920 lines
26 KiB
C
920 lines
26 KiB
C
#include <math.h>
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#include <stdlib.h>
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#include <string.h>
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#include <stdio.h>
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#include <complex.h>
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#ifdef complex
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#undef complex
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#endif
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#ifdef I
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#undef I
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#endif
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#if defined(_WIN64)
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typedef long long BLASLONG;
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typedef unsigned long long BLASULONG;
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#else
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typedef long BLASLONG;
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typedef unsigned long BLASULONG;
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#endif
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#ifdef LAPACK_ILP64
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typedef BLASLONG blasint;
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#if defined(_WIN64)
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#define blasabs(x) llabs(x)
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#else
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#define blasabs(x) labs(x)
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#endif
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#else
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typedef int blasint;
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#define blasabs(x) abs(x)
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#endif
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typedef blasint integer;
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typedef unsigned int uinteger;
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typedef char *address;
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typedef short int shortint;
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typedef float real;
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typedef double doublereal;
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typedef struct { real r, i; } complex;
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typedef struct { doublereal r, i; } doublecomplex;
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#ifdef _MSC_VER
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static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
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static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
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static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
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static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
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#else
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static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
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static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
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static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
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#endif
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#define pCf(z) (*_pCf(z))
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#define pCd(z) (*_pCd(z))
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typedef int logical;
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typedef short int shortlogical;
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typedef char logical1;
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typedef char integer1;
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#define TRUE_ (1)
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#define FALSE_ (0)
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/* Extern is for use with -E */
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#ifndef Extern
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#define Extern extern
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#endif
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/* I/O stuff */
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typedef int flag;
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typedef int ftnlen;
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typedef int ftnint;
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/*external read, write*/
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typedef struct
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{ flag cierr;
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ftnint ciunit;
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flag ciend;
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char *cifmt;
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ftnint cirec;
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} cilist;
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/*internal read, write*/
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typedef struct
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{ flag icierr;
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char *iciunit;
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flag iciend;
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char *icifmt;
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ftnint icirlen;
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ftnint icirnum;
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} icilist;
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/*open*/
