#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 DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE mat rices */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DGEGV + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, */ /* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) */ /* CHARACTER JOBVL, JOBVR */ /* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N */ /* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), */ /* $ B( LDB, * ), BETA( * ), VL( LDVL, * ), */ /* $ VR( LDVR, * ), WORK( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > This routine is deprecated and has been replaced by routine DGGEV. */ /* > */ /* > DGEGV computes the eigenvalues and, optionally, the left and/or right */ /* > eigenvectors of a real matrix pair (A,B). */ /* > Given two square matrices A and B, */ /* > the generalized nonsymmetric eigenvalue problem (GNEP) is to find the */ /* > eigenvalues lambda and corresponding (non-zero) eigenvectors x such */ /* > that */ /* > */ /* > A*x = lambda*B*x. */ /* > */ /* > An alternate form is to find the eigenvalues mu and corresponding */ /* > eigenvectors y such that */ /* > */ /* > mu*A*y = B*y. */ /* > */ /* > These two forms are equivalent with mu = 1/lambda and x = y if */ /* > neither lambda nor mu is zero. In order to deal with the case that */ /* > lambda or mu is zero or small, two values alpha and beta are returned */ /* > for each eigenvalue, such that lambda = alpha/beta and */ /* > mu = beta/alpha. */ /* > */ /* > The vectors x and y in the above equations are right eigenvectors of */ /* > the matrix pair (A,B). Vectors u and v satisfying */ /* > */ /* > u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B */ /* > */ /* > are left eigenvectors of (A,B). */ /* > */ /* > Note: this routine performs "full balancing" on A and B */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] JOBVL */ /* > \verbatim */ /* > JOBVL is CHARACTER*1 */ /* > = 'N': do not compute the left generalized eigenvectors; */ /* > = 'V': compute the left generalized eigenvectors (returned */ /* > in VL). */ /* > \endverbatim */ /* > */ /* > \param[in] JOBVR */ /* > \verbatim */ /* > JOBVR is CHARACTER*1 */ /* > = 'N': do not compute the right generalized eigenvectors; */ /* > = 'V': compute the right generalized eigenvectors (returned */ /* > in VR). */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrices A, B, VL, and VR. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is DOUBLE PRECISION array, dimension (LDA, N) */ /* > On entry, the matrix A. */ /* > If JOBVL = 'V' or JOBVR = 'V', then on exit A */ /* > contains the real Schur form of A from the generalized Schur */ /* > factorization of the pair (A,B) after balancing. */ /* > If no eigenvectors were computed, then only the diagonal */ /* > blocks from the Schur form will be correct. See DGGHRD and */ /* > DHGEQZ for details. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of A. LDA >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in,out] B */ /* > \verbatim */ /* > B is DOUBLE PRECISION array, dimension (LDB, N) */ /* > On entry, the matrix B. */ /* > If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the */ /* > upper triangular matrix obtained from B in the generalized */ /* > Schur factorization of the pair (A,B) after balancing. */ /* > If no eigenvectors were computed, then only those elements of */ /* > B corresponding to the diagonal blocks from the Schur form of */ /* > A will be correct. See DGGHRD and DHGEQZ for details. */ /* > \endverbatim */ /* > */ /* > \param[in] LDB */ /* > \verbatim */ /* > LDB is INTEGER */ /* > The leading dimension of B. LDB >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAR */ /* > \verbatim */ /* > ALPHAR is DOUBLE PRECISION array, dimension (N) */ /* > The real parts of each scalar alpha defining an eigenvalue of */ /* > GNEP. */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAI */ /* > \verbatim */ /* > ALPHAI is DOUBLE PRECISION array, dimension (N) */ /* > The imaginary parts of each scalar alpha defining an */ /* > eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th */ /* > eigenvalue is real; if positive, then the j-th and */ /* > (j+1)-st eigenvalues are a complex conjugate pair, with */ /* > ALPHAI(j+1) = -ALPHAI(j). */ /* > \endverbatim */ /* > */ /* > \param[out] BETA */ /* > \verbatim */ /* > BETA is DOUBLE PRECISION array, dimension (N) */ /* > The scalars beta that define the eigenvalues of GNEP. */ /* > */ /* > Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and */ /* > beta = BETA(j) represent the j-th eigenvalue of the matrix */ /* > pair (A,B), in one of the forms lambda = alpha/beta or */ /* > mu = beta/alpha. Since either lambda or mu may overflow, */ /* > they should not, in general, be computed. */ /* > \endverbatim */ /* > */ /* > \param[out] VL */ /* > \verbatim */ /* > VL is DOUBLE PRECISION array, dimension (LDVL,N) */ /* > If JOBVL = 'V', the left eigenvectors u(j) are stored */ /* > in the columns of VL, in the same order as their eigenvalues. */ /* > If the j-th eigenvalue is real, then u(j) = VL(:,j). */ /* > If the j-th and (j+1)-st eigenvalues form a complex conjugate */ /* > pair, then */ /* > u(j) = VL(:,j) + i*VL(:,j+1) */ /* > and */ /* > u(j+1) = VL(:,j) - i*VL(:,j+1). */ /* > */ /* > Each eigenvector is scaled so that its largest component has */ /* > abs(real part) + abs(imag. part) = 1, except for eigenvectors */ /* > corresponding to an eigenvalue with alpha = beta = 0, which */ /* > are set to zero. */ /* > Not referenced if JOBVL = 'N'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDVL */ /* > \verbatim */ /* > LDVL is INTEGER */ /* > The leading dimension of the matrix VL. LDVL >= 1, and */ /* > if JOBVL = 'V', LDVL >= N. */ /* > \endverbatim */ /* > */ /* > \param[out] VR */ /* > \verbatim */ /* > VR is DOUBLE PRECISION array, dimension (LDVR,N) */ /* > If JOBVR = 'V', the right eigenvectors x(j) are stored */ /* > in the columns of VR, in the same order as their eigenvalues. */ /* > If the j-th eigenvalue is real, then x(j) = VR(:,j). */ /* > If the j-th and (j+1)-st eigenvalues form a complex conjugate */ /* > pair, then */ /* > x(j) = VR(:,j) + i*VR(:,j+1) */ /* > and */ /* > x(j+1) = VR(:,j) - i*VR(:,j+1). */ /* > */ /* > Each eigenvector is scaled so that its largest component has */ /* > abs(real part) + abs(imag. part) = 1, except for eigenvalues */ /* > corresponding to an eigenvalue with alpha = beta = 0, which */ /* > are set to zero. */ /* > Not referenced if JOBVR = 'N'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDVR */ /* > \verbatim */ /* > LDVR is INTEGER */ /* > The leading dimension of the matrix VR. LDVR >= 1, and */ /* > if JOBVR = 'V', LDVR >= N. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */ /* > On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > The dimension of the array WORK. LWORK >= f2cmax(1,8*N). */ /* > For good performance, LWORK must generally be larger. */ /* > To compute the optimal value of LWORK, call ILAENV to get */ /* > blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: */ /* > NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; */ /* > The optimal LWORK is: */ /* > 2*N + MAX( 6*N, N*(NB+1) ). */ /* > */ /* > If LWORK = -1, then a workspace query is assumed; the routine */ /* > only calculates the optimal size of the WORK array, returns */ /* > this value as the first entry of the WORK array, and no error */ /* > message related to LWORK is issued by XERBLA. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value. */ /* > = 1,...,N: */ /* > The QZ iteration failed. No eigenvectors have been */ /* > calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) */ /* > should be correct for j=INFO+1,...,N. */ /* > > N: errors that usually indicate LAPACK problems: */ /* > =N+1: error return from DGGBAL */ /* > =N+2: error return from DGEQRF */ /* > =N+3: error return from DORMQR */ /* > =N+4: error return from DORGQR */ /* > =N+5: error return from DGGHRD */ /* > =N+6: error return from DHGEQZ (other than failed */ /* > iteration) */ /* > =N+7: error return from DTGEVC */ /* > =N+8: error return from DGGBAK (computing VL) */ /* > =N+9: error return from DGGBAK (computing VR) */ /* > =N+10: error return from DLASCL (various calls) */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup doubleGEeigen */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > Balancing */ /* > --------- */ /* > */ /* > This driver calls DGGBAL to both permute and scale rows and columns */ /* > of A and B. The permutations PL and PR are chosen so that PL*A*PR */ /* > and PL*B*R will be upper