#include #include #include #include #include #ifdef complex #undef complex #endif #ifdef I #undef I #endif #if defined(_WIN64) typedef long long BLASLONG; typedef unsigned long long BLASULONG; #else typedef long BLASLONG; typedef unsigned long BLASULONG; #endif #ifdef LAPACK_ILP64 typedef BLASLONG blasint; #if defined(_WIN64) #define blasabs(x) llabs(x) #else #define blasabs(x) labs(x) #endif #else typedef int blasint; #define blasabs(x) abs(x) #endif typedef blasint integer; typedef unsigned int uinteger; typedef char *address; typedef short int shortint; typedef float real; typedef double doublereal; typedef struct { real r, i; } complex; typedef struct { doublereal r, i; } doublecomplex; #ifdef _MSC_VER static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;} static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;} static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;} static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;} #else static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;} static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;} static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;} static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;} #endif #define pCf(z) (*_pCf(z)) #define pCd(z) (*_pCd(z)) typedef int logical; typedef short int shortlogical; typedef char logical1; typedef char integer1; #define TRUE_ (1) #define FALSE_ (0) /* Extern is for use with -E */ #ifndef Extern #define Extern extern #endif /* I/O stuff */ typedef int flag; typedef int ftnlen; typedef int ftnint; /*external read, write*/ typedef struct { flag cierr; ftnint ciunit; flag ciend; char *cifmt; ftnint cirec; } cilist; /*internal read, write*/ typedef struct { flag icierr; char *iciunit; flag iciend; char *icifmt; ftnint icirlen; ftnint icirnum; } icilist; /*open*/ typedef struct { flag oerr; ftnint ounit; char *ofnm; ftnlen ofnmlen; char *osta; char *oacc; char *ofm; ftnint orl; char *oblnk; } olist; /*close*/ typedef struct { flag cerr; ftnint cunit; char *csta; } cllist; /*rewind, backspace, endfile*/ typedef struct { flag aerr; ftnint aunit; } alist; /* inquire */ typedef struct { flag inerr; ftnint inunit; char *infile; ftnlen infilen; ftnint *inex; /*parameters in standard's order*/ ftnint *inopen; ftnint *innum; ftnint *innamed; char *inname; ftnlen innamlen; char *inacc; ftnlen inacclen; char *inseq; ftnlen inseqlen; char *indir; ftnlen indirlen; char *infmt; ftnlen infmtlen; char *inform; ftnint informlen; char *inunf; ftnlen inunflen; ftnint *inrecl; ftnint *innrec; char *inblank; ftnlen inblanklen; } inlist; #define VOID void union Multitype { /* for multiple entry points */ integer1 g; shortint h; integer i; /* longint j; */ real r; doublereal d; complex c; doublecomplex z; }; typedef union Multitype Multitype; struct Vardesc { /* for Namelist */ char *name; char *addr; ftnlen *dims; int type; }; typedef struct Vardesc Vardesc; struct Namelist { char *name; Vardesc **vars; int nvars; }; typedef struct Namelist Namelist; #define abs(x) ((x) >= 0 ? (x) : -(x)) #define dabs(x) (fabs(x)) #define f2cmin(a,b) ((a) <= (b) ? (a) : (b)) #define f2cmax(a,b) ((a) >= (b) ? (a) : (b)) #define dmin(a,b) (f2cmin(a,b)) #define dmax(a,b) (f2cmax(a,b)) #define bit_test(a,b) ((a) >> (b) & 1) #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b))) #define bit_set(a,b) ((a) | ((uinteger)1 << (b))) #define abort_() { sig_die("Fortran abort routine called", 1); } #define c_abs(z) (cabsf(Cf(z))) #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); } #ifdef _MSC_VER #define c_div(c, a, b) {Cf(c)._Val[0] = (Cf(a)._Val[0]/Cf(b)._Val[0]); Cf(c)._Val[1]=(Cf(a)._Val[1]/Cf(b)._Val[1]);} #define