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typedef struct
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{ flag oerr;
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ftnint ounit;
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char *ofnm;
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ftnlen ofnmlen;
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char *osta;
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char *oacc;
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char *ofm;
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ftnint orl;
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char *oblnk;
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} olist;
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/*close*/
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typedef struct
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{ flag cerr;
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ftnint cunit;
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char *csta;
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} cllist;
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/*rewind, backspace, endfile*/
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typedef struct
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{ flag aerr;
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ftnint aunit;
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} alist;
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/* inquire */
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typedef struct
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{ flag inerr;
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ftnint inunit;
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char *infile;
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ftnlen infilen;
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ftnint *inex; /*parameters in standard's order*/
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ftnint *inopen;
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ftnint *innum;
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ftnint *innamed;
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char *inname;
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ftnlen innamlen;
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char *inacc;
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ftnlen inacclen;
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char *inseq;
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ftnlen inseqlen;
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char *indir;
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ftnlen indirlen;
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char *infmt;
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ftnlen infmtlen;
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char *inform;
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ftnint informlen;
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char *inunf;
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ftnlen inunflen;
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ftnint *inrecl;
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ftnint *innrec;
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char *inblank;
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ftnlen inblanklen;
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} inlist;
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#define VOID void
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union Multitype { /* for multiple entry points */
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integer1 g;
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shortint h;
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integer i;
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/* longint j; */
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real r;
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doublereal d;
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complex c;
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doublecomplex z;
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};
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typedef union Multitype Multitype;
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struct Vardesc { /* for Namelist */
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char *name;
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char *addr;
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ftnlen *dims;
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int type;
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};
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typedef struct Vardesc Vardesc;
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struct Namelist {
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char *name;
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Vardesc **vars;
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int nvars;
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};
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typedef struct Namelist Namelist;
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#define abs(x) ((x) >= 0 ? (x) : -(x))
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#define dabs(x) (fabs(x))
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#define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
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#define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
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#define dmin(a,b) (f2cmin(a,b))
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#define dmax(a,b) (f2cmax(a,b))
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#define bit_test(a,b) ((a) >> (b) & 1)
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#define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
