#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 CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bound ed Bunch-Kaufman (rook) diagonal pivoting method. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download CLAHEF_RK + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE CLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, */ /* INFO ) */ /* CHARACTER UPLO */ /* INTEGER INFO, KB, LDA, LDW, N, NB */ /* INTEGER IPIV( * ) */ /* COMPLEX A( LDA, * ), E( * ), W( LDW, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > CLAHEF_RK computes a partial factorization of a complex Hermitian */ /* > matrix A using the bounded Bunch-Kaufman (rook) diagonal */ /* > pivoting method. The partial factorization has the form: */ /* > */ /* > A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: */ /* > ( 0 U22 ) ( 0 D ) ( U12**H U22**H ) */ /* > */ /* > A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L', */ /* > ( L21 I ) ( 0 A22 ) ( 0 I ) */ /* > */ /* > where the order of D is at most NB. The actual order is returned in */ /* > the argument KB, and is either NB or NB-1, or N if N <= NB. */ /* > */ /* > CLAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses */ /* > blocked code (calling Level 3 BLAS) to update the submatrix */ /* > A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] UPLO */ /* > \verbatim */ /* > UPLO is CHARACTER*1 */ /* > Specifies whether the upper or lower triangular part of the */ /* > Hermitian 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] NB */ /* > \verbatim */ /* > NB is INTEGER */ /* > The maximum number of columns of the matrix A that should be */ /* > factored. NB should be at least 2 to allow for 2-by-2 pivot */ /* > blocks. */ /* > \endverbatim */ /* > */ /* > \param[out] KB */ /* > \verbatim */ /* > KB is INTEGER */ /* > The number of columns of A that were actually factored. */ /* > KB is either NB-1 or NB, or N if N <= NB. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is COMPLEX array, dimension (LDA,N) */ /* > On entry, the Hermitian 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 Hermitian 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 COMPLEX array, dimension (N) */ /* > On exit, contains the superdiagonal (or subdiagonal) */ /* > elements of the Hermitian 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 Hermitian 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. */ /* > */ /* > 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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N). */ /* > If -IPIV(k-1) = k-1, no interchange occurred. */ /* > */ /* > c) In both cases a) and b) is 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 submatrix A(1:N,1:KB). */ /* > 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 submatrix A(1:N,1:KB). */ /* > 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 submatrix A(1:N,1:KB). */ /* > If -IPIV(k+1) = k+1, no interchange occurred. */ /* > */ /* > c) In both cases a) and b) is always ABS( IPIV(k) ) >= k. */ /* > */ /* > d) NOTE: Any entry IPIV(k) is always NONZERO on output. */ /* > \endverbatim */ /* > */ /* > \param[out] W */ /* > \verbatim */ /* > W is COMPLEX array, dimension (LDW,NB) */ /* > \endverbatim */ /* > */ /* > \param[in] LDW */ /* > \verbatim */ /* > LDW is INTEGER */ /* > The leading dimension of the array W. LDW >= f2cmax(1,N). */ /* > \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 complexHEcomputational */ /* > \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 */ /* > */ /* > \endverbatim */ /* ===================================================================== */ /* Subroutine */ int clahef_rk_(char *uplo, integer *n, integer *nb, integer *kb, complex *a, integer *lda, complex *e, integer *ipiv, complex *w, integer *ldw, integer *info) { /* System generated locals */ integer a_dim1, a_offset, w_dim1, w_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2; complex q__1, q__2, q__3, q__4, q__5; /* Local variables */ logical done; integer imax, jmax, j, k, p; real