTEGER */ /* > The leading dimension of the array AB. LDAB >= KL+KU+1. */ /* > \endverbatim */ /* > */ /* > \param[in,out] AFB */ /* > \verbatim */ /* > AFB is REAL array, dimension (LDAFB,N) */ /* > If FACT = 'F', then AFB is an input argument and on entry */ /* > contains details of the LU factorization of the band matrix */ /* > A, as computed by SGBTRF. U is stored as an upper triangular */ /* > band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, */ /* > and the multipliers used during the factorization are stored */ /* > in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is */ /* > the factored form of the equilibrated matrix A. */ /* > */ /* > If FACT = 'N', then AFB is an output argument and on exit */ /* > returns details of the LU factorization of A. */ /* > */ /* > If FACT = 'E', then AFB is an output argument and on exit */ /* > returns details of the LU factorization of the equilibrated */ /* > matrix A (see the description of AB for the form of the */ /* > equilibrated matrix). */ /* > \endverbatim */ /* > */ /* > \param[in] LDAFB */ /* > \verbatim */ /* > LDAFB is INTEGER */ /* > The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. */ /* > \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 = L*U */ /* > as computed by SGBTRF; 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 = 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 = 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. */ /* > \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. */ /* > \endverbatim */ /* > */ /* > \param[in,out] B */ /* > \verbatim */ /* > B is REAL array, dimension (LDB,NRHS) */ /* > On entry, the 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 or INFO = N+1, 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 */ /* > The estimate of the reciprocal condition number of the matrix */ /* > A after equilibration (if done). If RCOND is less than the */ /* > machine precision (in particular, if RCOND = 0), the matrix */ /* > is singular to working precision. This condition is */ /* > indicated by a return code of INFO > 0. */ /* > \endverbatim */ /* > */ /* > \param[out] FERR */ /* > \verbatim */ /* > FERR is REAL array, dimension (NRHS) */ /* > The estimated forward error bound for each solution vector */ /* > X(j) (the j-th column of the solution matrix X). */ /* > If XTRUE is the true solution corresponding to X(j), FERR(j) */ /* > is an estimated upper bound for the magnitude of the largest */ /* > element in (X(j) - XTRUE) divided by the magnitude of the */ /* > largest element in X(j). The estimate is as reliable as */ /* > the estimate for RCOND, and is almost always a slight */ /* > overestimate of the true error. */ /* > \endverbatim */ /* > */ /* > \param[out] BERR */ /* > \verbatim */ /* > BERR is REAL array, dimension (NRHS) */ /* > 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[out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension (3*N) */ /* > On exit, WORK(1) contains the reciprocal pivot growth */ /* > factor norm(A)/norm(U). The "f2cmax absolute element" norm is */ /* > used. If WORK(1) 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, condition */ /* > estimator RCOND, and forward error bound FERR could be */ /* > unreliable. If factorization fails with 0 WORK(1) contains the reciprocal pivot growth factor for the */ /* > leading INFO columns of A. */ /* > \endverbatim */ /* > */ /* > \param[out] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (N) */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value */ /* > > 0: if INFO = i, and i is */ /* > <= N: U(i,i) 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+1: U is nonsingular, but RCOND is less than machine */ /* > precision, meaning that the matrix is singular */ /* > to working precision. Nevertheless, the */ /* > solution and error bounds are computed because */ /* > there are a number of situations where the */ /* > computed solution can be more accurate than the */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date April 2012 */ /* > \ingroup realGBsolve */ /* ===================================================================== */ /* Subroutine */ int sgbsvx_(char *fact, char *trans, integer *n, integer *kl, integer *ku, integer *nrhs, real *ab, integer *ldab, real *afb, integer *ldafb, integer *ipiv, char *equed, real *r__, real *c__, real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *ferr, real *berr, real *work, integer *iwork, integer *info) { /* System generated locals */ integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2, r__3; /* Local variables */ real amax; char norm[1]; integer i__, j; extern logical lsame_(char *, char *); real rcmin, rcmax, anorm; logical equil; integer j1, j2; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); real colcnd; extern real slangb_(char *, integer *, integer *, integer *, real *, integer *, real *), slamch_(char *); extern /* Subroutine */ int slaqgb_(integer *, integer *, integer *, integer *, real *, integer *, real *, real *, real *, real *, real *, char *); logical nofact; extern /* Subroutine */ int sgbcon_(char *, integer *, integer *, integer *, real *, integer *, integer *, real *, real *, real *, integer * , integer *), xerbla_(char *, integer *, ftnlen); real bignum; extern real slantb_(char *, char *, char *, integer *, integer *, real *, integer *, real *); extern /* Subroutine */ int sgbequ_(integer *, integer *, integer *, integer *, real *, integer *, real *, real *, real *, real *, real *, integer *); integer infequ; logical colequ; extern /* Subroutine */ int sgbrfs_(char *, integer *, integer *, integer *, integer *, real *, integer *, real *, integer *, integer *, real *, integer *, real *, integer *, real *, real *, real *, integer *, integer *), sgbtrf_(integer *, integer *, integer *, integer *, real *, integer *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *); real rowcnd; logical notran; extern /* Subroutine */ int sgbtrs_(char *, integer *, integer *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *); real smlnum; logical rowequ; real rpvgrw; /* -- 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 */ /* ===================================================================== */ /* Moved setting of INFO = N+1 so INFO does not subsequently get */ /* overwritten. Sven, 17 Mar 05. */ /* ===================================================================== */ /* Parameter adjustments */ ab_dim1 = *ldab; ab_offset = 1 + ab_dim1 * 1; ab -= ab_offset; afb_dim1 = *ldafb; afb_offset = 1 + afb_dim1 * 1; afb -= afb_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; --ferr; --berr; --work; --iwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); notran = lsame_(trans, "N"); 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"); smlnum = slamch_("Safe minimum"); bignum = 1.f / smlnum; } /* Test the input parameters. */ 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 (*kl < 0) { *info = -4; } else if (*ku < 0) { *info = -5; } else if (*nrhs < 0) { *info = -6; } else if (*ldab < *kl + *ku + 1) { *info = -8; } else if (*ldafb < (*kl << 1) + *ku + 1) { *info = -10; } else if (lsame_(fact, "F") && ! (rowequ || colequ || lsame_(equed, "N"))) { *info = -12; } 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 = -13; } 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 = -14; } else if (*n > 0) { colcnd = f2cmax(rcmin,smlnum) / f2cmin(rcmax,bignum); } else { colcnd = 1.f; } } if (*info == 0) { if (*ldb < f2cmax(1,*n)) { *info = -16; } else if (*ldx < f2cmax(1,*n)) { *info = -18; } } } if (*info != 0) { i__1 = -(*info); xerbla_("SGBSVX", &i__1, (ftnlen)6); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ sgbequ_(n, n, kl, ku, &ab[ab_offset], ldab, &r__[1], &c__[1], &rowcnd, &colcnd, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ slaqgb_(n, n, kl, ku, &ab[ab_offset], ldab, &r__[1], &c__[1], & rowcnd, &colcnd, &amax, equed); rowequ = lsame_(equed, "R") || lsame_(equed, "B"); colequ = lsame_(equed, "C") || lsame_(equed, "B"); } } /* Scale the right hand side. */ if (notran) { if (rowequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { b[i__ + j * b_dim1] = r__[i__] * b[i__ + j * b_dim1]; /* L30: */ } /* L40: */ } } } else if (colequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { b[i__ + j * b_dim1] = c__[i__] * b[i__ + j * b_dim1]; /* L50: */ } /* L60: */ } } if (nofact || equil) { /* Compute the LU factorization of the band matrix A. */ i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MAX */ i__2 = j - *ku; j1 = f2cmax(i__2,1); /* Computing MIN */ i__2 = j + *kl; j2 = f2cmin(i__2,*n); i__2 = j2 - j1 + 1; scopy_(&i__2, &ab[*ku + 1 - j + j1 + j * ab_dim1], &c__1, &afb[* kl + *ku + 1 - j + j1 + j * afb_dim1], &c__1); /* L70: */ } sgbtrf_(n, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], info); /* Return if INFO is non-zero. */ if (*info > 0) { /* Compute the reciprocal pivot growth factor of the */ /* leading rank-deficient INFO columns of A. */ anorm = 0.f; i__1 = *info; for (j = 1; j <= i__1; ++j) { /* Computing MAX */ i__2 = *ku + 2 - j; /* Computing MIN */ i__4 = *n + *ku + 1 - j, i__5 = *kl + *ku + 1; i__3 = f2cmin(i__4,i__5); for (i__ = f2cmax(i__2,1); i__ <= i__3; ++i__) { /* Computing MAX */ r__2 = anorm, r__3 = (r__1 = ab[i__ + j * ab_dim1], abs( r__1)); anorm = f2cmax(r__2,r__3); /* L80: */ } /* L90: */ } /* Computing MIN */ i__3 = *info - 1, i__2 = *kl + *ku; i__1 = f2cmin(i__3,i__2); /* Computing MAX */ i__4 = 1, i__5 = *kl + *ku + 2 - *info; rpvgrw = slantb_("M", "U", "N", info, &i__1, &afb[f2cmax(i__4,i__5) + afb_dim1], ldafb, &work[1]); if (rpvgrw == 0.f) { rpvgrw = 1.f; } else { rpvgrw = anorm / rpvgrw; } work[1] = rpvgrw; *rcond = 0.f; return 0; } } /* Compute the norm of the matrix A and the */ /* reciprocal pivot growth factor RPVGRW. */ if (notran) { *(unsigned char *)norm = '1'; } else { *(unsigned char *)norm = 'I'; } anorm = slangb_(norm, n, kl, ku, &ab[ab_offset], ldab, &work[1]); i__1 = *kl + *ku; rpvgrw = slantb_("M", "U", "N", n, &i__1, &afb[afb_offset], ldafb, &work[ 1]); if (rpvgrw == 0.f) { rpvgrw = 1.f; } else { rpvgrw = slangb_("M", n, kl, ku, &ab[ab_offset], ldab, &work[1]) / rpvgrw; } /* Compute the reciprocal of the condition number of A. */ sgbcon_(norm, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], &anorm, rcond, &work[1], &iwork[1], info); /* Compute the solution matrix X. */ slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); sgbtrs_(trans, n, kl, ku, nrhs, &afb[afb_offset], ldafb, &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. */ sgbrfs_(trans, n, kl, ku, nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &ipiv[1], &b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], & berr[1], &work[1], &iwork[1], info); /* Transform the solution matrix X to a solution of the original */ /* system. */ if (notran) { if (colequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { x[i__ + j * x_dim1] = c__[i__] * x[i__ + j * x_dim1]; /* L100: */ } /* L110: */ } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] /= colcnd; /* L120: */ } } } else if (rowequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { x[i__ + j * x_dim1] = r__[i__] * x[i__ + j * x_dim1]; /* L130: */ } /* L140: */ } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] /= rowcnd; /* L150: */ } } /* Set INFO = N+1 if the matrix is singular to working precision. */ if (*rcond < slamch_("Epsilon")) { *info = *n + 1; } work[1] = rpvgrw; return 0; /* End of SGBSVX */ } /* sgbsvx_ */