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#define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
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#define abort_() { sig_die("Fortran abort routine called", 1); }
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#define c_abs(z) (cabsf(Cf(z)))
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#define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
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#ifdef _MSC_VER
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#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]);}
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#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]);}
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#else
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#define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
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#define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
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#endif
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#define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
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#define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
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#define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
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//#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
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#define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
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#define d_abs(x) (fabs(*(x)))
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#define d_acos(x) (acos(*(x)))
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#define d_asin(x) (asin(*(x)))
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#define d_atan(x) (atan(*(x)))
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#define d_atn2(x, y) (atan2(*(x),*(y)))
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#define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
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#define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
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#define d_cos(x) (cos(*(x)))
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#define d_cosh(x) (cosh(*(x)))
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#define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
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#define d_exp(x) (exp(*(x)))
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#define d_imag(z) (cimag(Cd(z)))
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#define r_imag(z) (cimagf(Cf(z)))
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#define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
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#define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
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#define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
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#define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
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#define d_log(x) (log(*(x)))
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#define d_mod(x, y) (fmod(*(x), *(y)))
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#define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
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#define d_nint(x) u_nint(*(x))
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#define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
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#define d_sign(a,b) u_sign(*(a),*(b))
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#define r_sign(a,b) u_sign(*(a),*(b))
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#define d_sin(x) (sin(*(x)))
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#define d_sinh(x) (sinh(*(x)))
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#define d_sqrt(x) (sqrt(*(x)))
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#define d_tan(x) (tan(*(x)))
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#define d_tanh(x) (tanh(*(x)))
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#define i_abs(x) abs(*(x))
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#define i_dnnt(x) ((integer)u_nint(*(x)))
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#define i_len(s, n) (n)
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#define i_nint(x) ((integer)u_nint(*(x)))
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#define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
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#define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
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#define pow_si(B,E) spow_ui(*(B),*(E))
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#define pow_ri(B,E) spow_ui(*(B),*(E))
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#define pow_di(B,E) dpow_ui(*(B),*(E))
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#define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
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#define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
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#define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
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#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++ = ' '; }
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#define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
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#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]; }
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#define sig_die(s, kill) { exit(1); }
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#define s_stop(s, n) {exit(0);}
