#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 DSTEMR */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DSTEMR + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, */ /* M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, */ /* IWORK, LIWORK, INFO ) */ /* CHARACTER JOBZ, RANGE */ /* LOGICAL TRYRAC */ /* INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N */ /* DOUBLE PRECISION VL, VU */ /* INTEGER ISUPPZ( * ), IWORK( * ) */ /* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) */ /* DOUBLE PRECISION Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DSTEMR computes selected eigenvalues and, optionally, eigenvectors */ /* > of a real symmetric tridiagonal matrix T. Any such unreduced matrix has */ /* > a well defined set of pairwise different real eigenvalues, the corresponding */ /* > real eigenvectors are pairwise orthogonal. */ /* > */ /* > The spectrum may be computed either completely or partially by specifying */ /* > either an interval (VL,VU] or a range of indices IL:IU for the desired */ /* > eigenvalues. */ /* > */ /* > Depending on the number of desired eigenvalues, these are computed either */ /* > by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are */ /* > computed by the use of various suitable L D L^T factorizations near clusters */ /* > of close eigenvalues (referred to as RRRs, Relatively Robust */ /* > Representations). An informal sketch of the algorithm follows. */ /* > */ /* > For each unreduced block (submatrix) of T, */ /* > (a) Compute T - sigma I = L D L^T, so that L and D */ /* > define all the wanted eigenvalues to high relative accuracy. */ /* > This means that small relative changes in the entries of D and L */ /* > cause only small relative changes in the eigenvalues and */ /* > eigenvectors. The standard (unfactored) representation of the */ /* > tridiagonal matrix T does not have this property in general. */ /* > (b) Compute the eigenvalues to suitable accuracy. */ /* > If the eigenvectors are desired, the algorithm attains full */ /* > accuracy of the computed eigenvalues only right before */ /* > the corresponding vectors have to be computed, see steps c) and d). */ /* > (c) For each cluster of close eigenvalues, select a new */ /* > shift close to the cluster, find a new factorization, and refine */ /* > the shifted eigenvalues to suitable accuracy. */ /* > (d) For each eigenvalue with a large enough relative separation compute */ /* > the corresponding eigenvector by forming a rank revealing twisted */ /* > factorization. Go back to (c) for any clusters that remain. */ /* > */ /* > For more details, see: */ /* > - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */ /* > to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */ /* > Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */ /* > - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */ /* > Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */ /* > 2004. Also LAPACK Working Note 154. */ /* > - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */ /* > tridiagonal eigenvalue/eigenvector problem", */ /* > Computer Science Division Technical Report No. UCB/CSD-97-971, */ /* > UC Berkeley, May 1997. */ /* > */ /* > Further Details */ /* > 1.DSTEMR works only on machines which follow IEEE-754 */ /* > floating-point standard in their handling of infinities and NaNs. */ /* > This permits the use of efficient inner loops avoiding a check for */ /* > zero divisors. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] JOBZ */ /* > \verbatim */ /* > JOBZ is CHARACTER*1 */ /* > = 'N': Compute eigenvalues only; */ /* > = 'V': Compute eigenvalues and eigenvectors. */ /* > \endverbatim */ /* > */ /* > \param[in] RANGE */ /* > \verbatim */ /* > RANGE is CHARACTER*1 */ /* > = 'A': all eigenvalues will be found. */ /* > = 'V': all eigenvalues in the half-open interval (VL,VU] */ /* > will be found. */ /* > = 'I': the IL-th through IU-th eigenvalues will be found. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrix. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] D */ /* > \verbatim */ /* > D is DOUBLE PRECISION array, dimension (N) */ /* > On entry, the N diagonal elements of the tridiagonal matrix */ /* > T. On exit, D is overwritten. */ /* > \endverbatim */ /* > */ /* > \param[in,out] E */ /* > \verbatim */ /* > E is DOUBLE PRECISION array, dimension (N) */ /* > On entry, the (N-1) subdiagonal elements of the tridiagonal */ /* > matrix T in elements 1 to N-1 of E. E(N) need not be set on */ /* > input, but is used internally as workspace. */ /* > On exit, E is overwritten. */ /* > \endverbatim */ /* > */ /* > \param[in] VL */ /* > \verbatim */ /* > VL is DOUBLE PRECISION */ /* > */ /* > If RANGE='V', the lower bound of the interval to */ /* > be searched for eigenvalues. VL < VU. */ /* > Not referenced if RANGE = 'A' or 'I'. */ /* > \endverbatim */ /* > */ /* > \param[in] VU */ /* > \verbatim */ /* > VU is DOUBLE PRECISION */ /* > */ /* > If RANGE='V', the upper bound of the interval to */ /* > be searched for eigenvalues. VL < VU. */ /* > Not referenced if RANGE = 'A' or 'I'. */ /* > \endverbatim */ /* > */ /* > \param[in] IL */ /* > \verbatim */ /* > IL is INTEGER */ /* > */ /* > If RANGE='I', the index of the */ /* > smallest eigenvalue to be returned. */ /* > 1 <= IL <= IU <= N, if N > 0. */ /* > Not referenced if RANGE = 'A' or 'V'. */ /* > \endverbatim */ /* > */ /* > \param[in] IU */ /* > \verbatim */ /* > IU is INTEGER */ /* > */ /* > If RANGE='I', the index of the */ /* > largest eigenvalue to be returned. */ /* > 1 <= IL <= IU <= N, if N > 0. */ /* > Not referenced if RANGE = 'A' or 'V'. */ /* > \endverbatim */ /* > */ /* > \param[out] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The total number of eigenvalues found. 0 <= M <= N. */ /* > If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */ /* > \endverbatim */ /* > */ /* > \param[out] W */ /* > \verbatim */ /* > W is DOUBLE PRECISION array, dimension (N) */ /* > The first M elements contain the selected eigenvalues in */ /* > ascending order. */ /* > \endverbatim */ /* > */ /* > \param[out] Z */ /* > \verbatim */ /* > Z is DOUBLE PRECISION array, dimension (LDZ, f2cmax(1,M) ) */ /* > If JOBZ = 'V', and if INFO = 0, then the first M columns of Z */ /* > contain the orthonormal eigenvectors of the matrix T */ /* > corresponding to the selected eigenvalues, with the i-th */ /* > column of Z holding the eigenvector associated with W(i). */ /* > If JOBZ = 'N', then Z is not referenced. */ /* > Note: the user must ensure that at least f2cmax(1,M) columns are */ /* > supplied in the array Z; if RANGE = 'V', the exact value of M */ /* > is not known in advance and can be computed with a workspace */ /* > query by setting NZC = -1, see below. */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. LDZ >= 1, and if */ /* > JOBZ = 'V', then LDZ >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[in] NZC */ /* > \verbatim */ /* > NZC is INTEGER */ /* > The number of eigenvectors to be held in the array Z. */ /* > If RANGE = 'A', then NZC >= f2cmax(1,N). */ /* > If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. */ /* > If RANGE = 'I', then NZC >= IU-IL+1. */ /* > If NZC = -1, then a workspace query is assumed; the */ /* > routine