#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 DTGSEN */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DTGSEN + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, */ /* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, */ /* PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO ) */ /* LOGICAL WANTQ, WANTZ */ /* INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, */ /* $ M, N */ /* DOUBLE PRECISION PL, PR */ /* LOGICAL SELECT( * ) */ /* INTEGER IWORK( * ) */ /* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), */ /* $ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), */ /* $ WORK( * ), Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DTGSEN reorders the generalized real Schur decomposition of a real */ /* > matrix pair (A, B) (in terms of an orthonormal equivalence trans- */ /* > formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues */ /* > appears in the leading diagonal blocks of the upper quasi-triangular */ /* > matrix A and the upper triangular B. The leading columns of Q and */ /* > Z form orthonormal bases of the corresponding left and right eigen- */ /* > spaces (deflating subspaces). (A, B) must be in generalized real */ /* > Schur canonical form (as returned by DGGES), i.e. A is block upper */ /* > triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper */ /* > triangular. */ /* > */ /* > DTGSEN also computes the generalized eigenvalues */ /* > */ /* > w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) */ /* > */ /* > of the reordered matrix pair (A, B). */ /* > */ /* > Optionally, DTGSEN computes the estimates of reciprocal condition */ /* > numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */ /* > (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */ /* > between the matrix pairs (A11, B11) and (A22,B22) that correspond to */ /* > the selected cluster and the eigenvalues outside the cluster, resp., */ /* > and norms of "projections" onto left and right eigenspaces w.r.t. */ /* > the selected cluster in the (1,1)-block. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] IJOB */ /* > \verbatim */ /* > IJOB is INTEGER */ /* > Specifies whether condition numbers are required for the */ /* > cluster of eigenvalues (PL and PR) or the deflating subspaces */ /* > (Difu and Difl): */ /* > =0: Only reorder w.r.t. SELECT. No extras. */ /* > =1: Reciprocal of norms of "projections" onto left and right */ /* > eigenspaces w.r.t. the selected cluster (PL and PR). */ /* > =2: Upper bounds on Difu and Difl. F-norm-based estimate */ /* > (DIF(1:2)). */ /* > =3: Estimate of Difu and Difl. 1-norm-based estimate */ /* > (DIF(1:2)). */ /* > About 5 times as expensive as IJOB = 2. */ /* > =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */ /* > version to get it all. */ /* > =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */ /* > \endverbatim */ /* > */ /* > \param[in] WANTQ */ /* > \verbatim */ /* > WANTQ is LOGICAL */ /* > .TRUE. : update the left transformation matrix Q; */ /* > .FALSE.: do not update Q. */ /* > \endverbatim */ /* > */ /* > \param[in] WANTZ */ /* > \verbatim */ /* > WANTZ is LOGICAL */ /* > .TRUE. : update the right transformation matrix Z; */ /* > .FALSE.: do not update Z. */ /* > \endverbatim */ /* > */ /* > \param[in] SELECT */ /* > \verbatim */ /* > SELECT is LOGICAL array, dimension (N) */ /* > SELECT specifies the eigenvalues in the selected cluster. */ /* > To select a real eigenvalue w(j), SELECT(j) must be set to */ /* > .TRUE.. To select a complex conjugate pair of eigenvalues */ /* > w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, */ /* > either SELECT(j) or SELECT(j+1) or both must be set to */ /* > .TRUE.; a complex conjugate pair of eigenvalues must be */ /* > either both included in the cluster or both excluded. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrices A and B. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is DOUBLE PRECISION array, dimension(LDA,N) */ /* > On entry, the upper quasi-triangular matrix A, with (A, B) in */ /* > generalized real Schur canonical form. */ /* > On exit, A is overwritten by the reordered matrix A. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in,out] B */ /* > \verbatim */ /* > B is DOUBLE PRECISION array, dimension(LDB,N) */ /* > On entry, the upper triangular matrix B, with (A, B) in */ /* > generalized real Schur canonical form. */ /* > On exit, B is overwritten by the reordered matrix B. */ /* > \endverbatim */ /* > */ /* > \param[in] LDB */ /* > \verbatim */ /* > LDB is INTEGER */ /* > The leading dimension of the array B. LDB >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAR */ /* > \verbatim */ /* > ALPHAR is DOUBLE PRECISION array, dimension (N) */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAI */ /* > \verbatim */ /* > ALPHAI is DOUBLE PRECISION array, dimension (N) */ /* > \endverbatim */ /* > */ /* > \param[out] BETA */ /* > \verbatim */ /* > BETA is DOUBLE PRECISION array, dimension (N) */ /* > */ /* > On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will */ /* > be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i */ /* > and BETA(j),j=1,...,N are the diagonals of the complex Schur */ /* > form (S,T) that would result if the 2-by-2 diagonal blocks of */ /* > the real generalized Schur form of (A,B) were further reduced */ /* > to triangular form using complex unitary transformations. */ /* > 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) negative. */ /* > \endverbatim */ /* > */ /* > \param[in,out] Q */ /* > \verbatim */ /* > Q is DOUBLE PRECISION array, dimension (LDQ,N) */ /* > On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */ /* > On exit, Q has been postmultiplied by the left orthogonal */ /* > transformation matrix which reorder (A, B); The leading M */ /* > columns of Q form orthonormal bases for the specified pair of */ /* > left eigenspaces (deflating subspaces). */ /* > If WANTQ = .FALSE., Q is not referenced. */ /* > \endverbatim */ /* > */ /* > \param[in] LDQ */ /* > \verbatim */ /* > LDQ is INTEGER */ /* > The leading dimension of the array Q. LDQ >= 1; */ /* > and if WANTQ = .TRUE., LDQ >= N. */ /* > \endverbatim */ /* > */ /* > \param[in,out] Z */ /* > \verbatim */ /* > Z is DOUBLE PRECISION array, dimension (LDZ,N) */ /* > On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */ /* > On exit, Z has been postmultiplied by the left orthogonal */ /* > transformation matrix which reorder (A, B); The leading M */ /* > columns of Z form orthonormal bases for the specified pair of */ /* > left eigenspaces (deflating subspaces). */ /* > If WANTZ = .FALSE., Z is not referenced. */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. LDZ >= 1; */ /* > If WANTZ = .TRUE., LDZ >= N. */ /* > \endverbatim */ /* > */ /* > \param[out] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The dimension of the specified pair of left and right eigen- */ /* > spaces (deflating subspaces). 0 <= M <= N. */ /* > \endverbatim */ /* > */ /* > \param[out] PL */ /* > \verbatim */ /* > PL is DOUBLE PRECISION */ /* > \endverbatim */ /* > */ /* > \param[out] PR */ /* > \verbatim */ /* > PR is DOUBLE PRECISION */ /* > */ /* > If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */ /* > reciprocal of the norm of "projections" onto left and right */ /* > eigenspaces with respect to the selected cluster. */ /* > 0 < PL, PR <= 1. */ /* > If M = 0 or M = N, PL = PR = 1. */ /* > If IJOB = 0, 2 or 3, PL and PR are not referenced. */ /* > \endverbatim */ /* > */ /* > \param[out] DIF */ /* > \verbatim */ /* > DIF is DOUBLE PRECISION array, dimension (2). */ /* > If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */ /* > If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */ /* > Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */ /* > estimates of Difu and Difl. */ /* > If M = 0 or N, DIF(1:2) = F-norm([A, B]). */ /* > If IJOB = 0 or 1, DIF is not referenced. */ /* > \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 >= 4*N+16. */ /* > If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). */ /* > If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). */ /* > */ /* > 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] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (MAX(1,LIWORK)) */ /* > On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */ /* > \endverbatim */ /* > */ /* > \param[in] LIWORK */ /* > \verbatim */ /* > LIWORK is INTEGER */ /* > The dimension of the array IWORK. LIWORK >= 1. */ /* > If IJOB = 1, 2 or 4, LIWORK >= N+6. */ /* > If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). */ /* > */ /* > If LIWORK = -1, then a workspace query is assumed; the */ /* > routine only calculates the optimal size of the IWORK array, */ /* > returns this value as the first entry of the IWORK array, and */ /* > no error message related to LIWORK 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: Reordering of (A, B) failed because the transformed */ /* > matrix pair (A, B) would be too far from generalized */ /* > Schur form; the problem is very ill-conditioned. */ /* > (A, B) may have been partially reordered. */ /* > If requested, 0 is returned in DIF(*), PL and PR. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2016 */ /* > \ingroup doubleOTHERcomputational */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > DTGSEN first collects the selected eigenvalues by computing */ /* > orthogonal U and W that move them to the top left corner of (A, B). */ /* > In other words, the selected eigenvalues are the eigenvalues of */ /* > (A11, B11) in: */ /* > */ /* > U**T*(A, B)*W = (A11 A12) (B11 B12) n1 */ /* > ( 0 A22),( 0 B22) n2 */ /* > n1 n2 n1 n2 */ /* > */ /* > where N = n1+n2 and U**T means the transpose of U. The first n1 columns */ /* > of U and W span the specified pair of left and right eigenspaces */ /* > (deflating subspaces) of (A, B). */ /* > */ /* > If (A, B) has been obtained from the generalized real Schur */ /* > decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the */ /* > reordered generalized real Schur form of (C, D) is given by */ /* > */ /* > (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T, */ /* > */ /* > and the first n1 columns of Q*U and Z*W span the corresponding */ /* > deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */ /* > */ /* > Note that if the selected eigenvalue is sufficiently ill-conditioned, */ /* > then its value may differ significantly from its value before */ /* > reordering. */ /* > */ /* > The reciprocal condition numbers of the left and right eigenspaces */ /* > spanned by the first n1 columns of U and W (or Q*U and Z*W) may */ /* > be returned in DIF(1:2), corresponding to Difu and Difl, resp. */ /* > */ /* > The Difu and Difl are defined as: */ /* > */ /* > Difu[(A11, B11), (A22, B22)] = sigma-f2cmin( Zu ) */ /* > and */ /* > Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */ /* > */ /* > where sigma-f2cmin(Zu) is the smallest singular value of the */ /* > (2*n1*n2)-by-(2*n1*n2) matrix */ /* > */ /* > Zu = [ kron(In2, A11) -kron(A22**T, In1) ] */ /* > [ kron(In2, B11) -kron(B22**T, In1) ]. */ /* > */ /* > Here, Inx is the identity matrix of size nx and A22**T is the */ /* > transpose of A22. kron(X, Y) is the Kronecker product between */ /* > the matrices X and Y. */ /* > */ /* > When DIF(2) is small, small changes in (A, B) can cause large changes */ /* > in the deflating subspace. An approximate (asymptotic) bound on the */ /* > maximum angular error in the computed deflating subspaces is */ /* > */ /* > EPS * norm((A, B)) / DIF(2), */ /* > */ /* > where EPS is the machine precision. */ /* > */ /* > The reciprocal norm of the projectors on the left and right */ /* > eigenspaces associated with (A11, B11) may be returned in PL and PR. */ /* > They are computed as follows. First we compute L and R so that */ /* > P*(A, B)*Q is block diagonal, where */ /* > */ /* > P = ( I -L ) n1 Q = ( I R ) n1 */ /* > ( 0 I ) n2 and ( 0 I ) n2 */ /* > n1 n2 n1 n2 */ /* > */ /* > and (L, R) is the solution to the generalized Sylvester equation */ /* > */ /* > A11*R - L*A22 = -A12 */ /* > B11*R - L*B22 = -B12 */ /* > */ /* > Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */ /* > An approximate (asymptotic) bound on the average absolute error of */ /* > the selected eigenvalues is */ /* > */ /* > EPS * norm((A, B)) / PL. */ /* > */ /* > There are also global error bounds which valid for perturbations up */ /* > to a certain restriction: A lower bound (x) on the smallest */ /* > F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */ /* > coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */ /* > (i.e. (A + E, B + F), is */ /* > */ /* > x = f2cmin(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*f2cmax(1/PL,1/PR)). */ /* > */ /* > An approximate bound on x can be computed from DIF(1:2), PL and PR. */ /* > */ /* > If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */ /* > (L', R') and unperturbed (L, R) left and right deflating subspaces */ /* > associated with the selected cluster in the (1,1)-blocks can be */ /* > bounded as */ /* > */ /* > f2cmax-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */ /* > f2cmax-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */ /* > */ /* > See LAPACK User's Guide section 4.11 or the following references */ /* > for more information. */ /* > */ /* > Note that if the default method for computing the Frobenius-norm- */ /* > based estimate DIF is not wanted (see DLATDF), then the parameter */ /* > IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF */ /* > (IJOB = 2 will be used)). See DTGSYL for more details. */ /* > \endverbatim */ /* > \par Contributors: */ /* ================== */ /* > */ /* > Bo Kagstrom and Peter Poromaa, Department of Computing Science, */ /* > Umea University, S-901 87 Umea, Sweden. */ /* > \par References: */ /* ================ */ /* > */ /* > \verbatim */ /* > */ /* > [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */ /* > Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */ /* > M.S. Moonen et al (eds), Linear Algebra for Large Scale and */ /* > Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */ /* > */ /* > [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */ /* > Eigenvalues of a Regular Matrix Pair (A, B) and