z_div(c, a, b) {Cd(c)._Val[0] = (Cd(a)._Val[0]/Cd(b)._Val[0]); Cd(c)._Val[1]=(Cd(a)._Val[1]/df(b)._Val[1]);} #else #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);} #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);} #endif #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));} #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));} #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));} //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));} #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));} #define d_abs(x) (fabs(*(x))) #define d_acos(x) (acos(*(x))) #define d_asin(x) (asin(*(x))) #define d_atan(x) (atan(*(x))) #define d_atn2(x, y) (atan2(*(x),*(y))) #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); } #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); } #define d_cos(x) (cos(*(x))) #define d_cosh(x) (cosh(*(x))) #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 ) #define d_exp(x) (exp(*(x))) #define d_imag(z) (cimag(Cd(z))) #define r_imag(z) (cimagf(Cf(z))) #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define d_log(x) (log(*(x))) #define d_mod(x, y) (fmod(*(x), *(y))) #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x))) #define d_nint(x) u_nint(*(x)) #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a))) #define d_sign(a,b) u_sign(*(a),*(b)) #define r_sign(a,b) u_sign(*(a),*(b)) #define d_sin(x) (sin(*(x))) #define d_sinh(x) (sinh(*(x))) #define d_sqrt(x) (sqrt(*(x))) #define d_tan(x) (tan(*(x))) #define d_tanh(x) (tanh(*(x))) #define i_abs(x) abs(*(x)) #define i_dnnt(x) ((integer)u_nint(*(x))) #define i_len(s, n) (n) #define i_nint(x) ((integer)u_nint(*(x))) #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b))) #define pow_dd(ap, bp) ( pow(*(ap), *(bp))) #define pow_si(B,E) spow_ui(*(B),*(E)) #define pow_ri(B,E) spow_ui(*(B),*(E)) #define pow_di(B,E) dpow_ui(*(B),*(E)) #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));} #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));} #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));} #define s_cat(lpp, rpp, rnp, np, llp) { ftnlen i, nc, ll; char *f__rp, *lp; ll = (llp); lp = (lpp); for(i=0; i < (int)*(np); ++i) { nc = ll; if((rnp)[i] < nc) nc = (rnp)[i]; ll -= nc; f__rp = (rpp)[i]; while(--nc >= 0) *lp++ = *(f__rp)++; } while(--ll >= 0) *lp++ = ' '; } #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d)))) #define s_copy(A,B,C,D) { int __i,__m; for (__i=0, __m=f2cmin((C),(D)); __i<__m && (B)[__i] != 0; ++__i) (A)[__i] = (B)[__i]; } #define sig_die(s, kill) { exit(1); } #define s_stop(s, n) {exit(0);} static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n"; #define z_abs(z) (cabs(Cd(z))) #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));} #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));} #define myexit_() break; #define mycycle() continue; #define myceiling(w) {ceil(w)} #define myhuge(w) {HUGE_VAL} //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);} #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)} /* procedure parameter types for -A and -C++ */ #define F2C_proc_par_types 1 #ifdef __cplusplus typedef logical (*L_fp)(...); #else typedef logical (*L_fp)(); #endif static float spow_ui(float x, integer n) { float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static double dpow_ui(double x, integer n) { double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #ifdef _MSC_VER static _Fcomplex cpow_ui(complex x, integer n) { complex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i; for(u = n; ; ) { if(u & 01) pow.r *= x.r, pow.i *= x.i; if(u >>= 1) x.r *= x.r, x.i *= x.i; else break; } } _Fcomplex p={pow.r, pow.i}; return p; } #else static _Complex float cpow_ui(_Complex float