#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 SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bu nch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SSYTF2_RK + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO ) */ /* CHARACTER UPLO */ /* INTEGER INFO, LDA, N */ /* INTEGER IPIV( * ) */ /* REAL A( LDA, * ), E ( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > SSYTF2_RK computes the factorization of a real symmetric matrix A */ /* > using the bounded Bunch-Kaufman (rook) diagonal pivoting method: */ /* > */ /* > A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), */ /* > */ /* > where U (or L) is unit upper (or lower) triangular matrix, */ /* > U**T (or L**T) is the transpose of U (or L), P is a permutation */ /* > matrix, P**T is the transpose of P, and D is symmetric and block */ /* > diagonal with 1-by-1 and 2-by-2 diagonal blocks. */ /* > */ /* > This is the unblocked version of the algorithm, calling Level 2 BLAS. */ /* > For more information see Further Details section. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] UPLO */ /* > \verbatim */ /* > UPLO is CHARACTER*1 */ /* > Specifies whether the upper or lower triangular part of the */ /* > symmetric matrix A is stored: */ /* > = 'U': Upper triangular */ /* > = 'L': Lower triangular */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrix A. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is REAL array, dimension (LDA,N) */ /* > On entry, the symmetric matrix A. */ /* > If UPLO = 'U': the leading N-by-N upper triangular part */ /* > of A contains the upper triangular part of the matrix A, */ /* > and the strictly lower triangular part of A is not */ /* > referenced. */ /* > */ /* > If UPLO = 'L': the leading N-by-N lower triangular part */ /* > of A contains the lower triangular part of the matrix A, */ /* > and the strictly upper triangular part of A is not */ /* > referenced. */ /* > */ /* > On exit, contains: */ /* > a) ONLY diagonal elements of the symmetric block diagonal */ /* > matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); */ /* > (superdiagonal (or subdiagonal) elements of D */ /* > are stored on exit in array E), and */ /* > b) If UPLO = 'U': factor U in the superdiagonal part of A. */ /* > If UPLO = 'L': factor L in the subdiagonal part of A. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] E */ /* > \verbatim */ /* > E is REAL array, dimension (N) */ /* > On exit, contains the superdiagonal (or subdiagonal) */ /* > elements of the symmetric block diagonal matrix D */ /* > with 1-by-1 or 2-by-2 diagonal blocks, where */ /* > If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; */ /* > If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. */ /* > */ /* > NOTE: For 1-by-1 diagonal block D(k), where */ /* > 1 <= k <= N, the element E(k) is set to 0 in both */ /* > UPLO = 'U' or UPLO = 'L' cases. */ /* > \endverbatim */ /* > */ /* > \param[out] IPIV */ /* > \verbatim */ /* > IPIV is INTEGER array, dimension (N) */ /* > IPIV describes the permutation matrix P in the factorization */ /* > of matrix A as follows. The absolute value of IPIV(k) */ /* > represents the index of row and column that were */ /* > interchanged with the k-th row and column. The value of UPLO */ /* > describes the order in which the interchanges were applied. */ /* > Also, the sign of IPIV represents the block structure of */ /* > the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 */ /* > diagonal blocks which correspond to 1 or 2 