t, alpha; extern /* Subroutine */ int cgemm_(char *, char *, integer *, integer *, integer *, complex *, complex *, integer *, complex *, integer *, complex *, complex *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int cgemv_(char *, integer *, integer *, complex * , complex *, integer *, complex *, integer *, complex *, complex * , integer *); real sfmin; extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *); integer itemp; extern /* Subroutine */ int cswap_(integer *, complex *, integer *, complex *, integer *); integer kstep; real stemp, r1; complex d11, d21, d22; integer jb, ii, jj, kk, kp; real absakk; extern /* Subroutine */ int clacgv_(integer *, complex *, integer *); integer kw; extern integer icamax_(integer *, complex *, integer *); extern real slamch_(char *); extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer *); real colmax, rowmax; integer kkw; /* -- 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 */ /* ===================================================================== */ /* Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; --e; --ipiv; w_dim1 = *ldw; w_offset = 1 + w_dim1 * 1; w -= w_offset; /* Function Body */ *info = 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 (lsame_(uplo, "U")) { /* Factorize the trailing columns of A using the upper triangle */ /* of A and working backwards, and compute the matrix W = U12*D */ /* for use in updating A11 (note that conjg(W) is actually stored) */ /* Initialize the first entry of array E, where superdiagonal */ /* elements of D are stored */ e[1].r = 0.f, e[1].i = 0.f; /* K is the main loop index, decreasing from N in steps of 1 or 2 */ k = *n; L10: /* KW is the column of W which corresponds to column K of A */ kw = *nb + k - *n; /* Exit from loop */ if (k <= *n - *nb + 1 && *nb < *n || k < 1) { goto L30; } kstep = 1; p = k; /* Copy column K of A to column KW of W and update it */ if (k > 1) { i__1 = k - 1; ccopy_(&i__1, &a[k * a_dim1 + 1], &c__1, &w[kw * w_dim1 + 1], & c__1); } i__1 = k + kw * w_dim1; i__2 = k + k * a_dim1; r__1 = a[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; if (k < *n) { i__1 = *n - k; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &k, &i__1, &q__1, &a[(k + 1) * a_dim1 + 1], lda, &w[k + (kw + 1) * w_dim1], ldw, &c_b1, &w[kw * w_dim1 + 1], &c__1); i__1 = k + kw * w_dim1; i__2 = k + kw * w_dim1; r__1 = w[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; } /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ i__1 = k + kw * w_dim1; absakk = (r__1 = w[i__1].r, 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 = icamax_(&i__1, &w[kw * w_dim1 + 1], &c__1); i__1 = imax + kw * w_dim1; colmax = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(&w[imax + kw * w_dim1]), abs(r__2)); } 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; i__1 = k + k * a_dim1; i__2 = k + kw * w_dim1; r__1 = w[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; if (k > 1) { i__1 = k - 1; ccopy_(&i__1, &w[kw * w_dim1 + 1], &c__1, &a[k * a_dim1 + 1], &c__1); } /* Set E( K ) to zero */ if (k > 1) { i__1 = k; e[i__1].r = 0.f, e[i__1].i = 0.f; } } else { /* ============================================================ */ /* BEGIN pivot search */ /* Case(1) */ /* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX */ /* (used to handle NaN and Inf) */ if (! (absakk < alpha * colmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { /* Lop until pivot found */ done = FALSE_; L12: /* BEGIN pivot search loop body */ /* Copy column IMAX to column KW-1 of W and update it */ if (imax > 1) { i__1 = imax - 1; ccopy_(&i__1, &a[imax * a_dim1 + 1], &c__1, &w[(kw - 1) * w_dim1 + 1], &c__1); } i__1 = imax + (kw - 1) * w_dim1; i__2 = imax + imax * a_dim1; r__1 = a[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; i__1 = k - imax; ccopy_(&i__1, &a[imax + (imax + 1) * a_dim1], lda, &w[imax + 1 + (kw - 1) * w_dim1], &c__1); i__1 = k - imax; clacgv_(&i__1, &w[imax + 1 + (kw - 1) * w_dim1], &c__1); if (k < *n) { i__1 = *n - k; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &k, &i__1, &q__1, &a[(k + 1) * a_dim1 + 1], lda, &w[imax + (kw + 1) * w_dim1], ldw, &c_b1, &w[(kw - 1) * w_dim1 + 1], &c__1); i__1 = imax + (kw - 1) * w_dim1; i__2 = imax + (kw - 1) * w_dim1; r__1 = w[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; } /* 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 + icamax_(&i__1, &w[imax + 1 + (kw - 1) * w_dim1], &c__1); i__1 = jmax + (kw - 1) * w_dim1; rowmax = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(& w[jmax + (kw - 1) * w_dim1]), abs(r__2)); } else { rowmax = 0.f; } if (imax > 1) { i__1 = imax - 1; itemp = icamax_(&i__1, &w[(kw - 1) * w_dim1 + 1], &c__1); i__1 = itemp + (kw - 1) * w_dim1; stemp = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(&w[ itemp + (kw - 1) * w_dim1]), abs(r__2)); if (stemp > rowmax) { rowmax = stemp; jmax = itemp; } } /* Case(2) */ /* Equivalent to testing for */ /* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX */ /* (used to handle NaN and Inf) */ i__1 = imax + (kw - 1) * w_dim1; if (! ((r__1 = w[i__1].r, abs(r__1)) < alpha * rowmax)) { /* interchange rows and columns K and IMAX, */ /* use 1-by-1 pivot block */ kp = imax; /* copy column KW-1 of W to column KW of W */ ccopy_(&k, &w[(kw - 1) * w_dim1 + 1], &c__1, &w[kw * w_dim1 + 1], &c__1); done = TRUE_; /* Case(3) */ /* 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_; /* Case(4) */ } else { /* Pivot not found: set params and repeat */ p = imax; colmax = rowmax; imax = jmax; /* Copy updated JMAXth (next IMAXth) column to Kth of W */ ccopy_(&k, &w[(kw - 1) * w_dim1 + 1], &c__1, &w[kw * w_dim1 + 1], &c__1); } /* END pivot search loop body */ if (! done) { goto L12; } } /* END pivot search */ /* ============================================================ */ /* KK is the column of A where pivoting step stopped */ kk = k - kstep + 1; /* KKW is the column of W which corresponds to column KK of A */ kkw = *nb + kk - *n; /* Interchange rows and columns P and K. */ /* Updated column P is already stored in column KW of W. */ if (kstep == 2 && p != k) { /* Copy non-updated column K to column P of submatrix A */ /* at step K. No need to copy element into columns */ /* K and K-1 of A for 2-by-2 pivot, since these columns */ /* will be later overwritten. */ i__1 = p + p * a_dim1; i__2 = k + k * a_dim1; r__1 = a[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; i__1 = k - 1 - p; ccopy_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + (p + 1) * a_dim1], lda); i__1 = k - 1 - p; clacgv_(&i__1, &a[p + (p + 1) * a_dim1], lda); if (p > 1) { i__1 = p - 1; ccopy_(&i__1, &a[k * a_dim1 + 1], &c__1, &a[p * a_dim1 + 1], &c__1); } /* Interchange rows K and P in the last K+1 to N columns of A */ /* (columns K and K-1 of A for 2-by-2 pivot will be */ /* later overwritten). Interchange rows K and P */ /* in last KKW to NB columns of W. */ if (k < *n) { i__1 = *n - k; cswap_(&i__1, &a[k + (k + 1) * a_dim1], lda, &a[p + (k + 1) * a_dim1], lda); } i__1 = *n - kk + 1; cswap_(&i__1, &w[k + kkw * w_dim1], ldw, &w[p + kkw * w_dim1], ldw); } /* Interchange rows and columns KP and KK. */ /* Updated column KP is already stored in column KKW of W. */ if (kp != kk) { /* Copy non-updated column KK to column KP of submatrix A */ /* at step K. No need to copy element into column K */ /* (or K and K-1 for 2-by-2 pivot) of A, since these columns */ /* will be later overwritten. */ i__1 = kp + kp * a_dim1; i__2 = kk + kk * a_dim1; r__1 = a[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; i__1 = kk - 1 - kp; ccopy_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (kp + 1) * a_dim1], lda); i__1 = kk - 1 - kp; clacgv_(&i__1, &a[kp + (kp + 1) * a_dim1], lda); if (kp > 1) { i__1 = kp - 1; ccopy_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1], &c__1); } /* Interchange rows KK and KP in last K+1 to N columns of A */ /* (columns K (or K and K-1 for 2-by-2 pivot) of A will be */ /* later overwritten). Interchange rows KK and KP */ /* in last KKW to NB columns