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static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
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#define z_abs(z) (cabs(Cd(z)))
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#define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
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#define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
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#define myexit_() break;
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#define mycycle() continue;
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#define myceiling(w) {ceil(w)}
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#define myhuge(w) {HUGE_VAL}
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//#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
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#define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
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/* procedure parameter types for -A and -C++ */
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#define F2C_proc_par_types 1
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#ifdef __cplusplus
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typedef logical (*L_fp)(...);
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#else
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typedef logical (*L_fp)();
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#endif
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static float spow_ui(float x, integer n) {
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float pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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static double dpow_ui(double x, integer n) {
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double pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#ifdef _MSC_VER
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static _Fcomplex cpow_ui(complex x, integer n) {
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complex pow={1.0,0.0}; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
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for(u = n; ; ) {
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if(u & 01) pow.r *= x.r, pow.i *= x.i;
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if(u >>= 1) x.r *= x.r, x.i *= x.i;
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else break;
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}
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}
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_Fcomplex p={pow.r, pow.i};
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return p;
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}
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#else
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static _Complex float cpow_ui(_Complex float x, integer n) {
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_Complex float pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#endif
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#ifdef _MSC_VER
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static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
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_Dcomplex pow={1.0,0.0}; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
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for(u = n; ; ) {
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if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
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if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
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else break;
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}
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}
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_Dcomplex p = {pow._Val[0], pow._Val[1]};
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return p;
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}
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#else
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static _Complex double zpow_ui(_Complex double x, integer n) {
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_Complex double pow=1.0; unsigned long int u;
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if(n != 0) {
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if(n < 0) n = -n, x = 1/x;
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for(u = n; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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#endif
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static integer pow_ii(integer x, integer n) {
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integer pow; unsigned long int u;
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if (n <= 0) {
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if (n == 0 || x == 1) pow = 1;
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else if (x != -1) pow = x == 0 ? 1/x : 0;
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else n = -n;
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}
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if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
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u = n;
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for(pow = 1; ; ) {
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if(u & 01) pow *= x;
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if(u >>= 1) x *= x;
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else break;
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}
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}
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return pow;
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}
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static integer dmaxloc_(double *w, integer s, integer e, integer *n)