x, integer n) { _Complex float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif #ifdef _MSC_VER static _Dcomplex zpow_ui(_Dcomplex x, integer n) { _Dcomplex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1]; for(u = n; ; ) { if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1]; if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1]; else break; } } _Dcomplex p = {pow._Val[0], pow._Val[1]}; return p; } #else static _Complex double zpow_ui(_Complex double x, integer n) { _Complex double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif static integer pow_ii(integer x, integer n) { integer pow; unsigned long int u; if (n <= 0) { if (n == 0 || x == 1) pow = 1; else if (x != -1) pow = x == 0 ? 1/x : 0; else n = -n; } if ((n > 0) || !(n == 0 || x == 1 || x != -1)) { u = n; for(pow = 1; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static integer dmaxloc_(double *w, integer s, integer e, integer *n) { double m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static integer smaxloc_(float *w, integer s, integer e, integer *n) { float m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) { integer n = *n_, incx = *incx_, incy = *incy_, i; #ifdef _MSC_VER _Fcomplex zdotc = {0.0, 0.0}; if (incx == 1 && incy == 1) { for (i=0;i \brief \b DSPTRF */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DSPTRF + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO ) */ /* CHARACTER UPLO */ /* INTEGER INFO, N */ /* INTEGER IPIV( * ) */ /* DOUBLE PRECISION AP( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DSPTRF computes the factorization of a real symmetric matrix A stored */ /* > in packed format using the Bunch-Kaufman diagonal pivoting method: */ /* > */ /* > A = U*D*U**T or A = L*D*L**T */ /* > */ /* > where U (or L) is a product of permutation and unit upper (lower) */ /* > triangular matrices, and D is symmetric and block diagonal with */ /* > 1-by-1 and 2-by-2 diagonal blocks. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \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 order of the matrix A. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] AP */ /* > \verbatim */ /* > AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) */ /* > On entry, the upper or lower triangle of the symmetric matrix */ /* > A, packed columnwise in a linear array. The j-th column of A */ /* > is stored in the array AP as follows: */ /* > if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */ /* > if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */ /* > */ /* > On exit, the block diagonal matrix D and the multipliers used */ /* > to obtain the factor U or L, stored as a packed triangular */ /* > matrix overwriting A (see below for further details). */ /* > \endverbatim */ /* > */ /* > \param[out] IPIV */ /* > \verbatim */ /* > IPIV is INTEGER array, dimension (N) */ /* > Details of the interchanges and the block structure of D. */ /* > If IPIV(k) > 0, then rows and columns k and IPIV(k) were */ /* > interchanged and D(k,k) is a 1-by-1 diagonal block. */ /* > If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and */ /* > columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) */ /* > is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = */ /* > IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were */ /* > interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */ /* > \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, D(i,i) is exactly zero. The factorization */ /* > has been completed, but the block diagonal matrix D is */ /* > exactly singular, and division by zero will occur if it */ /* > is used to solve a system of equations. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup doubleOTHERcomputational */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > If UPLO = 'U', then A = U*D*U**T, where */ /* > U = P(n)*U(n)* ... *P(k)U(k)* ..., */ /* > i.e., U is a product of terms P(k)*U(k), where k decreases from n to */ /* > 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 */ /* > and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as */ /* > defined by IPIV(k), and U(k) is a unit upper triangular matrix, such */ /* > that if the diagonal block D(k) is of order s (s = 1 or 2), then */ /* > */ /* > ( I v 0 ) k-s */ /* > U(k) = ( 0 I 0 ) s */ /* > ( 0 0 I ) n-k */ /* > k-s s n-k */ /* > */ /* > If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). */ /* > If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), */ /* > and A(k,k), and v overwrites A(1:k-2,k-1:k). */ /* > */ /* > If UPLO = 'L', then A = L*D*L**T, where */ /* > L = P(1)*L(1)* ... *P(k)*L(k)* ..., */ /* > i.e., L is a product of terms P(k)*L(k), where k increases from 1 to */ /* > n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 */ /* > and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as */ /* > defined by IPIV(k), and L(k) is a unit lower triangular matrix, such */ /* > that if the diagonal block D(k) is of order s (s = 1 or 2), then */ /* > */ /* > ( I 0 0 ) k-1 */ /* > L(k) = ( 0 I 0 ) s */ /* > ( 0 v I ) n-k-s+1 */ /* > k-1 s n-k-s+1 */ /* > */ /* > If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). */ /* > If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), */ /* > and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). */ /* > \endverbatim */ /* > \par Contributors: */ /* ================== */ /* > */ /* > J. Lewis, Boeing Computer Services Company */ /* > */ /* ===================================================================== */ /* Subroutine */ int dsptrf_(char *uplo, integer *n, doublereal *ap, integer * ipiv, integer *info) { /* System generated locals */ integer i__1, i__2; doublereal d__1, d__2, d__3; /* Local variables */ integer imax, jmax; extern /* Subroutine */ int dspr_(char *, integer *, doublereal *, doublereal *, integer *, doublereal *); integer i__, j, k; doublereal t, alpha; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *, doublereal *, integer *); integer kstep; logical upper; doublereal r1, d11, d12, d21, d22; integer kc, kk, kp; doublereal absakk, wk; integer kx; extern integer idamax_(integer *, doublereal *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); doublereal colmax, rowmax; integer knc, kpc, npp; doublereal wkm1, wkp1; /* -- LAPACK computational routine (version 3.7.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* December 2016 */ /* ===================================================================== */ /* Test the input parameters. */ /* Parameter adjustments */ --ipiv; --ap; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } if (*info != 0) { i__1 = -(*info); xerbla_("DSPTRF", &i__1, (ftnlen)6); return 0; } /* Initialize ALPHA for use in choosing pivot block size. */ alpha = (sqrt(17.) + 1.) / 8.; if (upper) { /* Factorize A as U*D*U**T using the upper triangle of A */ /* K is the main loop index, decreasing from N to 1 in steps of */ /* 1 or 2 */ k = *n; kc = (*n - 1) * *n / 2 + 1; L10: knc = kc; /* If K < 1, exit from loop */ if (k < 1) { goto L110; } kstep = 1; /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ absakk = (d__1 = ap[kc + k - 1], abs(d__1)); /* IMAX is the row-index of the largest off-diagonal element in */ /* column K, and COLMAX is its absolute value */ if (k > 1) { i__1 = k - 1; imax = idamax_(&i__1, &ap[kc], &c__1); colmax = (d__1 = ap[kc + imax - 1], abs(d__1)); } else { colmax = 0.; } if (f2cmax(absakk,colmax) == 0.) { /* Column