interchanges */ /* > at each factorization step. For more info see Further */ /* > Details section. */ /* > */ /* > If UPLO = 'U', */ /* > ( in factorization order, k decreases from N to 1 ): */ /* > a) A single positive entry IPIV(k) > 0 means: */ /* > D(k,k) is a 1-by-1 diagonal block. */ /* > If IPIV(k) != k, rows and columns k and IPIV(k) were */ /* > interchanged in the matrix A(1:N,1:N); */ /* > If IPIV(k) = k, no interchange occurred. */ /* > */ /* > b) A pair of consecutive negative entries */ /* > IPIV(k) < 0 and IPIV(k-1) < 0 means: */ /* > D(k-1:k,k-1:k) is a 2-by-2 diagonal block. */ /* > (NOTE: negative entries in IPIV appear ONLY in pairs). */ /* > 1) If -IPIV(k) != k, rows and columns */ /* > k and -IPIV(k) were interchanged */ /* > in the matrix A(1:N,1:N). */ /* > If -IPIV(k) = k, no interchange occurred. */ /* > 2) If -IPIV(k-1) != k-1, rows and columns */ /* > k-1 and -IPIV(k-1) were interchanged */ /* > in the matrix A(1:N,1:N). */ /* > If -IPIV(k-1) = k-1, no interchange occurred. */ /* > */ /* > c) In both cases a) and b), always ABS( IPIV(k) ) <= k. */ /* > */ /* > d) NOTE: Any entry IPIV(k) is always NONZERO on output. */ /* > */ /* > If UPLO = 'L', */ /* > ( in factorization order, k increases from 1 to N ): */ /* > a) A single positive entry IPIV(k) > 0 means: */ /* > D(k,k) is a 1-by-1 diagonal block. */ /* > If IPIV(k) != k, rows and columns k and IPIV(k) were */ /* > interchanged in the matrix A(1:N,1:N). */ /* > If IPIV(k) = k, no interchange occurred. */ /* > */ /* > b) A pair of consecutive negative entries */ /* > IPIV(k) < 0 and IPIV(k+1) < 0 means: */ /* > D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */ /* > (NOTE: negative entries in IPIV appear ONLY in pairs). */ /* > 1) If -IPIV(k) != k, rows and columns */ /* > k and -IPIV(k) were interchanged */ /* > in the matrix A(1:N,1:N). */ /* > If -IPIV(k) = k, no interchange occurred. */ /* > 2) If -IPIV(k+1) != k+1, rows and columns */ /* > k-1 and -IPIV(k-1) were interchanged */ /* > in the matrix A(1:N,1:N). */ /* > If -IPIV(k+1) = k+1, no interchange occurred. */ /* > */ /* > c) In both cases a) and b), always ABS( IPIV(k) ) >= k. */ /* > */ /* > d) NOTE: Any entry IPIV(k) is always NONZERO on output. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > */ /* > < 0: If INFO = -k, the k-th argument had an illegal value */ /* > */ /* > > 0: If INFO = k, the matrix A is singular, because: */ /* > If UPLO = 'U': column k in the upper */ /* > triangular part of A contains all zeros. */ /* > If UPLO = 'L': column k in the lower */ /* > triangular part of A contains all zeros. */ /* > */ /* > Therefore D(k,k) is exactly zero, and superdiagonal */ /* > elements of column k of U (or subdiagonal elements of */ /* > column k of L ) are all zeros. 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. */ /* > */ /* > NOTE: INFO only stores the first occurrence of */ /* > a singularity, any subsequent occurrence of singularity */ /* > is not stored in INFO even though the factorization */ /* > always completes. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup singleSYcomputational */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > TODO: put further details */ /* > \endverbatim */ /* > \par Contributors: */ /* ================== */ /* > */ /* > \verbatim */ /* > */ /* > December 2016, Igor Kozachenko, */ /* > Computer Science Division, */ /* > University of California, Berkeley */ /* > */ /* > September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, */ /* > School of Mathematics, */ /* > University of Manchester */ /* > */ /* > 01-01-96 - Based on modifications by */ /* > J. Lewis, Boeing Computer Services Company */ /* > A. Petitet, Computer Science Dept., */ /* > Univ. of Tenn., Knoxville abd , USA */ /* > \endverbatim */ /* ===================================================================== */ /* Subroutine */ int ssytf2_rk_(char *uplo, integer *n, real *a, integer * lda, real *e, integer *ipiv, integer *info) { /* System generated locals */ integer a_dim1, a_offset, i__1, i__2; real r__1; /* Local variables */ logical done; integer imax, jmax; extern /* Subroutine */ int ssyr_(char *, integer *, real *, real *, integer *, real *, integer *); integer i__, j, k, p; real t, alpha; extern logical lsame_(char *, char *); extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); real sfmin; integer itemp, kstep; real stemp; logical upper; extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, integer *); real d11, d12, d21, d22; integer ii, kk, kp; real absakk, wk; extern real slamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); extern integer isamax_(integer *, real *, integer *); real colmax, rowmax, 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 */ a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; --e; --ipiv; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < f2cmax(1,*n)) { *info = -4; } if (*info != 0) { i__1 = -(*info); xerbla_("SSYTF2_RK", &i__1, (ftnlen)9); return 0; } /* Initialize ALPHA for use in choosing pivot block size. */ alpha = (sqrt(17.f) + 1.f) / 8.f; /* Compute machine safe minimum */ sfmin = slamch_("S"); if (upper) { /* Factorize A as U*D*U**T using the upper triangle of A */ /* Initialize the first entry of array E, where superdiagonal */ /* elements of D are stored */ e[1] = 0.f; /* K is the main loop index, decreasing from N to 1 in steps of */ /* 1 or 2 */ k = *n; L10: /* If K < 1, exit from loop */ if (k < 1) { goto L34; } kstep = 1; p = k; /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ absakk = (r__1 = a[k + k * a_dim1], abs(r__1)); /* IMAX is the row-index of the largest off-diagonal element in */ /* column K, and COLMAX is its absolute value. */ /* Determine both COLMAX and IMAX. */ if (k > 1) { i__1 = k - 1; imax = isamax_(&i__1, &a[k * a_dim1 + 1], &c__1); colmax = (r__1 = a[imax + k * a_dim1], abs(r__1)); } else { colmax = 0.f; } if (f2cmax(absakk,colmax) == 0.f) { /* Column K is zero or underflow: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; /* Set E( K ) to zero */ if (k > 1) { e[k] = 0.f; } } else { /* Test for interchange */ /* Equivalent to testing for (used to handle NaN and Inf) */ /* ABSAKK.GE.ALPHA*COLMAX */ if (! (absakk < alpha * colmax)) { /* no interchange, */ /* use 1-by-1 pivot block */ kp = k; } else { done = FALSE_; /* Loop until pivot found */ L12: /* Begin pivot search loop body */ /* JMAX is the column-index of the largest off-diagonal */ /* element in row IMAX, and ROWMAX is its absolute value. */ /* Determine both ROWMAX and JMAX. */ if (imax != k) { i__1 = k - imax; jmax = imax + isamax_(&i__1, &a[imax + (imax + 1) * a_dim1], lda); rowmax = (r__1 = a[imax + jmax * a_dim1], abs(r__1)); } else { rowmax = 0.f; } if (imax > 1) { i__1 = imax - 1; itemp = isamax_(&i__1, &a[imax * a_dim1 + 1], &c__1); stemp = (r__1 = a[itemp + imax * a_dim1], abs(r__1)); if (stemp > rowmax) { rowmax = stemp; jmax = itemp; } } /* Equivalent to testing for (used to handle NaN and Inf) */ /* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX */ if (! ((r__1 = a[imax + imax * a_dim1], abs(r__1)) < alpha * rowmax)) { /* interchange rows and columns K and IMAX, */ /* use 1-by-1 pivot block */ kp = imax; done = TRUE_; /* Equivalent to testing for ROWMAX .EQ. COLMAX, */ /* used to handle NaN and Inf */ } else if (p == jmax || rowmax <= colmax) { /* interchange rows and columns K+1 and IMAX, */ /* use 2-by-2 pivot block */ kp = imax; kstep = 2; done = TRUE_; } else { /* Pivot NOT found, set variables and repeat */ p = imax; colmax = rowmax; imax = jmax; } /* End pivot search loop body */ if (! done) { goto L12; } } /* Swap TWO rows and TWO columns */ /* First swap */ if (kstep == 2 && p != k) { /* Interchange rows and column K and P in the leading */ /* submatrix A(1:k,1:k) if we have a 2-by-2 pivot */ if (p > 1) { i__1 = p - 1; sswap_(&i__1, &a[k * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); } if (p < k - 1) { i__1 = k - p - 1; sswap_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + (p + 1) * a_dim1], lda); } t = a[k + k * a_dim1]; a[k + k * a_dim1] = a[p + p * a_dim1]; a[p + p * a_dim1] = t; /* Convert upper triangle of A into U form by applying */ /* the interchanges in columns k+1:N. */ if (k < *n) { i__1 = *n - k; sswap_(&i__1, &a[k + (k + 1) * a_dim1], lda, &a[p + (k + 1) * a_dim1], lda); } } /* Second swap */ kk = k - kstep + 1; if (kp != kk) { /* Interchange rows and columns KK and KP in the leading */ /* submatrix A(1:k,1:k) */ if (kp > 1) { i__1 = kp - 1; sswap_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1], &c__1); } if (kk > 1 && kp < kk - 1) { i__1 = kk - kp - 1; sswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + ( kp + 1) * a_dim1], lda); } t = a[kk + kk * a_dim1]; a[kk + kk * a_dim1] = a[kp + kp * a_dim1]; a[kp + kp * a_dim1] = t; if (kstep == 2) { t = a[k - 1 + k * a_dim1]; a[k - 1 + k * a_dim1] = a[kp + k * a_dim1]; a[kp + k * a_dim1] = t; } /* Convert upper triangle of A into U form by applying */ /* the interchanges in columns k+1:N. */ if (k < *n) { i__1 = *n - k; sswap_(&i__1, &a[kk + (k + 1) * a_dim1], lda, &a[kp + (k + 1) * a_dim1], lda); } } /* 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 */ if (k > 1) { /* Perform a rank-1 update of A(1:k-1,1:k-1) and */ /* store U(k) in column k */ if ((r__1 = a[k + k * a_dim1], abs(r__1)) >= sfmin) { /* 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 */ d11 = 1.f / a[k + k * a_dim1]; i__1 = k - 1; r__1 = -d11; ssyr_(uplo, &i__1, &r__1, &a[k * a_dim1 + 1], &c__1, & a[a_offset], lda); /* Store U(k) in column k */ i__1 = k - 1; sscal_(&i__1, &d11, &a[k * a_dim1 + 1], &c__1); } else { /* Store L(k) in column K */ d11 = a[k + k * a_dim1]; i__1 = k - 1; for (ii = 1; ii <= i__1; ++ii) { a[ii + k * a_dim1] /= d11; /* L16: */ } /* Perform a rank-1 update of A(k+1:n,k+1:n) as */ /* A := A - U(k)*D(k)*U(k)**T */ /* = A - W(k)*(1/D(k))*W(k)**T */ /* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T */ i__1 = k - 1; r__1 = -d11; ssyr_(uplo, &i__1, &r__1, &a[k * a_dim1 + 1], &c__1, & a[a_offset], lda); } /* Store the superdiagonal element of D in array E */ e[k] = 0.f; } } 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 - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T */ /* and store L(k) and L(k+1) in columns k and k+1 */ if (k > 2) { d12 = a[k - 1 + k * a_dim1]; d22 = a[k - 1 + (k - 1) * a_dim1] / d12; d11 = a[k + k * a_dim1] / d12; t = 1.f / (d11 * d22 - 1.f); for (j = k - 2; j >= 1; --j) { wkm1 = t * (d11 * a[j + (k - 