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{
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double m; integer i, mi;
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for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
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if (w[i-1]>m) mi=i ,m=w[i-1];
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return mi-s+1;
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}
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static integer smaxloc_(float *w, integer s, integer e, integer *n)
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{
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float m; integer i, mi;
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for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
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if (w[i-1]>m) mi=i ,m=w[i-1];
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return mi-s+1;
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}
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static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
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integer n = *n_, incx = *incx_, incy = *incy_, i;
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#ifdef _MSC_VER
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_Fcomplex zdotc = {0.0, 0.0};
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
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zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
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zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
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}
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}
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pCf(z) = zdotc;
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}
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#else
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_Complex float zdotc = 0.0;
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
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}
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}
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pCf(z) = zdotc;
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}
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#endif
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static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
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integer n = *n_, incx = *incx_, incy = *incy_, i;
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#ifdef _MSC_VER
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_Dcomplex zdotc = {0.0, 0.0};
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if (incx == 1 && incy == 1) {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
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zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
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zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
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}
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} else {
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for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
|
|
zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
|
|
}
|
|
}
|
|
pCd(z) = zdotc;
|
|
}
|
|
#else
|
|
_Complex double zdotc = 0.0;
|
|
if (incx == 1 && incy == 1) {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
|
|
}
|
|
} else {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
|
|
}
|
|
}
|
|
pCd(z) = zdotc;
|
|
}
|
|
#endif
|
|
static inline void cdotu_(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<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
|
|
zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
|
|
}
|
|
} else {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
|
|
zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
|
|
}
|
|
}
|
|
pCf(z) = zdotc;
|
|
}
|
|
#else
|
|
_Complex float zdotc = 0.0;
|
|
if (incx == 1 && incy == 1) {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += Cf(&x[i]) * Cf(&y[i]);
|
|
}
|
|
} else {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
|
|
}
|
|
}
|
|
pCf(z) = zdotc;
|
|
}
|
|
#endif
|
|
static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
|
|
integer n = *n_, incx = *incx_, incy = *incy_, i;
|
|
#ifdef _MSC_VER
|
|
_Dcomplex zdotc = {0.0, 0.0};
|
|
if (incx == 1 && incy == 1) {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
|
|
zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
|
|
}
|
|
} else {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
|
|
zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
|
|
}
|
|
}
|
|
pCd(z) = zdotc;
|
|
}
|
|
#else
|
|
_Complex double zdotc = 0.0;
|
|
if (incx == 1 && incy == 1) {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += Cd(&x[i]) * Cd(&y[i]);
|
|
}
|
|
} else {
|
|
for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
|
|
zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
|
|
}
|
|
}
|
|
pCd(z) = zdotc;
|
|
}
|
|
#endif
|
|
/* -- translated by f2c (version 20000121).
|
|
You must link the resulting object file with the libraries:
|
|
-lf2c -lm (in that order)
|
|
*/
|
|
|
|
|
|
|
|
|
|
/* > \brief \b SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as nece
|
|
ssary to avoid over-/underflow. */
|
|
|
|
/* =========== DOCUMENTATION =========== */
|
|
|
|
/* Online html documentation available at */
|
|
/* http://www.netlib.org/lapack/explore-html/ */
|
|
|
|
/* > \htmlonly */
|
|
/* > Download SLAG2 + dependencies */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slag2.f
|
|
"> */
|
|
/* > [TGZ]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slag2.f
|
|
"> */
|
|
/* > [ZIP]</a> */
|
|
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slag2.f
|
|
"> */
|
|
/* > [TXT]</a> */
|
|
/* > \endhtmlonly */
|
|
|
|
/* Definition: */
|
|
/* =========== */
|
|
|
|
/* SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, */
|
|
/* WR2, WI ) */