calculates the number of columns of the array Z that */ /* > are needed to hold the eigenvectors. */ /* > This value is returned as the first entry of the Z array, and */ /* > no error message related to NZC is issued by XERBLA. */ /* > \endverbatim */ /* > */ /* > \param[out] ISUPPZ */ /* > \verbatim */ /* > ISUPPZ is INTEGER array, dimension ( 2*f2cmax(1,M) ) */ /* > The support of the eigenvectors in Z, i.e., the indices */ /* > indicating the nonzero elements in Z. The i-th computed eigenvector */ /* > is nonzero only in elements ISUPPZ( 2*i-1 ) through */ /* > ISUPPZ( 2*i ). This is relevant in the case when the matrix */ /* > is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] TRYRAC */ /* > \verbatim */ /* > TRYRAC is LOGICAL */ /* > If TRYRAC = .TRUE., indicates that the code should check whether */ /* > the tridiagonal matrix defines its eigenvalues to high relative */ /* > accuracy. If so, the code uses relative-accuracy preserving */ /* > algorithms that might be (a bit) slower depending on the matrix. */ /* > If the matrix does not define its eigenvalues to high relative */ /* > accuracy, the code can uses possibly faster algorithms. */ /* > If TRYRAC = .FALSE., the code is not required to guarantee */ /* > relatively accurate eigenvalues and can use the fastest possible */ /* > techniques. */ /* > On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix */ /* > does not define its eigenvalues to high relative accuracy. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is DOUBLE PRECISION array, dimension (LWORK) */ /* > On exit, if INFO = 0, WORK(1) returns the optimal */ /* > (and minimal) LWORK. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > The dimension of the array WORK. LWORK >= f2cmax(1,18*N) */ /* > if JOBZ = 'V', and LWORK >= f2cmax(1,12*N) if JOBZ = 'N'. */ /* > 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 (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 >= f2cmax(1,10*N) */ /* > if the eigenvectors are desired, and LIWORK >= f2cmax(1,8*N) */ /* > if only the eigenvalues are to be computed. */ /* > 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 */ /* > On exit, INFO */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value */ /* > > 0: if INFO = 1X, internal error in DLARRE, */ /* > if INFO = 2X, internal error in DLARRV. */ /* > Here, the digit X = ABS( IINFO ) < 10, where IINFO is */ /* > the nonzero error code returned by DLARRE or */ /* > DLARRV, respectively. */ /* > \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 Contributors: */ /* ================== */ /* > */ /* > Beresford Parlett, University of California, Berkeley, USA \n */ /* > Jim Demmel, University of California, Berkeley, USA \n */ /* > Inderjit Dhillon, University of Texas, Austin, USA \n */ /* > Osni Marques, LBNL/NERSC, USA \n */ /* > Christof Voemel, University of California, Berkeley, USA */ /* ===================================================================== */ /* Subroutine */ int dstemr_(char *jobz, char *range, integer *n, doublereal * d__, doublereal *e, doublereal *vl, doublereal *vu, integer *il, integer *iu, integer *m, doublereal *w, doublereal *z__, integer *ldz, integer *nzc, integer *isuppz, logical *tryrac, doublereal *work, integer *lwork, integer *iwork, integer *liwork, integer *info) { /* System generated locals */ integer z_dim1, z_offset, i__1, i__2; doublereal d__1, d__2; /* Local variables */ integer indd, iend, jblk, wend; doublereal rmin, rmax; integer itmp; doublereal tnrm; extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); integer inde2, itmp2; doublereal rtol1, rtol2; integer