#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 DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DLARRD + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL, IU, GERS, */ /* RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, */ /* M, W, WERR, WL, WU, IBLOCK, INDEXW, */ /* WORK, IWORK, INFO ) */ /* CHARACTER ORDER, RANGE */ /* INTEGER IL, INFO, IU, M, N, NSPLIT */ /* DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU */ /* INTEGER IBLOCK( * ), INDEXW( * ), */ /* $ ISPLIT( * ), IWORK( * ) */ /* DOUBLE PRECISION D( * ), E( * ), E2( * ), */ /* $ GERS( * ), W( * ), WERR( * ), WORK( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DLARRD computes the eigenvalues of a symmetric tridiagonal */ /* > matrix T to suitable accuracy. This is an auxiliary code to be */ /* > called from DSTEMR. */ /* > The user may ask for all eigenvalues, all eigenvalues */ /* > in the half-open interval (VL, VU], or the IL-th through IU-th */ /* > eigenvalues. */ /* > */ /* > To avoid overflow, the matrix must be scaled so that its */ /* > largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest */ /* > accuracy, it should not be much smaller than that. */ /* > */ /* > See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal */ /* > Matrix", Report CS41, Computer Science Dept., Stanford */ /* > University, July 21, 1966. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] RANGE */ /* > \verbatim */ /* > RANGE is CHARACTER*1 */ /* > = 'A': ("All") all eigenvalues will be found. */ /* > = 'V': ("Value") all eigenvalues in the half-open interval */ /* > (VL, VU] will be found. */ /* > = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */ /* > entire matrix) will be found. */ /* > \endverbatim */ /* > */ /* > \param[in] ORDER */ /* > \verbatim */ /* > ORDER is CHARACTER*1 */ /* > = 'B': ("By Block") the eigenvalues will be grouped by */ /* > split-off block (see IBLOCK, ISPLIT) and */ /* > ordered from smallest to largest within */ /* > the block. */ /* > = 'E': ("Entire matrix") */ /* > the eigenvalues for the entire matrix */ /* > will be ordered from smallest to */ /* > largest. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the tridiagonal matrix T. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] VL */ /* > \verbatim */ /* > VL is DOUBLE PRECISION */ /* > If RANGE='V', the lower bound of the interval to */ /* > be searched for eigenvalues. Eigenvalues less than or equal */ /* > to VL, or greater than VU, will not be returned. 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. Eigenvalues less than or equal */ /* > to VL, or greater than VU, will not be returned. 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; IL = 1 and IU = 0 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; IL = 1 and IU = 0 if N = 0. */ /* > Not referenced if RANGE = 'A' or 'V'. */ /* > \endverbatim */ /* > */ /* > \param[in] GERS */ /* > \verbatim */ /* > GERS is DOUBLE PRECISION array, dimension (2*N) */ /* > The N Gerschgorin intervals (the i-th Gerschgorin interval */ /* > is (GERS(2*i-1), GERS(2*i)). */ /* > \endverbatim */ /* > */ /* > \param[in] RELTOL */ /* > \verbatim */ /* > RELTOL is DOUBLE PRECISION */ /* > The minimum relative width of an interval. When an interval */ /* > is narrower than RELTOL times the larger (in */ /* > magnitude) endpoint, then it is considered to be */ /* > sufficiently small, i.e., converged. Note: this should */ /* > always be at least radix*machine epsilon. */ /* > \endverbatim */ /* > */ /* > \param[in] D */ /* > \verbatim */ /* > D is DOUBLE PRECISION array, dimension (N) */ /* > The n diagonal elements of the tridiagonal matrix T. */ /* > \endverbatim */ /* > */ /* > \param[in] E */ /* > \verbatim */ /* > E is DOUBLE PRECISION array, dimension (N-1) */ /* > The (n-1) off-diagonal elements of the tridiagonal matrix T. */ /* > \endverbatim */ /* > */ /* > \param[in] E2 */ /* > \verbatim */ /* > E2 is DOUBLE PRECISION array, dimension (N-1) */ /* > The (n-1) squared off-diagonal elements of the tridiagonal matrix T. */ /* > \endverbatim */ /* > */ /* > \param[in] PIVMIN */ /* > \verbatim */ /* > PIVMIN is DOUBLE PRECISION */ /* > The minimum pivot allowed in the Sturm sequence for T. */ /* > \endverbatim */ /* > */ /* > \param[in] NSPLIT */ /* > \verbatim */ /* > NSPLIT is INTEGER */ /* > The number of diagonal blocks in the matrix T. */ /* > 1 <= NSPLIT <= N. */ /* > \endverbatim */ /* > */ /* > \param[in] ISPLIT */ /* > \verbatim */ /* > ISPLIT is INTEGER array, dimension (N) */ /* > The splitting points, at which T breaks up into submatrices. */ /* > The first submatrix consists of rows/columns 1 to ISPLIT(1), */ /* > the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */ /* > etc., and the NSPLIT-th consists of rows/columns */ /* > ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */ /* > (Only the first NSPLIT elements will actually be used, but */ /* > since the user cannot know a priori what value NSPLIT will */ /* > have, N words must be reserved for ISPLIT.) */ /* > \endverbatim */ /* > */ /* > \param[out] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The actual number of eigenvalues found. 0 <= M <= N. */ /* > (See also the description of INFO=2,3.) */ /* > \endverbatim */ /* > */ /* > \param[out] W */ /* > \verbatim */ /* > W is DOUBLE PRECISION array, dimension (N) */ /* > On exit, the first M elements of W will contain the */ /* > eigenvalue approximations. DLARRD computes an interval */ /* > I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue */ /* > approximation is given as the interval midpoint */ /* > W(j)= ( a_j + b_j)/2. The corresponding error is bounded by */ /* > WERR(j) = abs( a_j - b_j)/2 */ /* > \endverbatim */ /* > */ /* > \param[out] WERR */ /* > \verbatim */ /* > WERR is DOUBLE PRECISION array, dimension (N) */ /* > The error bound on the corresponding eigenvalue approximation */ /* > in W. */ /* > \endverbatim */ /* > */ /* > \param[out] WL */ /* > \verbatim */ /* > WL is DOUBLE PRECISION */ /* > \endverbatim */ /* > */ /* > \param[out] WU */ /* > \verbatim */ /* > WU is DOUBLE PRECISION */ /* > The interval (WL, WU] contains all the wanted eigenvalues. */ /* > If RANGE='V', then WL=VL and WU=VU. */ /* > If RANGE='A', then WL and WU are the global Gerschgorin bounds */ /* > on the spectrum. */ /* > If RANGE='I', then WL and WU are computed by DLAEBZ from the */ /* > index range specified. */ /* > \endverbatim */ /* > */ /* > \param[out] IBLOCK */ /* > \verbatim */ /* > IBLOCK is INTEGER array, dimension (N) */ /* > At each row/column j where E(j) is zero or small, the */ /* > matrix T is considered to split into a block diagonal */ /* > matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which */ /* > block (from 1 to the number of blocks) the eigenvalue W(i) */ /* > belongs. (DLARRD may use the remaining N-M elements as */ /* > workspace.) */ /* > \endverbatim */ /* > */ /* > \param[out] INDEXW */ /* > \verbatim */ /* > INDEXW is INTEGER array, dimension (N) */ /* > The indices of the eigenvalues within each block (submatrix); */ /* > for example, INDEXW(i)= j and IBLOCK(i)=k imply that the */ /* > i-th eigenvalue W(i) is the j-th eigenvalue in block k. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is DOUBLE PRECISION array, dimension (4*N) */ /* > \endverbatim */ /* > */ /* > \param[out] IWORK */ /* > \verbatim */ /* > IWORK is INTEGER array, dimension (3*N) */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value */ /* > > 0: some or all of the eigenvalues failed to converge or */ /* > were not computed: */ /* > =1 or 3: Bisection failed to converge for some */ /* > eigenvalues; these eigenvalues are flagged by a */ /* > negative block number. The effect is that the */ /* > eigenvalues may not be as accurate as the */ /* > absolute and relative tolerances. This is */ /* > generally caused by unexpectedly inaccurate */ /* > arithmetic. */ /* > =2 or 3: RANGE='I' only: Not all of the eigenvalues */ /* > IL:IU were found. */ /* > Effect: M < IU+1-IL */ /* > Cause: non-monotonic arithmetic, causing the */ /* > Sturm sequence to be non-monotonic. */ /* > Cure: recalculate, using RANGE='A', and pick */ /* > out eigenvalues IL:IU. In some cases, */ /* > increasing the PARAMETER "FUDGE" may */ /* > make things work. */ /* > = 4: RANGE='I', and the Gershgorin interval */ /* > initially used was too small. No eigenvalues */ /* > were computed. */ /* > Probable cause: your machine has sloppy */ /* > floating-point arithmetic. */ /* > Cure: Increase the PARAMETER "FUDGE", */ /* > recompile, and try again. */ /* > \endverbatim */ /* > \par Internal Parameters: */ /* ========================= */ /* > */ /* > \verbatim */ /* > FUDGE DOUBLE PRECISION, default = 2 */ /* > A "fudge factor" to widen the Gershgorin intervals. Ideally, */ /* > a value of 1 should work, but on machines with sloppy */ /* > arithmetic, this needs to be larger. The default for */ /* > publicly released versions should be large enough to handle */ /* > the worst machine around. Note that this has no effect */ /* > on accuracy of the solution. */ /* > \endverbatim */ /* > */ /* > \par Contributors: */ /* ================== */ /* > */ /* > W. Kahan, University of California, Berkeley, USA \n */ /* > 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 \n */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2016 */ /* > \ingroup OTHERauxiliary */ /* ===================================================================== */ /* Subroutine */ int dlarrd_(char *range, char *order, integer *n, doublereal *vl, doublereal *vu, integer *il, integer *iu, doublereal *gers, doublereal *reltol, doublereal *d__, doublereal *e, doublereal *e2, doublereal *pivmin, integer *nsplit, integer *isplit, integer *m, doublereal *w, doublereal *werr, doublereal *wl, doublereal *wu, integer *iblock, integer *indexw, doublereal *work, integer *iwork, integer *info) { /* System generated locals */ integer i__1, i__2, i__3; doublereal d__1, d__2; /* Local variables */ integer iend, jblk, ioff, iout, itmp1, itmp2, i__, j, jdisc; extern logical lsame_(char *, char *); integer iinfo; doublereal atoli; integer iwoff, itmax; doublereal wkill, rtoli, uflow, tnorm; integer ib, ie, je, nb; doublereal gl; integer im, in; extern doublereal dlamch_(char *); doublereal gu; integer ibegin, iw; extern /* Subroutine */ int dlaebz_(integer *, integer *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); integer irange, idiscl, idumma[1]; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); integer idiscu; logical ncnvrg, toofew; integer jee; doublereal eps; integer nwl; doublereal wlu, wul; integer nwu; doublereal tmp1, tmp2; /* -- LAPACK auxiliary 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 */ /* ===================================================================== */ /* Parameter adjustments */ --iwork; --work; --indexw; --iblock; --werr; --w; --isplit; --e2; --e; --d__; --gers; /* Function Body */ *info = 0; /* Quick return if possible */ if (*n <= 0) { return 0; } /* Decode RANGE */ if (lsame_(range, "A")) { irange = 1; } else if (lsame_(range, "V")) { irange = 2; } else if (lsame_(range, "I")) { irange = 3; } else { irange = 0; } /* Check for Errors */ if (irange <= 0) { *info = -1; } else if (! (lsame_(order, "B") || lsame_(order, "E"))) { *info = -2; } else if (*n < 0) { *info = -3; } else if (irange == 2) { if (*vl >= *vu) { *info = -5; } } else if (irange == 3 && (*il < 1 || *il > f2cmax(1,*n))) { *info = -6; } else if (irange == 3 && (*iu < f2cmin(*n,*il) || *iu > *n)) { *info = -7; } if (*info != 0) { return 0; } /* Initialize error flags */ *info = 0; ncnvrg = FALSE_; toofew = FALSE_; /* Quick return if possible */ *m = 0; if (*n == 0) { return 0; } /* Simplification: */ if (irange == 3 && *il == 1 && *iu == *n) { irange = 1; } /* Get machine constants */ eps = dlamch_("P"); uflow = dlamch_("U"); /* Special Case when N=1 */ /* Treat case of 1x1 matrix for quick return */ if (*n == 1) { if (irange == 1 || irange == 2 && d__[1] > *vl && d__[1] <= *vu || irange == 3 && *il == 1 && *iu == 1) { *m = 1; w[1] = d__[1]; /* The computation error of the eigenvalue is zero */ werr[1] = 0.; iblock[1] = 1; indexw[1] = 1; } return 0; } /* NB is the minimum vector length for vector bisection, or 0 */ /* if only scalar is to be done. */ nb = ilaenv_(&c__1, "DSTEBZ", " ", n, &c_n1, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); if (nb <= 1) { nb = 0; } /* Find global spectral radius */ gl = d__[1]; gu = d__[1]; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { /* Computing MIN */ d__1 = gl, d__2 = gers[(i__ << 1) - 1]; gl = f2cmin(d__1,d__2); /* Computing MAX */ d__1 = gu, d__2 = gers[i__ * 2]; gu = f2cmax(d__1,d__2); /* L5: */ } /* Compute global Gerschgorin bounds and spectral diameter */ /* Computing MAX */ d__1 = abs(gl), d__2 = abs(gu); tnorm = f2cmax(d__1,d__2); gl = gl - tnorm * 2. * eps * *n - *pivmin * 4.; gu = gu + tnorm * 2. * eps * *n + *pivmin * 4.; /* [JAN/28/2009] remove the line below since SPDIAM variable not use */ /* SPDIAM = GU - GL */ /* Input arguments for DLAEBZ: */ /* The relative tolerance. An interval (a,b] lies within */ /* "relative tolerance" if b-a < RELTOL*f2cmax(|a|,|b|), */ rtoli = *reltol; /* Set the absolute tolerance for interval convergence to zero to force */ /* interval convergence based on relative size of the interval. */ /* This is dangerous because intervals might not converge when RELTOL is */ /* small. But at least a very small number should be selected so that for */ /* strongly graded matrices, the code can get relatively accurate */ /* eigenvalues. */ atoli = uflow * 4. + *pivmin * 4.; if (irange == 3) { /* RANGE='I': Compute an interval containing eigenvalues */ /* IL through IU. The initial interval [GL,GU] from the global */ /* Gerschgorin bounds GL and GU is refined by DLAEBZ. */ itmax = (integer) ((log(tnorm + *pivmin) - log(*pivmin)) / log(2.)) + 2; work[*n + 1] = gl; work[*n + 2] = gl; work[*n + 3] = gu; work[*n + 4] = gu; work[*n + 5] = gl; work[*n + 6] = gu; iwork[1] = -1; iwork[2] = -1; iwork[3] = *n + 1; iwork[4] = *n + 1; iwork[5] = *il - 1; iwork[6] = *iu; dlaebz_(&c__3, &itmax, n, &c__2, &c__2, &nb, &atoli, &rtoli, pivmin, & d__[1], &e[1], &e2[1], &iwork[5], &work[*n + 1], &work[*n + 5] , &iout, &iwork[1], &w[1], &iblock[1], &iinfo); if (iinfo != 0) { *info = iinfo; return 0; } /* On exit, output intervals may not be ordered by ascending negcount */ if (iwork[6] == *iu) { *wl = work[*n + 1]; wlu = work[*n + 3]; nwl = iwork[1]; *wu = work[*n + 4]; wul = work[*n + 2]; nwu = iwork[4]; } else { *wl = work[*n + 2]; wlu = work[*n + 4]; nwl = iwork[2]; *wu = work[*n + 3]; wul = work[*n + 1]; nwu = iwork[3]; } /* On exit, the interval [WL, WLU] contains a value with negcount NWL, */ /* and [WUL, WU] contains a value with negcount