|
|
|
|
/* INTEGER LDA, LDB */
|
|
/* REAL SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 */
|
|
/* REAL A( LDA, * ), B( LDB, * ) */
|
|
|
|
|
|
/* > \par Purpose: */
|
|
/* ============= */
|
|
/* > */
|
|
/* > \verbatim */
|
|
/* > */
|
|
/* > SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue */
|
|
/* > problem A - w B, with scaling as necessary to avoid over-/underflow. */
|
|
/* > */
|
|
/* > The scaling factor "s" results in a modified eigenvalue equation */
|
|
/* > */
|
|
/* > s A - w B */
|
|
/* > */
|
|
/* > where s is a non-negative scaling factor chosen so that w, w B, */
|
|
/* > and s A do not overflow and, if possible, do not underflow, either. */
|
|
/* > \endverbatim */
|
|
|
|
/* Arguments: */
|
|
/* ========== */
|
|
|
|
/* > \param[in] A */
|
|
/* > \verbatim */
|
|
/* > A is REAL array, dimension (LDA, 2) */
|
|
/* > On entry, the 2 x 2 matrix A. It is assumed that its 1-norm */
|
|
/* > is less than 1/SAFMIN. Entries less than */
|
|
/* > sqrt(SAFMIN)*norm(A) are subject to being treated as zero. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDA */
|
|
/* > \verbatim */
|
|
/* > LDA is INTEGER */
|
|
/* > The leading dimension of the array A. LDA >= 2. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] B */
|
|
/* > \verbatim */
|
|
/* > B is REAL array, dimension (LDB, 2) */
|
|
/* > On entry, the 2 x 2 upper triangular matrix B. It is */
|
|
/* > assumed that the one-norm of B is less than 1/SAFMIN. The */
|
|
/* > diagonals should be at least sqrt(SAFMIN) times the largest */
|
|
/* > element of B (in absolute value); if a diagonal is smaller */
|
|
/* > than that, then +/- sqrt(SAFMIN) will be used instead of */
|
|
/* > that diagonal. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] LDB */
|
|
/* > \verbatim */
|
|
/* > LDB is INTEGER */
|
|
/* > The leading dimension of the array B. LDB >= 2. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[in] SAFMIN */
|
|
/* > \verbatim */
|
|
/* > SAFMIN is REAL */
|
|
/* > The smallest positive number s.t. 1/SAFMIN does not */
|
|
/* > overflow. (This should always be SLAMCH('S') -- it is an */
|
|
/* > argument in order to avoid having to call SLAMCH frequently.) */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] SCALE1 */
|
|
/* > \verbatim */
|
|
/* > SCALE1 is REAL */
|
|
/* > A scaling factor used to avoid over-/underflow in the */
|
|
/* > eigenvalue equation which defines the first eigenvalue. If */
|
|
/* > the eigenvalues are complex, then the eigenvalues are */
|
|
/* > ( WR1 +/- WI i ) / SCALE1 (which may lie outside the */
|
|
/* > exponent range of the machine), SCALE1=SCALE2, and SCALE1 */
|
|
/* > will always be positive. If the eigenvalues are real, then */
|
|
/* > the first (real) eigenvalue is WR1 / SCALE1 , but this may */
|
|
/* > overflow or underflow, and in fact, SCALE1 may be zero or */
|
|
/* > less than the underflow threshold if the exact eigenvalue */
|
|
/* > is sufficiently large. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] SCALE2 */
|
|
/* > \verbatim */
|
|
/* > SCALE2 is REAL */
|
|
/* > A scaling factor used to avoid over-/underflow in the */
|
|
/* > eigenvalue equation which defines the second eigenvalue. If */
|
|
/* > the eigenvalues are complex, then SCALE2=SCALE1. If the */
|
|
/* > eigenvalues are real, then the second (real) eigenvalue is */
|
|
/* > WR2 / SCALE2 , but this may overflow or underflow, and in */
|
|
/* > fact, SCALE2 may be zero or less than the underflow */
|
|
/* > threshold if the exact eigenvalue is sufficiently large. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WR1 */
|
|
/* > \verbatim */
|
|
/* > WR1 is REAL */
|
|
/* > If the eigenvalue is real, then WR1 is SCALE1 times the */
|
|
/* > eigenvalue closest to the (2,2) element of A B**(-1). If the */
|
|
/* > eigenvalue is complex, then WR1=WR2 is SCALE1 times the real */
|
|
/* > part of the eigenvalues. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WR2 */
|
|
/* > \verbatim */
|
|
/* > WR2 is REAL */
|
|
/* > If the eigenvalue is real, then WR2 is SCALE2 times the */
|
|
/* > other eigenvalue. If the eigenvalue is complex, then */
|
|
/* > WR1=WR2 is SCALE1 times the real part of the eigenvalues. */
|
|
/* > \endverbatim */
|
|
/* > */
|
|
/* > \param[out] WI */
|
|
/* > \verbatim */
|
|
/* > WI is REAL */
|
|
/* > If the eigenvalue is real, then WI is zero. If the */
|
|
/* > eigenvalue is complex, then WI is SCALE1 times the imaginary */
|
|
/* > part of the eigenvalues. WI will always be non-negative. */
|
|
/* > \endverbatim */
|
|
|
|
/* Authors: */
|
|
/* ======== */
|
|
|
|
/* > \author Univ. of Tennessee */
|
|
/* > \author Univ. of California Berkeley */
|
|
/* > \author Univ. of Colorado Denver */
|
|
/* > \author NAG Ltd. */
|
|
|
|
/* > \date June 2016 */
|
|
|
|
/* > \ingroup realOTHERauxiliary */
|
|
|
|
/* ===================================================================== */
|
|
/* Subroutine */ void slag2_(real *a, integer *lda, real *b, integer *ldb,
|
|
real *safmin, real *scale1, real *scale2, real *wr1, real *wr2, real *
|
|
wi)
|
|
{
|
|
/* System generated locals */
|
|
integer a_dim1, a_offset, b_dim1, b_offset;
|
|
real r__1, r__2, r__3, r__4, r__5, r__6;
|
|
|
|
/* Local variables */
|
|
real diff, bmin, wbig, wabs, wdet, r__, binv11, binv22, discr, anorm,
|
|
bnorm, bsize, shift, c1, c2, c3, c4, c5, rtmin, rtmax, wsize, s1,
|
|
s2, a11, a12, a21, a22, b11, b12, b22, ascale, bscale, pp, qq, ss,
|
|
wscale, safmax, wsmall, as11, as12, as22, sum, abi22;
|
|
|
|
|
|
/* -- LAPACK auxiliary 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..-- */
|
|
/* June 2016 */
|
|
|
|
|
|
/* ===================================================================== */
|
|
|
|
|
|
/* Parameter adjustments */
|
|
a_dim1 = *lda;
|
|
a_offset = 1 + a_dim1 * 1;
|
|
a -= a_offset;
|
|
b_dim1 = *ldb;
|
|
b_offset = 1 + b_dim1 * 1;
|
|
b -= b_offset;
|
|
|
|
/* Function Body */
|
|
rtmin = sqrt(*safmin);
|
|
rtmax = 1.f / rtmin;
|
|
safmax = 1.f / *safmin;
|
|
|
|
/* Scale A */
|
|
|
|
