#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 DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DPBSVX + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, */ /* EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, */ /* WORK, IWORK, INFO ) */ /* CHARACTER EQUED, FACT, UPLO */ /* INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS */ /* DOUBLE PRECISION RCOND */ /* INTEGER IWORK( * ) */ /* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), */ /* $ BERR( * ), FERR( * ), S( * ), WORK( * ), */ /* $ X( LDX, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to */ /* > compute the solution to a real system of linear equations */ /* > A * X = B, */ /* > where A is an N-by-N symmetric positive definite band matrix 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: */ /* > */ /* > 1. If FACT = 'E', real scaling factors are computed to equilibrate */ /* > the system: */ /* > diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B. */ /* > */ /* > 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ /* > factor the matrix A (after equilibration if FACT = 'E') as */ /* > A = U**T * U, if UPLO = 'U', or */ /* > A = L * L**T, if UPLO = 'L', */ /* > where U is an upper triangular band matrix, and L is a lower */ /* > triangular band matrix. */ /* > */ /* > 3. If the leading i-by-i principal minor is not positive definite, */ /* > 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(S) 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 contains the factored form of A. */ /* > If EQUED = 'Y', the matrix A has been equilibrated */ /* > with scaling factors given by S. AB and AFB will not */ /* > be 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] UPLO */ /* > \verbatim */ /* > UPLO is CHARACTER*1 */ /* > = 'U': Upper triangle of A is stored; */ /* > = 'L': Lower triangle of A is stored. */ /* > \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] KD */ /* > \verbatim */ /* > KD is INTEGER */ /* > The number of superdiagonals of the matrix A if UPLO = 'U', */ /* > or the number of subdiagonals if UPLO = 'L'. KD >= 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 DOUBLE PRECISION array, dimension (LDAB,N) */ /* > On entry, the upper or lower triangle of the symmetric band */ /* > matrix A, stored in the first KD+1 rows of the array, except */ /* > if FACT = 'F' and EQUED = 'Y', then A must contain the */ /* > equilibrated matrix diag(S)*A*diag(S). The j-th column of A */ /* > is stored in the j-th column of the array AB as follows: */ /* > if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for f2cmax(1,j-KD)<=i<=j; */ /* > if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=f2cmin(N,j+KD). */ /* > See below for further details. */ /* > */ /* > On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ /* > diag(S)*A*diag(S). */ /* > \endverbatim */ /* > */ /* > \param[in] LDAB */ /* > \verbatim */ /* > LDAB is INTEGER */ /* > The leading dimension of the array A. LDAB >= KD+1. */ /* > \endverbatim */ /* > */ /* > \param[in,out] AFB */ /* > \verbatim */ /* > AFB is DOUBLE PRECISION array, dimension (LDAFB,N) */ /* > If FACT = 'F', then AFB is an input argument and on entry */ /* > contains the triangular factor U or L from the Cholesky */ /* > factorization A = U**T*U or A = L*L**T of the band matrix */ /* > A, in the same storage format as A (see AB). If EQUED = 'Y', */ /* > then AFB is the factored form of the equilibrated matrix A. */ /* > */ /* > If FACT = 'N', then AFB is an output argument and on exit */ /* > returns the triangular factor U or L from the Cholesky */ /* > factorization A = U**T*U or A = L*L**T. */ /* > */ /* > If FACT = 'E', then AFB is an output argument and on exit */ /* > returns the triangular factor U or L from the Cholesky */ /* > factorization A = U**T*U or A = L*L**T of the equilibrated */ /* > matrix A (see the description of A for the form of the */ /* > equilibrated matrix). */ /* > \endverbatim */ /* > */ /* > \param[in] LDAFB */ /* > \verbatim */ /* > LDAFB is INTEGER */ /* > The leading dimension of the array AFB. LDAFB >= KD+1. */ /* > \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'). */ /* > = 'Y': Equilibration was done, i.e., A has been replaced by */ /* > diag(S) * A * diag(S). */ /* > EQUED is an input argument if FACT = 'F'; otherwise, it is an */ /* > output argument. */ /* > \endverbatim */ /* > */ /* > \param[in,out] S */ /* > \verbatim */ /* > S is DOUBLE PRECISION array, dimension (N) */ /* > The scale factors for A; not accessed if EQUED = 'N'. S is */ /* > an input argument if FACT = 'F'; otherwise, S is an output */ /* > argument. If FACT = 'F' and EQUED = 'Y', each element of S */ /* > must be positive. */ /* > \endverbatim */ /* > */ /* > \param[in,out] B */ /* > \verbatim */ /* > B is DOUBLE PRECISION 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 EQUED = 'Y', */ /* > B is overwritten by diag(S) * 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 DOUBLE PRECISION 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 if EQUED = 'Y', */ /* > A and B are modified on exit, and the solution to the */ /* > equilibrated system is inv(diag(S))*X. */ /* > \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 DOUBLE PRECISION */ /* > 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N) */ /* > \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: the leading minor of order i of A is */ /* > not positive definite, so the factorization */ /* > could not be completed, and the solution has not */ /* > been 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 */ /* > value of RCOND would suggest. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date April 2012 */ /* > \ingroup doubleOTHERsolve */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > The band storage scheme is illustrated by the following example, when */ /* > N = 6, KD = 2, and UPLO = 'U': */ /* > */ /* > Two-dimensional storage of the symmetric matrix A: */ /* > */ /* > a11 a12 a13 */ /* > a22 a23 a24 */ /* > a33 a34 a35 */ /* > a44 a45 a46 */ /* > a55 a56 */ /* > (aij=conjg(aji)) a66 */ /* > */ /* > Band storage of the upper triangle of A: */ /* > */ /* > * * a13 a24 a35 a46 */ /* > * a12 a23 a34 a45 a56 */ /* > a11 a22 a33 a44 a55 a66 */ /* > */ /* > Similarly, if UPLO = 'L' the format of A is as follows: */ /* > */ /* > a11 a22 a33 a44 a55 a66 */ /* > a21 a32 a43 a54 a65 * */ /* > a31 a42 a53 a64 * * */ /* > */ /* > Array elements marked * are not used by the routine. */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int dpbsvx_(char *fact, char *uplo, integer *n, integer *kd, integer *nrhs, doublereal *ab, integer *ldab, doublereal *afb, integer *ldafb, char *equed, doublereal *s, doublereal *b, integer * ldb, doublereal *x, integer *ldx, doublereal *rcond, doublereal *ferr, doublereal *berr, doublereal *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; doublereal d__1, d__2; /* Local variables */ doublereal amax, smin, smax; integer i__, j; extern logical lsame_(char *, char *); doublereal scond, anorm; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); logical equil, rcequ, upper; integer j1, j2; extern doublereal dlamch_(char *), dlansb_(char *, char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dpbcon_(char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *), dlaqsb_(char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, char *); logical nofact; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *, ftnlen), dpbequ_(char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *); doublereal bignum; extern /* Subroutine */ int dpbrfs_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *), dpbtrf_(char *, integer *, integer *, doublereal *, integer *, integer *); integer infequ; extern /* Subroutine */ int dpbtrs_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); doublereal smlnum; /* -- 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 */ ab_dim1 = *ldab; ab_offset = 1 + ab_dim1 * 1; ab -= ab_offset; afb_dim1 = *ldafb; afb_offset = 1 + afb_dim1 * 1; afb -= afb_offset; --s; 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"); upper = lsame_(uplo, "U"); if (nofact || equil) { *(unsigned char *)equed = 'N'; rcequ = FALSE_; } else { rcequ = lsame_(equed, "Y"); smlnum = dlamch_("Safe minimum"); bignum = 1. / smlnum; } /* Test the input parameters. */ if (! nofact && ! equil && ! lsame_(fact, "F")) { *info = -1; } else if (! upper && ! lsame_(uplo, "L")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*kd < 0) { *info = -4; } else if (*nrhs < 0) { *info = -5; } else if (*ldab < *kd + 1) { *info = -7; } else if (*ldafb < *kd + 1) { *info = -9; } else if (lsame_(fact, "F") && ! (rcequ || lsame_( equed, "N"))) { *info = -10; } else { if (rcequ) { smin = bignum; smax = 0.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = smin, d__2 = s[j]; smin = f2cmin(d__1,d__2); /* Computing MAX */ d__1 = smax, d__2 = s[j]; smax = f2cmax(d__1,d__2); /* L10: */ } if (smin <= 0.) { *info = -11; } else if (*n > 0) { scond = f2cmax(smin,smlnum) / f2cmin(smax,bignum); } else { scond = 1.; } } if (*info == 0) { if (*ldb < f2cmax(1,*n)) { *info = -13; } else if (*ldx < f2cmax(1,*n)) { *info = -15; } } } if (*info != 0) { i__1 = -(*info); xerbla_("DPBSVX", &i__1, (ftnlen)6); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ dpbequ_(uplo, n, kd, &ab[ab_offset], ldab, &s[1], &scond, &amax, & infequ); if (infequ == 0) { /* Equilibrate the matrix. */ dlaqsb_(uplo, n, kd, &ab[ab_offset], ldab, &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right-hand side. */ if (rcequ) { 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] = s[i__] * b[i__ + j * b_dim1]; /* L20: */ } /* L30: */ } } if (nofact || equil) { /* Compute the Cholesky factorization A = U**T *U or A = L*L**T. */ if (upper) { i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MAX */ i__2 = j - *kd; j1 = f2cmax(i__2,1); i__2 = j - j1 + 1; dcopy_(&i__2, &ab[*kd + 1 - j + j1 + j * ab_dim1], &c__1, & afb[*kd + 1 - j + j1 + j * afb_dim1], &c__1); /* L40: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ i__2 = j + *kd; j2 = f2cmin(i__2,*n); i__2 = j2 - j + 1; dcopy_(&i__2, &ab[j * ab_dim1 + 1], &c__1, &afb[j * afb_dim1 + 1], &c__1); /* L50: */ } } dpbtrf_(uplo, n, kd, &afb[afb_offset], ldafb, info); /* Return if INFO is non-zero. */ if (*info > 0) { *rcond = 0.; return 0; } } /* Compute the norm of the matrix A. */ anorm = dlansb_("1", uplo, n, kd, &ab[ab_offset], ldab, &work[1]); /* Compute the reciprocal of the condition number of A. */ dpbcon_(uplo, n, kd, &afb[afb_offset], ldafb, &anorm, rcond, &work[1], & iwork[1], info); /* Compute the solution matrix X. */ dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); dpbtrs_(uplo, n, kd, nrhs, &afb[afb_offset], ldafb, &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ dpbrfs_(uplo, n, kd, nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &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 (rcequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1]; /* L60: */ } /* L70: */ } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] /= scond; /* L80: */ } } /* Set INFO = N+1 if the matrix is singular to working precision. */ if (*rcond < dlamch_("Epsilon")) { *info = *n + 1; } return 0; /* End of DPBSVX */ } /* dpbsvx_ */