i__, j; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); doublereal scale; integer indgp; extern logical lsame_(char *, char *); integer iinfo, iindw, ilast; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), dswap_(integer *, doublereal *, integer *, doublereal *, integer *); integer lwmin; logical wantz; doublereal r1, r2; extern /* Subroutine */ int dlaev2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); integer jj; doublereal cs; integer in; extern doublereal dlamch_(char *); logical alleig, indeig; integer ibegin, iindbl; doublereal sn, wl; logical valeig; extern /* Subroutine */ int dlarrc_(char *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *), dlarre_(char *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *); integer wbegin; doublereal safmin, wu; extern /* Subroutine */ int dlarrj_(integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *, ftnlen); doublereal bignum; integer inderr, iindwk, indgrs, offset; extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *); extern /* Subroutine */ int dlarrr_(integer *, doublereal *, doublereal *, integer *), dlarrv_(integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dlasrt_(char *, integer *, doublereal *, integer *); doublereal thresh; integer iinspl, ifirst, indwrk, liwmin, nzcmin; doublereal pivmin; integer nsplit; doublereal smlnum; logical lquery, zquery; integer iil, iiu; doublereal eps, tmp; /* -- 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 */ /* ===================================================================== */ /* Test the input parameters. */ /* Parameter adjustments */ --d__; --e; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --isuppz; --work; --iwork; /* Function Body */ wantz = lsame_(jobz, "V"); alleig = lsame_(range, "A"); valeig = lsame_(range, "V"); indeig = lsame_(range, "I"); lquery = *lwork == -1 || *liwork == -1; zquery = *nzc == -1; /* DSTEMR needs WORK of size 6*N, IWORK of size 3*N. */ /* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. */ /* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N. */ if (wantz) { lwmin = *n * 18; liwmin = *n * 10; } else { /* need less workspace if only the eigenvalues are wanted */ lwmin = *n * 12; liwmin = *n << 3; } wl = 0.; wu = 0.; iil = 0; iiu = 0; nsplit = 0; if (valeig) { /* We do not reference VL, VU in the cases RANGE = 'I','A' */ /* The interval (WL, WU] contains all the wanted eigenvalues. */ /* It is either given by the user or computed in DLARRE. */ wl = *vl; wu = *vu; } else if (indeig) { /* We do not reference IL, IU in the cases RANGE = 'V','A' */ iil = *il; iiu = *iu; } *info = 0; if (! (wantz || lsame_(jobz, "N"))) { *info = -1; } else if (! (alleig || valeig || indeig)) { *info = -2; } else if (*n < 0) { *info = -3; } else if (valeig && *n > 0 && wu <= wl) { *info = -7; } else if (indeig && (iil < 1 || iil > *n)) { *info = -8; } else if (indeig && (iiu < iil || iiu > *n)) { *info = -9; } else if (*ldz < 1 || wantz && *ldz < *n) { *info = -13; } else if (*lwork < lwmin && ! lquery) { *info = -17; } else if (*liwork < liwmin && ! lquery) { *info = -19; } /* Get machine constants. */ safmin = dlamch_("Safe minimum"); eps = dlamch_("Precision"); smlnum = safmin / eps; bignum = 1. / smlnum; rmin = sqrt(smlnum); /* Computing MIN */ d__1 = sqrt(bignum), d__2 = 1. / sqrt(sqrt(safmin)); rmax = f2cmin(d__1,d__2); if (*info == 0) { work[1] = (doublereal) lwmin; iwork[1] = liwmin; if (wantz && alleig) { nzcmin = *n; } else if (wantz && valeig) { dlarrc_("T", n, vl, vu, &d__[1], &e[1], &safmin, &nzcmin, &itmp, & itmp2, info); } else if (wantz && indeig) { nzcmin = iiu - iil + 1; } else { /* WANTZ .EQ. FALSE. */ nzcmin = 0; } if (zquery && *info == 0) { z__[z_dim1 + 1] = (doublereal) nzcmin; } else if (*nzc < nzcmin && ! zquery) { *info = -14; } } if (*info != 0) { i__1 = -(*info); xerbla_("DSTEMR", &i__1, (ftnlen)6); return 0; } else if (lquery || zquery) { return 0; } /* Handle N = 0, 1, and 2 cases immediately */ *m = 0; if (*n == 0) { return 0; } if (*n == 1) { if (alleig || indeig) { *m = 1; w[1] = d__[1]; } else { if (wl < d__[1] && wu >= d__[1]) { *m = 1; w[1] = d__[1]; } } if (wantz && ! zquery) { z__[z_dim1 + 1] = 1.; isuppz[1] = 1; isuppz[2] = 1; } return 0; } if (*n == 2) { if (! wantz) { dlae2_(&d__[1], &e[1], &d__[2], &r1, &r2); } else if (wantz && ! zquery) { dlaev2_(&d__[1], &e[1], &d__[2], &r1, &r2, &cs, &sn); } if (alleig || valeig && r2 > wl && r2 <= wu || indeig && iil == 1) { ++(*m); w[*m] = r2; if (wantz && ! zquery) { z__[*m * z_dim1 + 1] = -sn; z__[*m * z_dim1 + 2] = cs; /* Note: At most one of SN and CS can be zero. */ if (sn != 0.) { if (cs != 0.) { isuppz[(*m << 1) - 1] = 1; isuppz[*m * 2] = 2; } else { isuppz[(*m << 1) - 1] = 1; isuppz[*m * 2] = 1; } } else { isuppz[(*m << 1) - 1] = 2; isuppz[*m * 2] = 2; } } } if (alleig || valeig && r1 > wl && r1 <= wu || indeig && iiu == 2) { ++(*m); w[*m] = r1; if (wantz && ! zquery) { z__[*m * z_dim1 + 1] = cs; z__[*m * z_dim1 + 2] = sn; /* Note: At most one of SN and CS can be zero. */ if (sn != 0.) { if (cs != 0.) { isuppz[(*m << 1) - 1] = 1; isuppz[*m * 2] = 2; } else { isuppz[(*m << 1) - 1] = 1; isuppz[*m * 2] = 1; } } else { isuppz[(*m << 1) - 1] = 2; isuppz[*m * 2] = 2; } } } } else { /* Continue with general N */ indgrs = 1; inderr = (*n << 1) + 1; indgp = *n * 3 + 1; indd = (*n << 2) + 1; inde2 = *n * 5 + 1; indwrk = *n * 6 + 1; iinspl = 1; iindbl = *n + 1; iindw = (*n << 1) + 1; iindwk = *n * 3 + 1; /* Scale matrix to allowable range, if necessary. */ /* The allowable range is related to the PIVMIN parameter; see the */ /* comments in DLARRD. The preference for scaling small values */ /* up is heuristic; we expect users' matrices not to be close to the */ /* RMAX threshold. */ scale = 1.; tnrm = dlanst_("M", n, &d__[1], &e[1]); if (tnrm > 0. && tnrm < rmin) { scale = rmin / tnrm; } else if (tnrm > rmax) { scale = rmax / tnrm; } if (scale != 1.) { dscal_(n, &scale, &d__[1], &c__1); i__1 = *n - 1; dscal_(&i__1, &scale, &e[1], &c__1); tnrm *= scale; if (valeig) { /* If eigenvalues in interval have to be found, */ /* scale (WL, WU] accordingly */ wl *= scale; wu *= scale; } } /* Compute the desired eigenvalues of the tridiagonal after splitting */ /* into smaller subblocks if the corresponding off-diagonal elements */ /* are small */ /* THRESH is the splitting parameter for DLARRE */ /* A negative THRESH forces the old splitting criterion based on the */ /* size of the off-diagonal. A positive THRESH switches to splitting */ /* which preserves relative accuracy. */ if (*tryrac) { /* Test whether the matrix warrants the more expensive relative approach. */ dlarrr_(n, &d__[1], &e[1], &iinfo); } else { /* The user does not care about relative accurately eigenvalues */ iinfo = -1; } /* Set the splitting criterion */ if (iinfo == 0) { thresh = eps; } else { thresh = -eps; /* relative accuracy is desired but T does not guarantee it */ *tryrac = FALSE_; } if (*tryrac) { /* Copy original diagonal, needed to guarantee relative accuracy */ dcopy_(n, &d__[1], &c__1, &work[indd], &c__1); } /* Store the squares of the offdiagonal values of T */ i__1 = *n - 1; for (j = 1; j <= i__1; ++j) { /* Computing 2nd power */ d__1 = e[j]; work[inde2 + j - 1] = d__1 * d__1; /* L5: */ } /* Set the tolerance parameters for bisection */ if (! wantz) { /* DLARRE computes the eigenvalues to full precision. */ rtol1 = eps * 4.; rtol2 = eps * 4.; } else { /* DLARRE computes the eigenvalues to less than full precision. */ /* DLARRV will refine the eigenvalue approximations, and we can */ /* need less accurate initial bisection in DLARRE. */ /* Note: these settings do only affect the subset case and DLARRE */ rtol1 = sqrt(eps); /* Computing MAX */ d__1 = sqrt(eps) * .005, d__2 = eps * 4.; rtol2 = f2cmax(d__1,d__2); } dlarre_(range, n, &wl, &wu, &iil, &iiu, &d__[1], &e[1], &work[inde2], &rtol1, &rtol2, &thresh, &nsplit, &iwork[iinspl], m, &w[1], & work[inderr], &work[indgp], &iwork[iindbl], &iwork[iindw], & work[indgrs], &pivmin, &work[indwrk], &iwork[iindwk], &iinfo); if (iinfo != 0) { *info = abs(iinfo) + 10; return 0; } /* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired */ /* part of the spectrum. All desired eigenvalues are contained in */ /* (WL,WU] */ if (wantz) { /* Compute the desired eigenvectors corresponding to the computed */ /* eigenvalues */ dlarrv_(n, &wl, &wu, &d__[1], &e[1], &pivmin, &iwork[iinspl], m, & c__1, m, &c_b18, &rtol1, &rtol2, &w[1], &work[inderr], & work[indgp], &iwork[iindbl], &iwork[iindw], &work[indgrs], &z__[z_offset], ldz, &isuppz[1], &work[indwrk], &iwork[ iindwk], &iinfo); if (iinfo != 0) { *info = abs(iinfo) + 20; return 0; } } else { /* DLARRE computes eigenvalues of the (shifted) root representation */ /* DLARRV returns the eigenvalues of the unshifted matrix. */ /* However, if the eigenvectors are not desired by the user, we need */ /* to apply the corresponding shifts from DLARRE to obtain the */ /* eigenvalues of the original matrix. */ i__1 = *m; for (j = 1; j <= i__1; ++j) { itmp = iwork[iindbl + j - 1]; w[j] += e[iwork[iinspl + itmp - 1]]; /* L20: */ } } if (*tryrac) { /* Refine computed eigenvalues so that they are relatively accurate */ /* with respect to the original matrix T. */ ibegin = 1; wbegin = 1; i__1 = iwork[iindbl + *m - 1]; for (jblk = 1; jblk <= i__1; ++jblk) { iend = iwork[iinspl + jblk - 1]; in = iend - ibegin + 1; wend = wbegin - 1; /* check if any eigenvalues have to be refined in this block */ L36: if (wend < *m) { if (iwork[iindbl + wend] == jblk) { ++wend; goto L36; } } if (wend < wbegin) { ibegin = iend + 1; goto L39; } offset = iwork[iindw + wbegin - 1] - 1; ifirst = iwork[iindw + wbegin - 1]; ilast = iwork[iindw + wend - 1]; rtol2 = eps * 4.; dlarrj_(&in, &work[indd + ibegin - 1], &work[inde2 + ibegin - 1], &ifirst, &ilast, &rtol2, &offset, &w[wbegin], & work[inderr + wbegin - 1], &work[indwrk], &iwork[ iindwk], &pivmin, &tnrm, &iinfo); ibegin = iend + 1; wbegin = wend + 1; L39: ; } } /* If matrix was scaled, then rescale eigenvalues appropriately. */ if (scale != 1.) { d__1 = 1. / scale; dscal_(m, &d__1, &w[1], &c__1); } } /* If eigenvalues are not in increasing order, then sort them, */ /* possibly along with eigenvectors. */ if (nsplit > 1 || *n == 2) { if (! wantz) { dlasrt_("I", m, &w[1], &iinfo); if (iinfo != 0) { *info = 3; return 0; } } else { i__1 = *m - 1; for (j = 1; j <= i__1; ++j) { i__ = 0; tmp = w[j]; i__2 = *m; for (jj = j + 1; jj <= i__2; ++jj) { if (w[jj] < tmp) { i__ = jj; tmp = w[jj]; } /* L50: */ } if (i__ != 0) { w[i__] = w[j]; w[j] = tmp; if (wantz) { dswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1], &c__1); itmp = isuppz[(i__ << 1) - 1]; isuppz[(i__ << 1) - 1] = isuppz[(j << 1) - 1]; isuppz[(j << 1) - 1] = itmp; itmp = isuppz[i__ * 2]; isuppz[i__ * 2] = isuppz[j * 2]; isuppz[j * 2] = itmp; } } /* L60: */ } } } work[1] = (doublereal) lwmin; iwork[1] = liwmin; return 0; /* End of DSTEMR */ } /* dstemr_ */