1) * a_dim1] - a[j + k * a_dim1]); wk = t * (d22 * a[j + k * a_dim1] - a[j + (k - 1) * a_dim1]); for (i__ = j; i__ >= 1; --i__) { a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__ + k * a_dim1] / d12 * wk - a[i__ + (k - 1) * a_dim1] / d12 * wkm1; /* L20: */ } /* Store U(k) and U(k-1) in cols k and k-1 for row J */ a[j + k * a_dim1] = wk / d12; a[j + (k - 1) * a_dim1] = wkm1 / d12; /* L30: */ } } /* Copy superdiagonal elements of D(K) to E(K) and */ /* ZERO out superdiagonal entry of A */ e[k] = a[k - 1 + k * a_dim1]; e[k - 1] = 0.f; a[k - 1 + k * a_dim1] = 0.f; } /* End column K is nonsingular */ } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -p; ipiv[k - 1] = -kp; } /* Decrease K and return to the start of the main loop */ k -= kstep; goto L10; L34: ; } else { /* Factorize A as L*D*L**T using the lower triangle of A */ /* Initialize the unused last entry of the subdiagonal array E. */ e[*n] = 0.f; /* K is the main loop index, increasing from 1 to N in steps of */ /* 1 or 2 */ k = 1; L40: /* If K > N, exit from loop */ if (k > *n) { goto L64; } kstep = 1; p = k; /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ absakk = (r__1 = a[k + k * a_dim1], abs(r__1)); /* IMAX is the row-index of the largest off-diagonal element in */ /* column K, and COLMAX is its absolute value. */ /* Determine both COLMAX and IMAX. */ if (k < *n) { i__1 = *n - k; imax = k + isamax_(&i__1, &a[k + 1 + k * a_dim1], &c__1); colmax = (r__1 = a[imax + k * a_dim1], abs(r__1)); } else { colmax = 0.f; } if (f2cmax(absakk,colmax) == 0.f) { /* Column K is zero or underflow: set INFO and continue */ if (*info == 0) { *info = k; } kp = k; /* Set E( K ) to zero */ if (k < *n) { e[k] = 0.f; } } else { /* Test for interchange */ /* Equivalent to testing for (used to handle NaN and Inf) */ /* ABSAKK.GE.ALPHA*COLMAX */ if (! (absakk < alpha * colmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { done = FALSE_; /* Loop until pivot found */ L42: /* Begin pivot search loop body */ /* JMAX is the column-index of the largest off-diagonal */ /* element in row IMAX, and ROWMAX is its absolute value. */ /* Determine both ROWMAX and JMAX. */ if (imax != k) { i__1 = imax - k; jmax = k - 1 + isamax_(&i__1, &a[imax + k * a_dim1], lda); rowmax = (r__1 = a[imax + jmax * a_dim1], abs(r__1)); } else { rowmax = 0.f; } if (imax < *n) { i__1 = *n - imax; itemp = imax + isamax_(&i__1, &a[imax + 1 + imax * a_dim1] , &c__1); stemp = (r__1 = a[itemp + imax * a_dim1], abs(r__1)); if (stemp > rowmax) { rowmax = stemp; jmax = itemp; } } /* Equivalent to testing for (used to handle NaN and Inf) */ /* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX */ if (! ((r__1 = a[imax + imax * a_dim1], abs(r__1)) < alpha * rowmax)) { /* interchange rows and columns K and IMAX, */ /* use 1-by-1 pivot block */ kp = imax; done = TRUE_; /* Equivalent to testing for ROWMAX .EQ. COLMAX, */ /* used to handle NaN and Inf */ } else if (p == jmax || rowmax <= colmax) { /* interchange rows and columns K+1 and IMAX, */ /* use 2-by-2 pivot block */ kp = imax; kstep = 2; done = TRUE_; } else { /* Pivot NOT found, set variables and repeat */ p = imax; colmax = rowmax; imax = jmax; } /* End pivot search loop body */ if (! done) { goto L42; } } /* Swap TWO rows and TWO columns */ /* First swap */ if (kstep == 2 && p != k) { /* Interchange rows and column K and P in the trailing */ /* submatrix A(k:n,k:n) if we have a 2-by-2 pivot */ if (p < *n) { i__1 = *n - p; sswap_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + 1 + p * a_dim1], &c__1); } if (p > k + 1) { i__1 = p - k - 1; sswap_(&i__1, &a[k + 1 + k * a_dim1], &c__1, &a[p + (k + 1) * a_dim1], lda); } t = a[k + k * a_dim1]; a[k + k * a_dim1] = a[p + p * a_dim1]; a[p + p * a_dim1] = t; /* Convert lower triangle of A into L form by applying */ /* the interchanges in columns 1:k-1. */ if (k > 1) { i__1 = k - 1; sswap_(&i__1, &a[k + a_dim1], lda, &a[p + a_dim1], lda); } } /* Second swap */ kk = k + kstep - 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; sswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1 + kp * a_dim1], &c__1); } if (kk < *n && kp > kk + 1) { i__1 = kp - kk - 1; sswap_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + ( kk + 1) * a_dim1], lda); } t = a[kk + kk * a_dim1]; a[kk + kk * a_dim1] = a[kp + kp * a_dim1]; a[kp + kp * a_dim1] = t; if (kstep == 2) { t = a[k + 1 + k * a_dim1]; a[k + 1 + k * a_dim1] = a[kp + k * a_dim1]; a[kp + k * a_dim1] = t; } /* Convert lower triangle of A into L form by applying */ /* the interchanges in columns 1:k-1. */ if (k > 1) { i__1 = k - 1; sswap_(&i__1, &a[kk + a_dim1], lda, &a[kp + a_dim1], lda); } } /* 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) and */ /* store L(k) in column k */ if ((r__1 = a[k + k * a_dim1], abs(r__1)) >= sfmin) { /* 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 */ d11 = 1.f / a[k + k * a_dim1]; i__1 = *n - k; r__1 = -d11; ssyr_(uplo, &i__1, &r__1, &a[k + 1 + k * a_dim1], & c__1, &a[k + 1 + (k + 1) * a_dim1], lda); /* Store L(k) in column k */ i__1 = *n - k; sscal_(&i__1, &d11, &a[k + 1 + k * a_dim1], &c__1); } else { /* Store L(k) in column k */ d11 = a[k + k * a_dim1]; i__1 = *n; for (ii = k + 1; ii <= i__1; ++ii) { a[ii + k * a_dim1] /= d11; /* L46: */ } /* 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 */ /* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T */ i__1 = *n - k; r__1 = -d11; ssyr_(uplo, &i__1, &r__1, &a[k + 1 + k * a_dim1], & c__1, &a[k + 1 + (k + 1) * a_dim1], lda); } /* Store the subdiagonal element of D in array E */ e[k] = 0.f; } } 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 */ /* 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 - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T */ /* and store L(k) and L(k+1) in columns k and k+1 */ if (k < *n - 1) { d21 = a[k + 1 + k * a_dim1]; d11 = a[k + 1 + (k + 1) * a_dim1] / d21; d22 = a[k + k * a_dim1] / d21; t = 1.f / (d11 * d22 - 1.f); i__1 = *n; for (j = k + 2; j <= i__1; ++j) { /* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J */ wk = t * (d11 * a[j + k * a_dim1] - a[j + (k + 1) * a_dim1]); wkp1 = t * (d22 * a[j + (k + 1) * a_dim1] - a[j + k * a_dim1]); /* Perform a rank-2 update of A(k+2:n,k+2:n) */ i__2 = *n; for (i__ = j; i__ <= i__2; ++i__) { a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__ + k * a_dim1] / d21 * wk - a[i__ + (k + 1) * a_dim1] / d21 * wkp1; /* L50: */ } /* Store L(k) and L(k+1) in cols k and k+1 for row J */ a[j + k * a_dim1] = wk / d21; a[j + (k + 1) * a_dim1] = wkp1 / d21; /* L60: */ } } /* Copy subdiagonal elements of D(K) to E(K) and */ /* ZERO out subdiagonal entry of A */ e[k] = a[k + 1 + k * a_dim1]; e[k + 1] = 0.f; a[k + 1 + k * a_dim1] = 0.f; } /* End column K is nonsingular */ } /* Store details of the interchanges in IPIV */ if (kstep == 1) { ipiv[k] = kp; } else { ipiv[k] = -p; ipiv[k + 1] = -kp; } /* Increase K and return to the start of the main loop */ k += kstep; goto L40; L64: ; } return 0; /* End of SSYTF2_RK */ } /* ssytf2_rk__ */