triangular except for the diagonal blocks */ /* > A(i:j,i:j) and B(i:j,i:j), with i and j as close together as */ /* > possible. The diagonal scaling matrices DL and DR are chosen so */ /* > that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to */ /* > one (except for the elements that start out zero.) */ /* > */ /* > After the eigenvalues and eigenvectors of the balanced matrices */ /* > have been computed, DGGBAK transforms the eigenvectors back to what */ /* > they would have been (in perfect arithmetic) if they had not been */ /* > balanced. */ /* > */ /* > Contents of A and B on Exit */ /* > -------- -- - --- - -- ---- */ /* > */ /* > If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or */ /* > both), then on exit the arrays A and B will contain the real Schur */ /* > form[*] of the "balanced" versions of A and B. If no eigenvectors */ /* > are computed, then only the diagonal blocks will be correct. */ /* > */ /* > [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", */ /* > by Golub & van Loan, pub. by Johns Hopkins U. Press. */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int dgegv_(char *jobvl, char *jobvr, integer *n, doublereal * a, integer *lda, doublereal *b, integer *ldb, doublereal *alphar, doublereal *alphai, doublereal *beta, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, doublereal *work, integer *lwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2; doublereal d__1, d__2, d__3, d__4; /* Local variables */ doublereal absb, anrm, bnrm; integer itau; doublereal temp; logical ilvl, ilvr; integer lopt; doublereal anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta; extern logical lsame_(char *, char *); integer ileft, iinfo, icols, iwork, irows, jc; extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); integer nb; extern /* Subroutine */ int dggbal_(char *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *); integer in; extern doublereal dlamch_(char *), dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); integer jr; doublereal salfai; extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); doublereal salfar; extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); doublereal safmin; extern /* Subroutine */ int dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *); doublereal safmax; char chtemp[1]; logical ldumma[1]; extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dtgevc_(char *, char *, logical *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *), xerbla_(char *, integer *); integer ijobvl, iright; logical ilimit; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); integer ijobvr; extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *); doublereal onepls; integer lwkmin, nb1, nb2, nb3; extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); integer lwkopt; logical lquery; integer ihi, ilo; doublereal eps; logical ilv; /* -- LAPACK driver routine (version 3.7.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* December 2016 */ /* ===================================================================== */ /* Decode the input arguments */ /* 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; --alphar; --alphai; --beta; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1 * 1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1 * 1; vr -= vr_offset; --work; /* Function Body */ if (lsame_(jobvl, "N")) { ijobvl = 1; ilvl = FALSE_; } else if (lsame_(jobvl, "V")) { ijobvl = 2; ilvl = TRUE_; } else { ijobvl = -1; ilvl = FALSE_; } if (lsame_(jobvr, "N")) { ijobvr = 1; ilvr = FALSE_; } else if (lsame_(jobvr, "V")) { ijobvr = 2; ilvr = TRUE_; } else { ijobvr = -1; ilvr = FALSE_; } ilv = ilvl || ilvr; /* Test the input arguments */ /* Computing MAX */ i__1 = *n << 3; lwkmin = f2cmax(i__1,1); lwkopt = lwkmin; work[1] = (doublereal) lwkopt; lquery = *lwork == -1; *info = 0; if (ijobvl <= 0) { *info = -1; } else if (ijobvr <= 0) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*lda < f2cmax(1,*n)) { *info = -5; } else if (*ldb < f2cmax(1,*n)) { *info = -7; } else if (*ldvl < 1 || ilvl && *ldvl < *n) { *info = -12; } else if (*ldvr < 1 || ilvr && *ldvr < *n) { *info = -14; } else if (*lwork < lwkmin && ! lquery) { *info = -16; } if (*info == 0) { nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1, (ftnlen)6, ( ftnlen)1); /* Computing MAX */ i__1 = f2cmax(nb1,nb2); nb = f2cmax(i__1,nb3); /* Computing MAX */ i__1 = *n * 6, i__2 = *n * (nb + 1); lopt = (*n << 1) + f2cmax(i__1,i__2); work[1] = (doublereal) lopt; } if (*info != 0) { i__1 = -(*info); xerbla_("DGEGV ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = dlamch_("E") * dlamch_("B"); safmin = dlamch_("S"); safmin += safmin; safmax = 1. / safmin; onepls = eps * 4 + 1.; /* Scale A */ anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]); anrm1 = anrm; anrm2 = 1.; if (anrm < 1.) { if (safmax * anrm < 1.) { anrm1 = safmin; anrm2 = safmax * anrm; } } if (anrm > 0.) { dlascl_("G", &c_n1, &c_n1, &anrm, &c_b27, n, n, &a[a_offset], lda, & iinfo); if (iinfo != 0) { *info = *n + 10; return 0; } } /* Scale B */ bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]); bnrm1 = bnrm; bnrm2 = 1.; if (bnrm < 1.) { if (safmax * bnrm < 1.) { bnrm1 = safmin; bnrm2 = safmax * bnrm; } } if (bnrm > 0.) { dlascl_("G", &c_n1, &c_n1, &bnrm, &c_b27, n, n, &b[b_offset], ldb, & iinfo); if (iinfo != 0) { *info = *n + 10; return 0; } } /* Permute the matrix to make it more nearly triangular */ /* Workspace layout: (8*N words -- "work" requires 6*N words) */ /* left_permutation, right_permutation, work... */ ileft = 1; iright = *n + 1; iwork = iright + *n; dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[ ileft], &work[iright], &work[iwork], &iinfo); if (iinfo != 0) { *info = *n + 1; goto L120; } /* Reduce B to triangular form, and initialize VL and/or VR */ /* Workspace layout: ("work..." must have at least N words) */ /* left_permutation, right_permutation, tau, work... */ irows = ihi + 1 - ilo; if (ilv) { icols = *n + 1 - ilo; } else { icols = irows; } itau = iwork; iwork = itau + irows; i__1 = *lwork + 1 - iwork; dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[ iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = f2cmax(i__1,i__2); } if (iinfo != 0) { *info = *n + 2; goto L120; } i__1 = *lwork + 1 - iwork; dormqr_("L", "T", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, & work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwork], &i__1, & iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = f2cmax(i__1,i__2); } if (iinfo != 0) { *info = *n + 3; goto L120; } if (ilvl) { dlaset_("Full", n, n, &c_b38, &c_b27, &vl[vl_offset], ldvl) ; i__1 = irows - 1; i__2 = irows - 1; dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo + 1 + ilo * vl_dim1], ldvl); i__1 = *lwork + 1 - iwork; dorgqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[ itau], &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = f2cmax(i__1,i__2); } if (iinfo != 0) { *info = *n + 4; goto L120; } } if (ilvr) { dlaset_("Full", n, n, &c_b38, &c_b27, &vr[vr_offset], ldvr) ; } /* Reduce to generalized Hessenberg form */ if (ilv) { /* Eigenvectors requested -- work on whole matrix. */ dgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo); } else { dgghrd_("N", "N", &irows, &c__1, &irows, &a[ilo + ilo * a_dim1], lda, &b[ilo + ilo * b_dim1], ldb, &vl[vl_offset], ldvl, &vr[ vr_offset], ldvr, &iinfo); } if (iinfo != 0) { *info = *n + 5; goto L120; } /* Perform QZ algorithm */ /* Workspace layout: ("work..." must have at least 1 word) */ /* left_permutation, right_permutation, work... */ iwork = itau; if (ilv) { *(unsigned char *)chtemp = 'S'; } else { *(unsigned char *)chtemp = 'E'; } i__1 = *lwork + 1 - iwork; dhgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[ b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &work[iwork], &i__1, &iinfo); if (iinfo >= 0) { /* Computing MAX */ i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1; lwkopt = f2cmax(i__1,i__2); } if (iinfo != 0) { if (iinfo > 0 && iinfo <= *n) { *info = iinfo; } else if (iinfo > *n && iinfo <= *n << 1) { *info = iinfo - *n; } else { *info = *n + 6; } goto L120; } if (ilv) { /* Compute Eigenvectors (DTGEVC requires 6*N words of workspace) */ if (ilvl) { if (ilvr) { *(unsigned char *)chtemp = 'B'; } else { *(unsigned char *)chtemp = 'L'; } } else { *(unsigned char *)chtemp = 'R'; } dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[ iwork], &iinfo); if (iinfo != 0) { *info = *n + 7; goto L120; } /* Undo balancing on VL and VR, rescale */ if (ilvl) { dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, & vl[vl_offset], ldvl, &iinfo); if (iinfo != 0) { *info = *n + 8; goto L120; } i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { if (alphai[jc] < 0.) { goto L50; } temp = 0.; if (alphai[jc] == 0.) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ d__2 = temp, d__3 = (d__1 = vl[jr + jc * vl_dim1], abs(d__1)); temp = f2cmax(d__2,d__3); /* L10: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ d__3 = temp, d__4 = (d__1 = vl[jr + jc * vl_dim1], abs(d__1)) + (d__2 = vl[jr + (jc + 1) * vl_dim1], abs(d__2)); temp = f2cmax(d__3,d__4); /* L20: */ } } if (temp < safmin) { goto L50; } temp = 1. / temp; if (alphai[jc] == 0.) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl[jr + jc * vl_dim1] *= temp; /* L30: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl[jr + jc * vl_dim1] *= temp; vl[jr + (jc + 1) * vl_dim1] *= temp; /* L40: */ } } L50: ; } } if (ilvr) { dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, & vr[vr_offset], ldvr, &iinfo); if (iinfo != 0) { *info = *n + 9; goto L120; } i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { if (alphai[jc] < 0.) { goto L100; } temp = 0.; if (alphai[jc] == 0.) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ d__2 = temp, d__3 = (d__1 = vr[jr + jc * vr_dim1], abs(d__1)); temp = f2cmax(d__2,d__3); /* L60: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { /* Computing MAX */ d__3 = temp, d__4 = (d__1 = vr[jr + jc * vr_dim1], abs(d__1)) + (d__2 = vr[jr + (jc + 1) * vr_dim1], abs(d__2)); temp = f2cmax(d__3,d__4); /* L70: */ } } if (temp < safmin) { goto L100; } temp = 1. / temp; if (alphai[jc] == 0.) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr[jr + jc * vr_dim1] *= temp; /* L80: */ } } else { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr[jr + jc * vr_dim1] *= temp; vr[jr + (jc + 1) * vr_dim1] *= temp; /* L90: */ } } L100: ; } } /* End of eigenvector calculation */ } /* Undo scaling in alpha, beta */ /* Note: this does not give the alpha and beta for the unscaled */ /* problem. */ /* Un-scaling is limited to avoid underflow in alpha and beta */ /* if they are significant. */ i__1 = *n; for (jc = 1; jc <= i__1; ++jc) { absar = (d__1 = alphar[jc], abs(d__1)); absai = (d__1 = alphai[jc], abs(d__1)); absb = (d__1 = beta[jc], abs(d__1)); salfar = anrm * alphar[jc]; salfai = anrm * alphai[jc]; sbeta = bnrm * beta[jc]; ilimit = FALSE_; scale = 1.; /* Check for significant underflow in ALPHAI */ /* Computing MAX */ d__1 = safmin, d__2 = eps * absar, d__1 = f2cmax(d__1,d__2), d__2 = eps * absb; if (abs(salfai) < safmin && absai >= f2cmax(d__1,d__2)) { ilimit = TRUE_; /* Computing MAX */ d__1 = onepls * safmin, d__2 = anrm2 * absai; scale = onepls * safmin / anrm1 / f2cmax(d__1,d__2); } else if (salfai == 0.) { /* If insignificant underflow in ALPHAI, then make the */ /* conjugate eigenvalue real. */ if (alphai[jc] < 0. && jc > 1) { alphai[jc - 1] = 0.; } else if (alphai[jc] > 0. && jc < *n) { alphai[jc + 1] = 0.; } } /* Check for significant underflow in ALPHAR */ /* Computing MAX */ d__1 = safmin, d__2 = eps * absai, d__1 = f2cmax(d__1,d__2), d__2 = eps * absb; if (abs(salfar) < safmin && absar >= f2cmax(d__1,d__2)) { ilimit = TRUE_; /* Computing MAX */ /* Computing MAX */ d__3 = onepls * safmin, d__4 = anrm2 * absar; d__1 = scale, d__2 = onepls * safmin / anrm1 / f2cmax(d__3,d__4); scale = f2cmax(d__1,d__2); } /* Check for significant underflow in BETA */ /* Computing MAX */ d__1 = safmin, d__2 = eps * absar, d__1 = f2cmax(d__1,d__2), d__2 = eps * absai; if (abs(sbeta) < safmin && absb >= f2cmax(d__1,d__2)) { ilimit = TRUE_; /* Computing MAX */ /* Computing MAX */ d__3 = onepls * safmin, d__4 = bnrm2 * absb; d__1 = scale, d__2 = onepls * safmin / bnrm1 / f2cmax(d__3,d__4); scale = f2cmax(d__1,d__2); } /* Check for possible overflow when limiting scaling */ if (ilimit) { /* Computing MAX */ d__1 = abs(salfar), d__2 = abs(salfai), d__1 = f2cmax(d__1,d__2), d__2 = abs(sbeta); temp = scale * safmin * f2cmax(d__1,d__2); if (temp > 1.) { scale /= temp; } if (scale < 1.) { ilimit = FALSE_; } } /* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary. */ if (ilimit) { salfar = scale * alphar[jc] * anrm; salfai = scale * alphai[jc] * anrm; sbeta = scale * beta[jc] * bnrm; } alphar[jc] = salfar; alphai[jc] = salfai; beta[jc] = sbeta; /* L110: */ } L120: work[1] = (doublereal) lwkopt; return 0; /* End of DGEGV */ } /* dgegv_ */