K is zero: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; } else { if (absakk >= alpha * colmax) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { rowmax = 0.; jmax = imax; kx = imax * (imax + 1) / 2 + imax; i__1 = k; for (j = imax + 1; j <= i__1; ++j) { if ((d__1 = ap[kx], abs(d__1)) > rowmax) { rowmax = (d__1 = ap[kx], abs(d__1)); jmax = j; } kx += j; /* L20: */ } kpc = (imax - 1) * imax / 2 + 1; if (imax > 1) { i__1 = imax - 1; jmax = idamax_(&i__1, &ap[kpc], &c__1); /* Computing MAX */ d__2 = rowmax, d__3 = (d__1 = ap[kpc + jmax - 1], abs( d__1)); rowmax = f2cmax(d__2,d__3); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else if ((d__1 = ap[kpc + imax - 1], abs(d__1)) >= alpha * rowmax) { /* interchange rows and columns K and IMAX, use 1-by-1 */ /* pivot block */ kp = imax; } else { /* interchange rows and columns K-1 and IMAX, use 2-by-2 */ /* pivot block */ kp = imax; kstep = 2; } } kk = k - kstep + 1; if (kstep == 2) { knc = knc - k + 1; } if (kp != kk) { /* Interchange rows and columns KK and KP in the leading */ /* submatrix A(1:k,1:k) */ i__1 = kp - 1; dswap_(&i__1, &ap[knc], &c__1, &ap[kpc], &c__1); kx = kpc + kp - 1; i__1 = kk - 1; for (j = kp + 1; j <= i__1; ++j) { kx = kx + j - 1; t = ap[knc + j - 1]; ap[knc + j - 1] = ap[kx]; ap[kx] = t; /* L30: */ } t = ap[knc + kk - 1]; ap[knc + kk - 1] = ap[kpc + kp - 1]; ap[kpc + kp - 1] = t; if (kstep == 2) { t = ap[kc + k - 2]; ap[kc + k - 2] = ap[kc + kp - 1]; ap[kc + kp - 1] = t; } } /* Update the leading submatrix */ if (kstep == 1) { /* 1-by-1 pivot block D(k): column k now holds */ /* W(k) = U(k)*D(k) */ /* where U(k) is the k-th column of U */ /* Perform a rank-1 update of A(1:k-1,1:k-1) as */ /* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T */ r1 = 1. / ap[kc + k - 1]; i__1 = k - 1; d__1 = -r1; dspr_(uplo, &i__1, &d__1, &ap[kc], &c__1, &ap[1]); /* Store U(k) in column k */ i__1 = k - 1; dscal_(&i__1, &r1, &ap[kc], &c__1); } else { /* 2-by-2 pivot block D(k): columns k and k-1 now hold */ /* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) */ /* where U(k) and U(k-1) are the k-th and (k-1)-th columns */ /* of U */ /* Perform a rank-2 update of A(1:k-2,1:k-2) as */ /* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T */ /* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T */ if (k > 2) { d12 = ap[k - 1 + (k - 1) * k / 2]; d22 = ap[k - 1 + (k - 2) * (k - 1) / 2] / d12; d11 = ap[k + (k - 1) * k / 2] / d12; t = 1. / (d11 * d22 - 1.); d12 = t / d12; for (j = k - 2; j >= 1; --j) { wkm1 = d12 * (d11 * ap[j + (k - 2) * (k - 1) / 2] - ap[j + (k - 1) * k / 2]); wk = d12 * (d22 * ap[j + (k - 1) * k / 2] - ap[j + (k - 2) * (k - 1) / 2]); for (i__ = j; i__ >= 1; --i__) { ap[i__ + (j - 1) * j / 2] = ap[i__ + (j - 1) * j / 2] - ap[i__ + (k - 1) * k / 2] * wk - ap[ i__ + (k - 2) * (k - 1) / 2] * wkm1; /* L40: */ } ap[j + (k - 1) * k / 2] = wk; ap[j + (k - 2) * (k - 1) / 2] = wkm1; /* L50: */ } } } } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -kp; ipiv[k - 1] = -kp; } /* Decrease K and return to the start of the main loop */ k -= kstep; kc = knc - k; goto L10; } else { /* Factorize A as L*D*L**T using the lower triangle of A */ /* K is the main loop index, increasing from 1 to N in steps of */ /* 1 or 2 */ k = 1; kc = 1; npp = *n * (*n + 1) / 2; L60: knc = kc; /* If K > N, exit from loop */ if (k > *n) { goto L110; } kstep = 1; /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ absakk = (d__1 = ap[kc], abs(d__1)); /* IMAX is the row-index of the largest off-diagonal element in */ /* column K, and COLMAX is its absolute value */ if (k < *n) { i__1 = *n - k; imax = k + idamax_(&i__1, &ap[kc + 1], &c__1); colmax = (d__1 = ap[kc + imax - k], abs(d__1)); } else { colmax = 0.; } if (f2cmax(absakk,colmax) == 0.) { /* Column K is zero: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; } else { if (absakk >= alpha * colmax) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { /* JMAX is the column-index of the largest off-diagonal */ /* element in row IMAX, and ROWMAX is its absolute value */ rowmax = 0.; kx = kc + imax - k; i__1 = imax - 1; for (j = k; j <= i__1; ++j) { if ((d__1 = ap[kx], abs(d__1)) > rowmax) { rowmax = (d__1 = ap[kx], abs(d__1)); jmax = j; } kx = kx + *n - j; /* L70: */ } kpc = npp - (*n - imax + 1) * (*n - imax + 2) / 2 + 1; if (imax < *n) { i__1 = *n - imax; jmax = imax + idamax_(&i__1, &ap[kpc + 1], &c__1); /* Computing MAX */ d__2 = rowmax, d__3 = (d__1 = ap[kpc + jmax - imax], abs( d__1)); rowmax = f2cmax(d__2,d__3); } if (absakk >= alpha * colmax * (colmax / rowmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else if ((d__1 = ap[kpc], abs(d__1)) >= alpha * rowmax) { /* interchange rows and columns K and IMAX, use 1-by-1 */ /* pivot block */ kp = imax; } else { /* interchange rows and columns K+1 and IMAX, use 2-by-2 */ /* pivot block */ kp = imax; kstep = 2; } } kk = k + kstep - 1; if (kstep == 2) { knc = knc + *n - k + 1; } if (kp != kk) { /* Interchange rows and columns KK and KP in the trailing */ /* submatrix A(k:n,k:n) */ if (kp < *n) { i__1 = *n - kp; dswap_(&i__1, &ap[knc + kp - kk + 1], &c__1, &ap[kpc + 1], &c__1); } kx = knc + kp - kk; i__1 = kp - 1; for (j = kk + 1; j <= i__1; ++j) { kx = kx + *n - j + 1; t = ap[knc + j - kk]; ap[knc + j - kk] = ap[kx]; ap[kx] = t; /* L80: */ } t = ap[knc]; ap[knc] = ap[kpc]; ap[kpc] = t; if (kstep == 2) { t = ap[kc + 1]; ap[kc + 1] = ap[kc + kp - k]; ap[kc + kp - k] = t; } } /* Update the trailing submatrix */ if (kstep == 1) { /* 1-by-1 pivot block D(k): column k now holds */ /* W(k) = L(k)*D(k) */ /* where L(k) is the k-th column of L */ if (k < *n) { /* Perform a rank-1 update of A(k+1:n,k+1:n) as */ /* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T */ r1 = 1. / ap[kc]; i__1 = *n - k; d__1 = -r1; dspr_(uplo, &i__1, &d__1, &ap[kc + 1], &c__1, &ap[kc + *n - k + 1]); /* Store L(k) in column K */ i__1 = *n - k; dscal_(&i__1, &r1, &ap[kc + 1], &c__1); } } else { /* 2-by-2 pivot block D(k): columns K and K+1 now hold */ /* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) */ /* where L(k) and L(k+1) are the k-th and (k+1)-th columns */ /* of L */ if (k < *n - 1) { /* Perform a rank-2 update of A(k+2:n,k+2:n) as */ /* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T */ /* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T */ /* where L(k) and L(k+1) are the k-th and (k+1)-th */ /* columns of L */ d21 = ap[k + 1 + (k - 1) * ((*n << 1) - k) / 2]; d11 = ap[k + 1 + k * ((*n << 1) - k - 1) / 2] / d21; d22 = ap[k + (k - 1) * ((*n << 1) - k) / 2] / d21; t = 1. / (d11 * d22 - 1.); d21 = t / d21; i__1 = *n; for (j = k + 2; j <= i__1; ++j) { wk = d21 * (d11 * ap[j + (k - 1) * ((*n << 1) - k) / 2] - ap[j + k * ((*n << 1) - k - 1) / 2]); wkp1 = d21 * (d22 * ap[j + k * ((*n << 1) - k - 1) / 2] - ap[j + (k - 1) * ((*n << 1) - k) / 2]); i__2 = *n; for (i__ = j; i__ <= i__2; ++i__) { ap[i__ + (j - 1) * ((*n << 1) - j) / 2] = ap[i__ + (j - 1) * ((*n << 1) - j) / 2] - ap[i__ + (k - 1) * ((*n << 1) - k) / 2] * wk - ap[i__ + k * ((*n << 1) - k - 1) / 2] * wkp1; /* L90: */ } ap[j + (k - 1) * ((*n << 1) - k) / 2] = wk; ap[j + k * ((*n << 1) - k - 1) / 2] = wkp1; /* L100: */ } } } } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -kp; ipiv[k + 1] = -kp; } /* Increase K and return to the start of the main loop */ k += kstep; kc = knc + *n - k + 2; goto L60; } L110: return 0; /* End of DSPTRF */ } /* dsptrf_ */