Condition */ /* > Estimation: Theory, Algorithms and Software, */ /* > Report UMINF - 94.04, Department of Computing Science, Umea */ /* > University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */ /* > Note 87. To appear in Numerical Algorithms, 1996. */ /* > */ /* > [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */ /* > for Solving the Generalized Sylvester Equation and Estimating the */ /* > Separation between Regular Matrix Pairs, Report UMINF - 93.23, */ /* > Department of Computing Science, Umea University, S-901 87 Umea, */ /* > Sweden, December 1993, Revised April 1994, Also as LAPACK Working */ /* > Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */ /* > 1996. */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int dtgsen_(integer *ijob, logical *wantq, logical *wantz, logical *select, integer *n, doublereal *a, integer *lda, doublereal * b, integer *ldb, doublereal *alphar, doublereal *alphai, doublereal * beta, doublereal *q, integer *ldq, doublereal *z__, integer *ldz, integer *m, doublereal *pl, doublereal *pr, doublereal *dif, doublereal *work, integer *lwork, integer *iwork, integer *liwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1, z_offset, i__1, i__2; doublereal d__1; /* Local variables */ integer kase; logical pair; integer ierr; doublereal dsum; logical swap; extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); integer i__, k, isave[3]; logical wantd; integer lwmin; logical wantp; integer n1, n2; extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); logical wantd1, wantd2; integer kk; extern doublereal dlamch_(char *); doublereal dscale; integer ks; doublereal rdscal; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *, ftnlen), dtgexc_(logical *, logical *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *, integer *), dlassq_(integer *, doublereal *, integer *, doublereal *, doublereal *); integer liwmin; extern /* Subroutine */ int dtgsyl_(char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *); doublereal smlnum; integer mn2; logical lquery; integer ijb; doublereal eps; /* -- LAPACK computational routine (version 3.7.1) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* June 2016 */ /* ===================================================================== */ /* Decode and test the input parameters */ /* Parameter adjustments */ --select; 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; q_dim1 = *ldq; q_offset = 1 + q_dim1 * 1; q -= q_offset; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --dif; --work; --iwork; /* Function Body */ *info = 0; lquery = *lwork == -1 || *liwork == -1; if (*ijob < 0 || *ijob > 5) { *info = -1; } else if (*n < 0) { *info = -5; } else if (*lda < f2cmax(1,*n)) { *info = -7; } else if (*ldb < f2cmax(1,*n)) { *info = -9; } else if (*ldq < 1 || *wantq && *ldq < *n) { *info = -14; } else if (*ldz < 1 || *wantz && *ldz < *n) { *info = -16; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSEN", &i__1, (ftnlen)6); return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; ierr = 0; wantp = *ijob == 1 || *ijob >= 4; wantd1 = *ijob == 2 || *ijob == 4; wantd2 = *ijob == 3 || *ijob == 5; wantd = wantd1 || wantd2; /* Set M to the dimension of the specified pair of deflating */ /* subspaces. */ *m = 0; pair = FALSE_; if (! lquery || *ijob != 0) { i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (a[k + 1 + k * a_dim1] == 0.) { if (select[k]) { ++(*m); } } else { pair = TRUE_; if (select[k] || select[k + 1]) { *m += 2; } } } else { if (select[*n]) { ++(*m); } } } /* L10: */ } } if (*ijob == 1 || *ijob == 2 || *ijob == 4) { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16, i__1 = f2cmax(i__1,i__2), i__2 = (*m << 1) * (*n - *m); lwmin = f2cmax(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = *n + 6; liwmin = f2cmax(i__1,i__2); } else if (*ijob == 3 || *ijob == 5) { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16, i__1 = f2cmax(i__1,i__2), i__2 = (*m << 2) * (*n - *m); lwmin = f2cmax(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = f2cmax(i__1,i__2), i__2 = *n + 6; liwmin = f2cmax(i__1,i__2); } else { /* Computing MAX */ i__1 = 1, i__2 = (*n << 2) + 16; lwmin = f2cmax(i__1,i__2); liwmin = 1; } work[1] = (doublereal) lwmin; iwork[1] = liwmin; if (*lwork < lwmin && ! lquery) { *info = -22; } else if (*liwork < liwmin && ! lquery) { *info = -24; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGSEN", &i__1, (ftnlen)6); return 0; } else if (lquery) { return 0; } /* Quick return if possible. */ if (*m == *n || *m == 0) { if (wantp) { *pl = 1.; *pr = 1.; } if (wantd) { dscale = 0.; dsum = 1.; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { dlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum); dlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum); /* L20: */ } dif[1] = dscale * sqrt(dsum); dif[2] = dif[1]; } goto L60; } /* Collect the selected blocks at the top-left corner of (A, B). */ ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { swap = select[k]; if (k < *n) { if (a[k + 1 + k * a_dim1] != 0.) { pair = TRUE_; swap = swap || select[k + 1]; } } if (swap) { ++ks; /* Swap the K-th block to position KS. */ /* Perform the reordering of diagonal blocks in (A, B) */ /* by orthogonal transformation matrices and update */ /* Q and Z accordingly (if requested): */ kk = k; if (k != ks) { dtgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb, &q[q_offset], ldq, &z__[z_offset], ldz, &kk, &ks, &work[1], lwork, &ierr); } if (ierr > 0) { /* Swap is rejected: exit. */ *info = 1; if (wantp) { *pl = 0.; *pr = 0.; } if (wantd) { dif[1] = 0.; dif[2] = 0.; } goto L60; } if (pair) { ++ks; } } } /* L30: */ } if (wantp) { /* Solve generalized Sylvester equation for R and L */ /* and compute PL and PR. */ n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 0; dlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1); dlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 + 1], &n1); i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1] , lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], & work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr); /* Estimate the reciprocal of norms of "projections" onto left */ /* and right eigenspaces. */ rdscal = 0.; dsum = 1.; i__1 = n1 * n2; dlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum); *pl = rdscal * sqrt(dsum); if (*pl == 0.) { *pl = 1.; } else { *pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl)); } rdscal = 0.; dsum = 1.; i__1 = n1 * n2; dlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum); *pr = rdscal * sqrt(dsum); if (*pr == 0.) { *pr = 1.; } else { *pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr)); } } if (wantd) { /* Compute estimates of Difu and Difl. */ if (wantd1) { n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 3; /* Frobenius norm-based Difu-estimate. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, & dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & ierr); /* Frobenius norm-based Difl-estimate. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[ a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], & ierr); } else { /* Compute 1-norm-based estimates of Difu and Difl using */ /* reversed communication with DLACN2. In each step a */ /* generalized Sylvester equation or a transposed variant */ /* is solved. */ kase = 0; n1 = *m; n2 = *n - *m; i__ = n1 + 1; ijb = 0; mn2 = (n1 << 1) * n2; /* 1-norm-based estimate of Difu. */ L40: dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[1], &kase, isave); if (kase != 0) { if (kase == 1) { /* Solve generalized Sylvester equation. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } else { /* Solve the transposed variant. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("T", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } goto L40; } dif[1] = dscale / dif[1]; /* 1-norm-based estimate of Difl. */ L50: dlacn2_(&mn2, &work[mn2 + 1], &work[1], &iwork[1], &dif[2], &kase, isave); if (kase != 0) { if (kase == 1) { /* Solve generalized Sylvester equation. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } else { /* Solve the transposed variant. */ i__1 = *lwork - (n1 << 1) * n2; dtgsyl_("T", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1 << 1) * n2 + 1], &i__1, &iwork[1], &ierr); } goto L50; } dif[2] = dscale / dif[2]; } } L60: /* Compute generalized eigenvalues of reordered pair (A, B) and */ /* normalize the generalized Schur form. */ pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (a[k + 1 + k * a_dim1] != 0.) { pair = TRUE_; } } if (pair) { /* Compute the eigenvalue(s) at position K. */ work[1] = a[k + k * a_dim1]; work[2] = a[k + 1 + k * a_dim1]; work[3] = a[k + (k + 1) * a_dim1]; work[4] = a[k + 1 + (k + 1) * a_dim1]; work[5] = b[k + k * b_dim1]; work[6] = b[k + 1 + k * b_dim1]; work[7] = b[k + (k + 1) * b_dim1]; work[8] = b[k + 1 + (k + 1) * b_dim1]; d__1 = smlnum * eps; dlag2_(&work[1], &c__2, &work[5], &c__2, &d__1, &beta[k], & beta[k + 1], &alphar[k], &alphar[k + 1], &alphai[k]); alphai[k + 1] = -alphai[k]; } else { if (d_sign(&c_b28, &b[k + k * b_dim1]) < 0.) { /* If B(K,K) is negative, make it positive */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { a[k + i__ * a_dim1] = -a[k + i__ * a_dim1]; b[k + i__ * b_dim1] = -b[k + i__ * b_dim1]; if (*wantq) { q[i__ + k * q_dim1] = -q[i__ + k * q_dim1]; } /* L70: */ } } alphar[k] = a[k + k * a_dim1]; alphai[k] = 0.; beta[k] = b[k + k * b_dim1]; } } /* L80: */ } work[1] = (doublereal) lwmin; iwork[1] = liwmin; return 0; /* End of DTGSEN */ } /* dtgsen_ */