of W. */ if (k < *n) { i__1 = *n - k; cswap_(&i__1, &a[kk + (k + 1) * a_dim1], lda, &a[kp + (k + 1) * a_dim1], lda); } i__1 = *n - kk + 1; cswap_(&i__1, &w[kk + kkw * w_dim1], ldw, &w[kp + kkw * w_dim1], ldw); } if (kstep == 1) { /* 1-by-1 pivot block D(k): column kw of W now holds */ /* W(kw) = U(k)*D(k), */ /* where U(k) is the k-th column of U */ /* (1) Store subdiag. elements of column U(k) */ /* and 1-by-1 block D(k) in column k of A. */ /* (NOTE: Diagonal element U(k,k) is a UNIT element */ /* and not stored) */ /* A(k,k) := D(k,k) = W(k,kw) */ /* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) */ /* (NOTE: No need to use for Hermitian matrix */ /* A( K, K ) = REAL( W( K, K) ) to separately copy diagonal */ /* element D(k,k) from W (potentially saves only one load)) */ ccopy_(&k, &w[kw * w_dim1 + 1], &c__1, &a[k * a_dim1 + 1], & c__1); if (k > 1) { /* (NOTE: No need to check if A(k,k) is NOT ZERO, */ /* since that was ensured earlier in pivot search: */ /* case A(k,k) = 0 falls into 2x2 pivot case(3)) */ /* Handle division by a small number */ i__1 = k + k * a_dim1; t = a[i__1].r; if (abs(t) >= sfmin) { r1 = 1.f / t; i__1 = k - 1; csscal_(&i__1, &r1, &a[k * a_dim1 + 1], &c__1); } else { i__1 = k - 1; for (ii = 1; ii <= i__1; ++ii) { i__2 = ii + k * a_dim1; i__3 = ii + k * a_dim1; q__1.r = a[i__3].r / t, q__1.i = a[i__3].i / t; a[i__2].r = q__1.r, a[i__2].i = q__1.i; /* L14: */ } } /* (2) Conjugate column W(kw) */ i__1 = k - 1; clacgv_(&i__1, &w[kw * w_dim1 + 1], &c__1); /* Store the superdiagonal element of D in array E */ i__1 = k; e[i__1].r = 0.f, e[i__1].i = 0.f; } } else { /* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold */ /* ( W(kw-1) W(kw) ) = ( 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 */ /* (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 */ /* block D(k-1:k,k-1:k) in columns k-1 and k of A. */ /* (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT */ /* block and not stored) */ /* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) */ /* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = */ /* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) */ if (k > 2) { /* Factor out the columns of the inverse of 2-by-2 pivot */ /* block D, so that each column contains 1, to reduce the */ /* number of FLOPS when we multiply panel */ /* ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1). */ /* D**(-1) = ( d11 cj(d21) )**(-1) = */ /* ( d21 d22 ) */ /* = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) = */ /* ( (-d21) ( d11 ) ) */ /* = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) * */ /* * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) = */ /* ( ( -1 ) ( d11/conj(d21) ) ) */ /* = 1/(|d21|**2) * 1/(D22*D11-1) * */ /* * ( d21*( D11 ) conj(d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* Handle division by a small number. (NOTE: order of */ /* operations is important) */ /* = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) ) */ /* ( (( -1 ) ) (( D22 ) ) ), */ /* where D11 = d22/d21, */ /* D22 = d11/conj(d21), */ /* D21 = d21, */ /* T = 1/(D22*D11-1). */ /* (NOTE: No need to check for division by ZERO, */ /* since that was ensured earlier in pivot search: */ /* (a) d21 != 0 in 2x2 pivot case(4), */ /* since |d21| should be larger than |d11| and |d22|; */ /* (b) (D22*D11 - 1) != 0, since from (a), */ /* both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.) */ i__1 = k - 1 + kw * w_dim1; d21.r = w[i__1].r, d21.i = w[i__1].i; r_cnjg(&q__2, &d21); c_div(&q__1, &w[k + kw * w_dim1], &q__2); d11.r = q__1.r, d11.i = q__1.i; c_div(&q__1, &w[k - 1 + (kw - 1) * w_dim1], &d21); d22.r = q__1.r, d22.i = q__1.i; q__1.r = d11.r * d22.r - d11.i * d22.i, q__1.i = d11.r * d22.i + d11.i * d22.r; t = 1.f / (q__1.r - 1.f); /* Update elements in columns A(k-1) and A(k) as */ /* dot products of rows of ( W(kw-1) W(kw) ) and columns */ /* of D**(-1) */ i__1 = k - 2; for (j = 1; j <= i__1; ++j) { i__2 = j + (k - 1) * a_dim1; i__3 = j + (kw - 1) * w_dim1; q__4.r = d11.r * w[i__3].r - d11.i * w[i__3].i, q__4.i = d11.r * w[i__3].i + d11.i * w[i__3] .r; i__4 = j + kw * w_dim1; q__3.r = q__4.r - w[i__4].r, q__3.i = q__4.i - w[i__4] .i; c_div(&q__2, &q__3, &d21); q__1.r = t * q__2.r, q__1.i = t * q__2.i; a[i__2].r = q__1.r, a[i__2].i = q__1.i; i__2 = j + k * a_dim1; i__3 = j + kw * w_dim1; q__4.r = d22.r * w[i__3].r - d22.i * w[i__3].i, q__4.i = d22.r * w[i__3].i + d22.i * w[i__3] .r; i__4 = j + (kw - 1) * w_dim1; q__3.r = q__4.r - w[i__4].r, q__3.i = q__4.i - w[i__4] .i; r_cnjg(&q__5, &d21); c_div(&q__2, &q__3, &q__5); q__1.r = t * q__2.r, q__1.i = t * q__2.i; a[i__2].r = q__1.r, a[i__2].i = q__1.i; /* L20: */ } } /* Copy diagonal elements of D(K) to A, */ /* copy superdiagonal element of D(K) to E(K) and */ /* ZERO out superdiagonal entry of A */ i__1 = k - 1 + (k - 1) * a_dim1; i__2 = k - 1 + (kw - 1) * w_dim1; a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i; i__1 = k - 1 + k * a_dim1; a[i__1].r = 0.f, a[i__1].i = 0.f; i__1 = k + k * a_dim1; i__2 = k + kw * w_dim1; a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i; i__1 = k; i__2 = k - 1 + kw * w_dim1; e[i__1].r = w[i__2].r, e[i__1].i = w[i__2].i; i__1 = k - 1; e[i__1].r = 0.f, e[i__1].i = 0.f; /* (2) Conjugate columns W(kw) and W(kw-1) */ i__1 = k - 1; clacgv_(&i__1, &w[kw * w_dim1 + 1], &c__1); i__1 = k - 2; clacgv_(&i__1, &w[(kw - 1) * w_dim1 + 1], &c__1); } /* 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; L30: /* Update the upper triangle of A11 (= A(1:k,1:k)) as */ /* A11 := A11 - U12*D*U12**H = A11 - U12*W**H */ /* computing blocks of NB columns at a time (note that conjg(W) is */ /* actually stored) */ i__1 = -(*nb); for (j = (k - 1) / *nb * *nb + 1; i__1 < 0 ? j >= 1 : j <= 1; j += i__1) { /* Computing MIN */ i__2 = *nb, i__3 = k - j + 1; jb = f2cmin(i__2,i__3); /* Update the upper triangle of the diagonal block */ i__2 = j + jb - 1; for (jj = j; jj <= i__2; ++jj) { i__3 = jj + jj * a_dim1; i__4 = jj + jj * a_dim1; r__1 = a[i__4].r; a[i__3].r = r__1, a[i__3].i = 0.f; i__3 = jj - j + 1; i__4 = *n - k; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &i__3, &i__4, &q__1, &a[j + (k + 1) * a_dim1], lda, &w[jj + (kw + 1) * w_dim1], ldw, &c_b1, &a[j + jj * a_dim1], &c__1); i__3 = jj + jj * a_dim1; i__4 = jj + jj * a_dim1; r__1 = a[i__4].r; a[i__3].r = r__1, a[i__3].i = 0.f; /* L40: */ } /* Update the rectangular superdiagonal block */ if (j >= 2) { i__2 = j - 1; i__3 = *n - k; q__1.r = -1.f, q__1.i = 0.f; cgemm_("No transpose", "Transpose", &i__2, &jb, &i__3, &q__1, &a[(k + 1) * a_dim1 + 1], lda, &w[j + (kw + 1) * w_dim1], ldw, &c_b1, &a[j * a_dim1 + 1], lda); } /* L50: */ } /* Set KB to the number of columns factorized */ *kb = *n - k; } else { /* Factorize the leading columns of A using the lower triangle */ /* of A and working forwards, and compute the matrix W = L21*D */ /* for use in updating A22 (note that conjg(W) is actually stored) */ /* Initialize the unused last entry of the subdiagonal array E. */ i__1 = *n; e[i__1].r = 0.f, e[i__1].i = 0.f; /* K is the main loop index, increasing from 1 in steps of 1 or 2 */ k = 1; L70: /* Exit from loop */ if (k >= *nb && *nb < *n || k > *n) { goto L90; } kstep = 1; p = k; /* Copy column K of A to column K of W and update column K of W */ i__1 = k + k * w_dim1; i__2 = k + k * a_dim1; r__1 = a[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; if (k < *n) { i__1 = *n - k; ccopy_(&i__1, &a[k + 1 + k * a_dim1], &c__1, &w[k + 1 + k * w_dim1], &c__1); } if (k > 1) { i__1 = *n - k + 1; i__2 = k - 1; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &i__1, &i__2, &q__1, &a[k + a_dim1], lda, & w[k + w_dim1], ldw, &c_b1, &w[k + k * w_dim1], &c__1); i__1 = k + k * w_dim1; i__2 = k + k * w_dim1; r__1 = w[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; } /* Determine rows and columns to be interchanged and whether */ /* a 1-by-1 or 2-by-2 pivot block will be used */ i__1 = k + k * w_dim1; absakk = (r__1 = w[i__1].r, 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 + icamax_(&i__1, &w[k + 1 + k * w_dim1], &c__1); i__1 = imax + k * w_dim1; colmax = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(&w[imax + k * w_dim1]), abs(r__2)); } 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; i__1 = k + k * a_dim1; i__2 = k + k * w_dim1; r__1 = w[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; if (k < *n) { i__1 = *n - k; ccopy_(&i__1, &w[k + 1 + k * w_dim1], &c__1, &a[k + 1 + k * a_dim1], &c__1); } /* Set E( K ) to zero */ if (k < *n) { i__1 = k; e[i__1].r = 0.f, e[i__1].i = 0.f; } } else { /* ============================================================ */ /* BEGIN pivot search */ /* Case(1) */ /* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX */ /* (used to handle NaN and Inf) */ if (! (absakk < alpha * colmax)) { /* no interchange, use 1-by-1 pivot block */ kp = k; } else { done = FALSE_; /* Loop until pivot found */ L72: /* BEGIN pivot search loop body */ /* Copy column IMAX to column k+1 of W and update it */ i__1 = imax - k; ccopy_(&i__1, &a[imax + k * a_dim1], lda, &w[k + (k + 1) * w_dim1], &c__1); i__1 = imax - k; clacgv_(&i__1, &w[k + (k + 1) * w_dim1], &c__1); i__1 = imax + (k + 1) * w_dim1; i__2 = imax + imax * a_dim1; r__1 = a[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; if (imax < *n) { i__1 = *n - imax; ccopy_(&i__1, &a[imax + 1 + imax * a_dim1], &c__1, &w[ imax + 1 + (k + 1) * w_dim1], &c__1); } if (k > 1) { i__1 = *n - k + 1; i__2 = k - 1; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &i__1, &i__2, &q__1, &a[k + a_dim1] , lda, &w[imax + w_dim1], ldw, &c_b1, &w[k + (k + 1) * w_dim1], &c__1); i__1 = imax + (k + 1) * w_dim1; i__2 = imax + (k + 1) * w_dim1; r__1 = w[i__2].r; w[i__1].r = r__1, w[i__1].i = 0.f; } /* 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 + icamax_(&i__1, &w[k + (k + 1) * w_dim1], & c__1); i__1 = jmax + (k + 1) * w_dim1; rowmax = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(& w[jmax + (k + 1) * w_dim1]), abs(r__2)); } else { rowmax = 0.f; } if (imax < *n) { i__1 = *n - imax; itemp = imax + icamax_(&i__1, &w[imax + 1 + (k + 1) * w_dim1], &c__1); i__1 = itemp + (k + 1) * w_dim1; stemp = (r__1 = w[i__1].r, abs(r__1)) + (r__2 = r_imag(&w[ itemp + (k + 1) * w_dim1]), abs(r__2)); if (stemp > rowmax) { rowmax = stemp; jmax = itemp; } } /* Case(2) */ /* Equivalent to testing for */ /* ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX */ /* (used to handle NaN and Inf) */ i__1 = imax + (k + 1) * w_dim1; if (! ((r__1 = w[i__1].r, abs(r__1)) < alpha * rowmax)) { /* interchange rows and columns K and IMAX, */ /* use 1-by-1 pivot block */ kp = imax; /* copy column K+1 of W to column K of W */ i__1 = *n - k + 1; ccopy_(&i__1, &w[k + (k + 1) * w_dim1], &c__1, &w[k + k * w_dim1], &c__1); done = TRUE_; /* Case(3) */ /* 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_; /* Case(4) */ } else { /* Pivot not found: set params and repeat */ p = imax; colmax = rowmax; imax = jmax; /* Copy updated JMAXth (next IMAXth) column to Kth of W */ i__1 = *n - k + 1; ccopy_(&i__1, &w[k + (k + 1) * w_dim1], &c__1, &w[k + k * w_dim1], &c__1); } /* End pivot search loop body */ if (! done) { goto L72; } } /* END pivot search */ /* ============================================================ */ /* KK is the column of A where pivoting step stopped */ kk = k + kstep - 1; /* Interchange rows and columns P and K (only for 2-by-2 pivot). */ /* Updated column P is already stored in column K of W. */ if (kstep == 2 && p != k) { /* Copy non-updated column KK-1 to column P of submatrix A */ /* at step K. No need to copy element into columns */ /* K and K+1 of A for 2-by-2 pivot, since these columns */ /* will be later overwritten. */ i__1 = p + p * a_dim1; i__2 = k + k * a_dim1; r__1 = a[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; i__1 = p - k - 1; ccopy_(&i__1, &a[k + 1 + k * a_dim1], &c__1, &a[p + (k + 1) * a_dim1], lda); i__1 = p - k - 1; clacgv_(&i__1, &a[p + (k + 1) * a_dim1], lda); if (p < *n) { i__1 = *n - p; ccopy_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + 1 + p * a_dim1], &c__1); } /* Interchange rows K and P in first K-1 columns of A */ /* (columns K and K+1 of A for 2-by-2 pivot will be */ /* later overwritten). Interchange rows K and P */ /* in first KK columns of W. */ if (k > 1) { i__1 = k - 1; cswap_(&i__1, &a[k + a_dim1], lda, &a[p + a_dim1], lda); } cswap_(&kk, &w[k + w_dim1], ldw, &w[p + w_dim1], ldw); } /* Interchange rows and columns KP and KK. */ /* Updated column KP is already stored in column KK of W. */ if (kp != kk) { /* Copy non-updated column KK to column KP of submatrix A */ /* at step K. No need to copy element into column K */ /* (or K and K+1 for 2-by-2 pivot) of A, since these columns */ /* will be later overwritten. */ i__1 = kp + kp * a_dim1; i__2 = kk + kk * a_dim1; r__1 = a[i__2].r; a[i__1].r = r__1, a[i__1].i = 0.f; i__1 = kp - kk - 1; ccopy_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + (kk + 1) * a_dim1], lda); i__1 = kp - kk - 1; clacgv_(&i__1, &a[kp + (kk + 1) * a_dim1], lda); if (kp < *n) { i__1 = *n - kp; ccopy_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1 + kp * a_dim1], &c__1); } /* Interchange rows KK and KP in first K-1 columns of A */ /* (column K (or K and K+1 for 2-by-2 pivot) of A will be */ /* later overwritten). Interchange rows KK and KP */ /* in first KK columns of W. */ if (k > 1) { i__1 = k - 1; cswap_(&i__1, &a[kk + a_dim1], lda, &a[kp + a_dim1], lda); } cswap_(&kk, &w[kk + w_dim1], ldw, &w[kp + w_dim1], ldw); } if (kstep == 1) { /* 1-by-1 pivot block D(k): column k of W now holds */ /* W(k) = L(k)*D(k), */ /* where L(k) is the k-th column of L */ /* (1) Store subdiag. elements of column L(k) */ /* and 1-by-1 block D(k) in column k of A. */ /* (NOTE: Diagonal element L(k,k) is a UNIT element */ /* and not stored) */ /* A(k,k) := D(k,k) = W(k,k) */ /* A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k) */ /* (NOTE: No need to use for Hermitian matrix */ /* A( K, K ) = REAL( W( K, K) ) to separately copy diagonal */ /* element D(k,k) from W (potentially saves only one load)) */ i__1 = *n - k + 1; ccopy_(&i__1, &w[k + k * w_dim1], &c__1, &a[k + k * a_dim1], & c__1); if (k < *n) { /* (NOTE: No need to check if A(k,k) is NOT ZERO, */ /* since that was ensured earlier in pivot search: */ /* case A(k,k) = 0 falls into 2x2 pivot case(3)) */ /* Handle division by a small number */ i__1 = k + k * a_dim1; t = a[i__1].r; if (abs(t) >= sfmin) { r1 = 1.f / t; i__1 = *n - k; csscal_(&i__1, &r1, &a[k + 1 + k * a_dim1], &c__1); } else { i__1 = *n; for (ii = k + 1; ii <= i__1; ++ii) { i__2 = ii + k * a_dim1; i__3 = ii + k * a_dim1; q__1.r = a[i__3].r / t, q__1.i = a[i__3].i / t; a[i__2].r = q__1.r, a[i__2].i = q__1.i; /* L74: */ } } /* (2) Conjugate column W(k) */ i__1 = *n - k; clacgv_(&i__1, &w[k + 1 + k * w_dim1], &c__1); /* Store the subdiagonal element of D in array E */ i__1 = k; e[i__1].r = 0.f, e[i__1].i = 0.f; } } else { /* 2-by-2 pivot block D(k): columns k and k+1 of W 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 */ /* (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2 */ /* block D(k:k+1,k:k+1) in columns k and k+1 of A. */ /* NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT */ /* block and not stored. */ /* A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1) */ /* A(k+2:N,k:k+1) := L(k+2:N,k:k+1) = */ /* = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) ) */ if (k < *n - 1) { /* Factor out the columns of the inverse of 2-by-2 pivot */ /* block D, so that each column contains 1, to reduce the */ /* number of FLOPS when we multiply panel */ /* ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1). */ /* D**(-1) = ( d11 cj(d21) )**(-1) = */ /* ( d21 d22 ) */ /* = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) = */ /* ( (-d21) ( d11 ) ) */ /* = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) * */ /* * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) = */ /* ( ( -1 ) ( d11/conj(d21) ) ) */ /* = 1/(|d21|**2) * 1/(D22*D11-1) * */ /* * ( d21*( D11 ) conj(d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) = */ /* ( ( -1 ) ( D22 ) ) */ /* Handle division by a small number. (NOTE: order of */ /* operations is important) */ /* = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) ) */ /* ( (( -1 ) ) (( D22 ) ) ), */ /* where D11 = d22/d21, */ /* D22 = d11/conj(d21), */ /* D21 = d21, */ /* T = 1/(D22*D11-1). */ /* (NOTE: No need to check for division by ZERO, */ /* since that was ensured earlier in pivot search: */ /* (a) d21 != 0 in 2x2 pivot case(4), */ /* since |d21| should be larger than |d11| and |d22|; */ /* (b) (D22*D11 - 1) != 0, since from (a), */ /* both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.) */ i__1 = k + 1 + k * w_dim1; d21.r = w[i__1].r, d21.i = w[i__1].i; c_div(&q__1, &w[k + 1 + (k + 1) * w_dim1], &d21); d11.r = q__1.r, d11.i = q__1.i; r_cnjg(&q__2, &d21); c_div(&q__1, &w[k + k * w_dim1], &q__2); d22.r = q__1.r, d22.i = q__1.i; q__1.r = d11.r * d22.r - d11.i * d22.i, q__1.i = d11.r * d22.i + d11.i * d22.r; t = 1.f / (q__1.r - 1.f); /* Update elements in columns A(k) and A(k+1) as */ /* dot products of rows of ( W(k) W(k+1) ) and columns */ /* of D**(-1) */ i__1 = *n; for (j = k + 2; j <= i__1; ++j) { i__2 = j + k * a_dim1; i__3 = j + k * w_dim1; q__4.r = d11.r * w[i__3].r - d11.i * w[i__3].i, q__4.i = d11.r * w[i__3].i + d11.i * w[i__3] .r; i__4 = j + (k + 1) * w_dim1; q__3.r = q__4.r - w[i__4].r, q__3.i = q__4.i - w[i__4] .i; r_cnjg(&q__5, &d21); c_div(&q__2, &q__3, &q__5); q__1.r = t * q__2.r, q__1.i = t * q__2.i; a[i__2].r = q__1.r, a[i__2].i = q__1.i; i__2 = j + (k + 1) * a_dim1; i__3 = j + (k + 1) * w_dim1; q__4.r = d22.r * w[i__3].r - d22.i * w[i__3].i, q__4.i = d22.r * w[i__3].i + d22.i * w[i__3] .r; i__4 = j + k * w_dim1; q__3.r = q__4.r - w[i__4].r, q__3.i = q__4.i - w[i__4] .i; c_div(&q__2, &q__3, &d21); q__1.r = t * q__2.r, q__1.i = t * q__2.i; a[i__2].r = q__1.r, a[i__2].i = q__1.i; /* L80: */ } } /* Copy diagonal elements of D(K) to A, */ /* copy subdiagonal element of D(K) to E(K) and */ /* ZERO out subdiagonal entry of A */ i__1 = k + k * a_dim1; i__2 = k + k * w_dim1; a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i; i__1 = k + 1 + k * a_dim1; a[i__1].r = 0.f, a[i__1].i = 0.f; i__1 = k + 1 + (k + 1) * a_dim1; i__2 = k + 1 + (k + 1) * w_dim1; a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i; i__1 = k; i__2 = k + 1 + k * w_dim1; e[i__1].r = w[i__2].r, e[i__1].i = w[i__2].i; i__1 = k + 1; e[i__1].r = 0.f, e[i__1].i = 0.f; /* (2) Conjugate columns W(k) and W(k+1) */ i__1 = *n - k; clacgv_(&i__1, &w[k + 1 + k * w_dim1], &c__1); i__1 = *n - k - 1; clacgv_(&i__1, &w[k + 2 + (k + 1) * w_dim1], &c__1); } /* 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 L70; L90: /* Update the lower triangle of A22 (= A(k:n,k:n)) as */ /* A22 := A22 - L21*D*L21**H = A22 - L21*W**H */ /* computing blocks of NB columns at a time (note that conjg(W) is */ /* actually stored) */ i__1 = *n; i__2 = *nb; for (j = k; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) { /* Computing MIN */ i__3 = *nb, i__4 = *n - j + 1; jb = f2cmin(i__3,i__4); /* Update the lower triangle of the diagonal block */ i__3 = j + jb - 1; for (jj = j; jj <= i__3; ++jj) { i__4 = jj + jj * a_dim1; i__5 = jj + jj * a_dim1; r__1 = a[i__5].r; a[i__4].r = r__1, a[i__4].i = 0.f; i__4 = j + jb - jj; i__5 = k - 1; q__1.r = -1.f, q__1.i = 0.f; cgemv_("No transpose", &i__4, &i__5, &q__1, &a[jj + a_dim1], lda, &w[jj + w_dim1], ldw, &c_b1, &a[jj + jj * a_dim1] , &c__1); i__4 = jj + jj * a_dim1; i__5 = jj + jj * a_dim1; r__1 = a[i__5].r; a[i__4].r = r__1, a[i__4].i = 0.f; /* L100: */ } /* Update the rectangular subdiagonal block */ if (j + jb <= *n) { i__3 = *n - j - jb + 1; i__4 = k - 1; q__1.r = -1.f, q__1.i = 0.f; cgemm_("No transpose", "Transpose", &i__3, &jb, &i__4, &q__1, &a[j + jb + a_dim1], lda, &w[j + w_dim1], ldw, &c_b1, &a[j + jb + j * a_dim1], lda); } /* L110: */ } /* Set KB to the number of columns factorized */ *kb = k - 1; } return 0; /* End of CLAHEF_RK */ } /* clahef_rk__ */