/* Computing MAX */
|
|
r__5 = (r__1 = a[a_dim1 + 1], abs(r__1)) + (r__2 = a[a_dim1 + 2], abs(
|
|
r__2)), r__6 = (r__3 = a[(a_dim1 << 1) + 1], abs(r__3)) + (r__4 =
|
|
a[(a_dim1 << 1) + 2], abs(r__4)), r__5 = f2cmax(r__5,r__6);
|
|
anorm = f2cmax(r__5,*safmin);
|
|
ascale = 1.f / anorm;
|
|
a11 = ascale * a[a_dim1 + 1];
|
|
a21 = ascale * a[a_dim1 + 2];
|
|
a12 = ascale * a[(a_dim1 << 1) + 1];
|
|
a22 = ascale * a[(a_dim1 << 1) + 2];
|
|
|
|
/* Perturb B if necessary to insure non-singularity */
|
|
|
|
b11 = b[b_dim1 + 1];
|
|
b12 = b[(b_dim1 << 1) + 1];
|
|
b22 = b[(b_dim1 << 1) + 2];
|
|
/* Computing MAX */
|
|
r__1 = abs(b11), r__2 = abs(b12), r__1 = f2cmax(r__1,r__2), r__2 = abs(b22),
|
|
r__1 = f2cmax(r__1,r__2);
|
|
bmin = rtmin * f2cmax(r__1,rtmin);
|
|
if (abs(b11) < bmin) {
|
|
b11 = r_sign(&bmin, &b11);
|
|
}
|
|
if (abs(b22) < bmin) {
|
|
b22 = r_sign(&bmin, &b22);
|
|
}
|
|
|
|
/* Scale B */
|
|
|
|
/* Computing MAX */
|
|
r__1 = abs(b11), r__2 = abs(b12) + abs(b22), r__1 = f2cmax(r__1,r__2);
|
|
bnorm = f2cmax(r__1,*safmin);
|
|
/* Computing MAX */
|
|
r__1 = abs(b11), r__2 = abs(b22);
|
|
bsize = f2cmax(r__1,r__2);
|
|
bscale = 1.f / bsize;
|
|
b11 *= bscale;
|
|
b12 *= bscale;
|
|
b22 *= bscale;
|
|
|
|
/* Compute larger eigenvalue by method described by C. van Loan */
|
|
|
|
/* ( AS is A shifted by -SHIFT*B ) */
|
|
|
|
binv11 = 1.f / b11;
|
|
binv22 = 1.f / b22;
|
|
s1 = a11 * binv11;
|
|
s2 = a22 * binv22;
|
|
if (abs(s1) <= abs(s2)) {
|
|
as12 = a12 - s1 * b12;
|
|
as22 = a22 - s1 * b22;
|
|
ss = a21 * (binv11 * binv22);
|
|
abi22 = as22 * binv22 - ss * b12;
|
|
pp = abi22 * .5f;
|
|
shift = s1;
|
|
} else {
|
|
as12 = a12 - s2 * b12;
|
|
as11 = a11 - s2 * b11;
|
|
ss = a21 * (binv11 * binv22);
|
|
abi22 = -ss * b12;
|
|
pp = (as11 * binv11 + abi22) * .5f;
|
|
shift = s2;
|
|
}
|
|
qq = ss * as12;
|
|
if ((r__1 = pp * rtmin, abs(r__1)) >= 1.f) {
|
|
/* Computing 2nd power */
|
|
r__1 = rtmin * pp;
|
|
discr = r__1 * r__1 + qq * *safmin;
|
|
r__ = sqrt((abs(discr))) * rtmax;
|
|
} else {
|
|
/* Computing 2nd power */
|
|
r__1 = pp;
|
|
if (r__1 * r__1 + abs(qq) <= *safmin) {
|
|
/* Computing 2nd power */
|
|
r__1 = rtmax * pp;
|
|
discr = r__1 * r__1 + qq * safmax;
|
|
r__ = sqrt((abs(discr))) * rtmin;
|
|
} else {
|
|
/* Computing 2nd power */
|
|
r__1 = pp;
|
|
discr = r__1 * r__1 + qq;
|
|
r__ = sqrt((abs(discr)));
|
|
}
|
|
}
|
|
|
|
/* Note: the test of R in the following IF is to cover the case when */
|
|
/* DISCR is small and negative and is flushed to zero during */
|
|
/* the calculation of R. On machines which have a consistent */
|
|
/* flush-to-zero threshold and handle numbers above that */
|
|
/* threshold correctly, it would not be necessary. */
|
|
|
|
if (discr >= 0.f || r__ == 0.f) {
|
|
sum = pp + r_sign(&r__, &pp);
|
|
diff = pp - r_sign(&r__, &pp);
|
|
wbig = shift + sum;
|
|
|
|
/* Compute smaller eigenvalue */
|
|
|
|
wsmall = shift + diff;
|
|
/* Computing MAX */
|
|
r__1 = abs(wsmall);
|
|
if (abs(wbig) * .5f > f2cmax(r__1,*safmin)) {
|
|
wdet = (a11 * a22 - a12 * a21) * (binv11 * binv22);
|
|
wsmall = wdet / wbig;
|
|
}
|
|
|
|
/* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1) */
|
|
/* for WR1. */
|
|
|
|
if (pp > abi22) {
|
|
*wr1 = f2cmin(wbig,wsmall);
|
|
*wr2 = f2cmax(wbig,wsmall);
|
|
} else {
|
|
*wr1 = f2cmax(wbig,wsmall);
|
|
*wr2 = f2cmin(wbig,wsmall);
|
|
}
|
|
*wi = 0.f;
|
|
} else {
|
|
|
|
/* Complex eigenvalues */
|
|
|
|
*wr1 = shift + pp;
|
|
*wr2 = *wr1;
|
|
*wi = r__;
|
|
}
|
|
|
|
/* Further scaling to avoid underflow and overflow in computing */
|
|
/* SCALE1 and overflow in computing w*B. */
|
|
|
|
/* This scale factor (WSCALE) is bounded from above using C1 and C2, */
|
|
/* and from below using C3 and C4. */
|
|
/* C1 implements the condition s A must never overflow. */
|
|
/* C2 implements the condition w B must never overflow. */
|
|
/* C3, with C2, */
|
|
/* implement the condition that s A - w B must never overflow. */
|
|
/* C4 implements the condition s should not underflow. */
|
|
/* C5 implements the condition f2cmax(s,|w|) should be at least 2. */
|
|
|
|
c1 = bsize * (*safmin * f2cmax(1.f,ascale));
|
|
c2 = *safmin * f2cmax(1.f,bnorm);
|
|
c3 = bsize * *safmin;
|
|
if (ascale <= 1.f && bsize <= 1.f) {
|
|
/* Computing MIN */
|
|
r__1 = 1.f, r__2 = ascale / *safmin * bsize;
|
|
c4 = f2cmin(r__1,r__2);
|
|
} else {
|
|
c4 = 1.f;
|
|
}
|
|
if (ascale <= 1.f || bsize <= 1.f) {
|
|
/* Computing MIN */
|
|
r__1 = 1.f, r__2 = ascale * bsize;
|
|
c5 = f2cmin(r__1,r__2);
|
|
} else {
|
|
c5 = 1.f;
|
|
}
|
|
|
|
/* Scale first eigenvalue */
|
|
|
|
wabs = abs(*wr1) + abs(*wi);
|
|
/* Computing MAX */
|
|
/* Computing MIN */
|
|
r__3 = c4, r__4 = f2cmax(wabs,c5) * .5f;
|
|
r__1 = f2cmax(*safmin,c1), r__2 = (wabs * c2 + c3) * 1.0000100000000001f,
|
|
r__1 = f2cmax(r__1,r__2), r__2 = f2cmin(r__3,r__4);
|
|
wsize = f2cmax(r__1,r__2);
|
|
if (wsize != 1.f) {
|
|
wscale = 1.f / wsize;
|
|
if (wsize > 1.f) {
|
|
*scale1 = f2cmax(ascale,bsize) * wscale * f2cmin(ascale,bsize);
|
|
} else {
|
|
*scale1 = f2cmin(ascale,bsize) * wscale * f2cmax(ascale,bsize);
|
|
}
|
|
*wr1 *= wscale;
|
|
if (*wi != 0.f) {
|
|
*wi *= wscale;
|
|
*wr2 = *wr1;
|
|
*scale2 = *scale1;
|
|
}
|
|
} else {
|
|
*scale1 = ascale * bsize;
|
|
*scale2 = *scale1;
|
|
}
|
|
|
|
/* Scale second eigenvalue (if real) */
|
|
|
|
if (*wi == 0.f) {
|
|
/* Computing MAX */
|
|
/* Computing MIN */
|
|
/* Computing MAX */
|
|
r__5 = abs(*wr2);
|
|
r__3 = c4, r__4 = f2cmax(r__5,c5) * .5f;
|
|
r__1 = f2cmax(*safmin,c1), r__2 = (abs(*wr2) * c2 + c3) *
|
|
1.0000100000000001f, r__1 = f2cmax(r__1,r__2), r__2 = f2cmin(r__3,
|
|
r__4);
|
|
wsize = f2cmax(r__1,r__2);
|
|
if (wsize != 1.f) {
|
|
wscale = 1.f / wsize;
|
|
if (wsize > 1.f) {
|
|
*scale2 = f2cmax(ascale,bsize) * wscale * f2cmin(ascale,bsize);
|
|
} else {
|
|
*scale2 = f2cmin(ascale,bsize) * wscale * f2cmax(ascale,bsize);
|
|
}
|
|
*wr2 *= wscale;
|
|
} else {
|
|
*scale2 = ascale * bsize;
|
|
}
|
|
}
|
|
|
|
/* End of SLAG2 */
|
|
|
|
return;
|
|
} /* slag2_ */
|
|
|