NWU. */ if (nwl < 0 || nwl >= *n || nwu < 1 || nwu > *n) { *info = 4; return 0; } } else if (irange == 2) { *wl = *vl; *wu = *vu; } else if (irange == 1) { *wl = gl; *wu = gu; } /* Find Eigenvalues -- Loop Over blocks and recompute NWL and NWU. */ /* NWL accumulates the number of eigenvalues .le. WL, */ /* NWU accumulates the number of eigenvalues .le. WU */ *m = 0; iend = 0; *info = 0; nwl = 0; nwu = 0; i__1 = *nsplit; for (jblk = 1; jblk <= i__1; ++jblk) { ioff = iend; ibegin = ioff + 1; iend = isplit[jblk]; in = iend - ioff; if (in == 1) { /* 1x1 block */ if (*wl >= d__[ibegin] - *pivmin) { ++nwl; } if (*wu >= d__[ibegin] - *pivmin) { ++nwu; } if (irange == 1 || *wl < d__[ibegin] - *pivmin && *wu >= d__[ ibegin] - *pivmin) { ++(*m); w[*m] = d__[ibegin]; werr[*m] = 0.; /* The gap for a single block doesn't matter for the later */ /* algorithm and is assigned an arbitrary large value */ iblock[*m] = jblk; indexw[*m] = 1; } /* Disabled 2x2 case because of a failure on the following matrix */ /* RANGE = 'I', IL = IU = 4 */ /* Original Tridiagonal, d = [ */ /* -0.150102010615740E+00 */ /* -0.849897989384260E+00 */ /* -0.128208148052635E-15 */ /* 0.128257718286320E-15 */ /* ]; */ /* e = [ */ /* -0.357171383266986E+00 */ /* -0.180411241501588E-15 */ /* -0.175152352710251E-15 */ /* ]; */ /* ELSE IF( IN.EQ.2 ) THEN */ /* * 2x2 block */ /* DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 ) */ /* TMP1 = HALF*(D(IBEGIN)+D(IEND)) */ /* L1 = TMP1 - DISC */ /* IF( WL.GE. L1-PIVMIN ) */ /* $ NWL = NWL + 1 */ /* IF( WU.GE. L1-PIVMIN ) */ /* $ NWU = NWU + 1 */ /* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE. */ /* $ L1-PIVMIN ) ) THEN */ /* M = M + 1 */ /* W( M ) = L1 */ /* * The uncertainty of eigenvalues of a 2x2 matrix is very small */ /* WERR( M ) = EPS * ABS( W( M ) ) * TWO */ /* IBLOCK( M ) = JBLK */ /* INDEXW( M ) = 1 */ /* ENDIF */ /* L2 = TMP1 + DISC */ /* IF( WL.GE. L2-PIVMIN ) */ /* $ NWL = NWL + 1 */ /* IF( WU.GE. L2-PIVMIN ) */ /* $ NWU = NWU + 1 */ /* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE. */ /* $ L2-PIVMIN ) ) THEN */ /* M = M + 1 */ /* W( M ) = L2 */ /* * The uncertainty of eigenvalues of a 2x2 matrix is very small */ /* WERR( M ) = EPS * ABS( W( M ) ) * TWO */ /* IBLOCK( M ) = JBLK */ /* INDEXW( M ) = 2 */ /* ENDIF */ } else { /* General Case - block of size IN >= 2 */ /* Compute local Gerschgorin interval and use it as the initial */ /* interval for DLAEBZ */ gu = d__[ibegin]; gl = d__[ibegin]; tmp1 = 0.; i__2 = iend; for (j = ibegin; j <= i__2; ++j) { /* Computing MIN */ d__1 = gl, d__2 = gers[(j << 1) - 1]; gl = f2cmin(d__1,d__2); /* Computing MAX */ d__1 = gu, d__2 = gers[j * 2]; gu = f2cmax(d__1,d__2); /* L40: */ } /* [JAN/28/2009] */ /* change SPDIAM by TNORM in lines 2 and 3 thereafter */ /* line 1: remove computation of SPDIAM (not useful anymore) */ /* SPDIAM = GU - GL */ /* GL = GL - FUDGE*SPDIAM*EPS*IN - FUDGE*PIVMIN */ /* GU = GU + FUDGE*SPDIAM*EPS*IN + FUDGE*PIVMIN */ gl = gl - tnorm * 2. * eps * in - *pivmin * 2.; gu = gu + tnorm * 2. * eps * in + *pivmin * 2.; if (irange > 1) { if (gu < *wl) { /* the local block contains none of the wanted eigenvalues */ nwl += in; nwu += in; goto L70; } /* refine search interval if possible, only range (WL,WU] matters */ gl = f2cmax(gl,*wl); gu = f2cmin(gu,*wu); if (gl >= gu) { goto L70; } } /* Find negcount of initial interval boundaries GL and GU */ work[*n + 1] = gl; work[*n + in + 1] = gu; dlaebz_(&c__1, &c__0, &in, &in, &c__1, &nb, &atoli, &rtoli, pivmin, &d__[ibegin], &e[ibegin], &e2[ibegin], idumma, & work[*n + 1], &work[*n + (in << 1) + 1], &im, &iwork[1], & w[*m + 1], &iblock[*m + 1], &iinfo); if (iinfo != 0) { *info = iinfo; return 0; } nwl += iwork[1]; nwu += iwork[in + 1]; iwoff = *m - iwork[1]; /* Compute Eigenvalues */ itmax = (integer) ((log(gu - gl + *pivmin) - log(*pivmin)) / log( 2.)) + 2; dlaebz_(&c__2, &itmax, &in, &in, &c__1, &nb, &atoli, &rtoli, pivmin, &d__[ibegin], &e[ibegin], &e2[ibegin], idumma, & work[*n + 1], &work[*n + (in << 1) + 1], &iout, &iwork[1], &w[*m + 1], &iblock[*m + 1], &iinfo); if (iinfo != 0) { *info = iinfo; return 0; } /* Copy eigenvalues into W and IBLOCK */ /* Use -JBLK for block number for unconverged eigenvalues. */ /* Loop over the number of output intervals from DLAEBZ */ i__2 = iout; for (j = 1; j <= i__2; ++j) { /* eigenvalue approximation is middle point of interval */ tmp1 = (work[j + *n] + work[j + in + *n]) * .5; /* semi length of error interval */ tmp2 = (d__1 = work[j + *n] - work[j + in + *n], abs(d__1)) * .5; if (j > iout - iinfo) { /* Flag non-convergence. */ ncnvrg = TRUE_; ib = -jblk; } else { ib = jblk; } i__3 = iwork[j + in] + iwoff; for (je = iwork[j] + 1 + iwoff; je <= i__3; ++je) { w[je] = tmp1; werr[je] = tmp2; indexw[je] = je - iwoff; iblock[je] = ib; /* L50: */ } /* L60: */ } *m += im; } L70: ; } /* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU */ /* If NWL+1 < IL or NWU > IU, discard extra eigenvalues. */ if (irange == 3) { idiscl = *il - 1 - nwl; idiscu = nwu - *iu; if (idiscl > 0) { im = 0; i__1 = *m; for (je = 1; je <= i__1; ++je) { /* Remove some of the smallest eigenvalues from the left so that */ /* at the end IDISCL =0. Move all eigenvalues up to the left. */ if (w[je] <= wlu && idiscl > 0) { --idiscl; } else { ++im; w[im] = w[je]; werr[im] = werr[je]; indexw[im] = indexw[je]; iblock[im] = iblock[je]; } /* L80: */ } *m = im; } if (idiscu > 0) { /* Remove some of the largest eigenvalues from the right so that */ /* at the end IDISCU =0. Move all eigenvalues up to the left. */ im = *m + 1; for (je = *m; je >= 1; --je) { if (w[je] >= wul && idiscu > 0) { --idiscu; } else { --im; w[im] = w[je]; werr[im] = werr[je]; indexw[im] = indexw[je]; iblock[im] = iblock[je]; } /* L81: */ } jee = 0; i__1 = *m; for (je = im; je <= i__1; ++je) { ++jee; w[jee] = w[je]; werr[jee] = werr[je]; indexw[jee] = indexw[je]; iblock[jee] = iblock[je]; /* L82: */ } *m = *m - im + 1; } if (idiscl > 0 || idiscu > 0) { /* Code to deal with effects of bad arithmetic. (If N(w) is */ /* monotone non-decreasing, this should never happen.) */ /* Some low eigenvalues to be discarded are not in (WL,WLU], */ /* or high eigenvalues to be discarded are not in (WUL,WU] */ /* so just kill off the smallest IDISCL/largest IDISCU */ /* eigenvalues, by marking the corresponding IBLOCK = 0 */ if (idiscl > 0) { wkill = *wu; i__1 = idiscl; for (jdisc = 1; jdisc <= i__1; ++jdisc) { iw = 0; i__2 = *m; for (je = 1; je <= i__2; ++je) { if (iblock[je] != 0 && (w[je] < wkill || iw == 0)) { iw = je; wkill = w[je]; } /* L90: */ } iblock[iw] = 0; /* L100: */ } } if (idiscu > 0) { wkill = *wl; i__1 = idiscu; for (jdisc = 1; jdisc <= i__1; ++jdisc) { iw = 0; i__2 = *m; for (je = 1; je <= i__2; ++je) { if (iblock[je] != 0 && (w[je] >= wkill || iw == 0)) { iw = je; wkill = w[je]; } /* L110: */ } iblock[iw] = 0; /* L120: */ } } /* Now erase all eigenvalues with IBLOCK set to zero */ im = 0; i__1 = *m; for (je = 1; je <= i__1; ++je) { if (iblock[je] != 0) { ++im; w[im] = w[je]; werr[im] = werr[je]; indexw[im] = indexw[je]; iblock[im] = iblock[je]; } /* L130: */ } *m = im; } if (idiscl < 0 || idiscu < 0) { toofew = TRUE_; } } if (irange == 1 && *m != *n || irange == 3 && *m != *iu - *il + 1) { toofew = TRUE_; } /* If ORDER='B', do nothing the eigenvalues are already sorted by */ /* block. */ /* If ORDER='E', sort the eigenvalues from smallest to largest */ if (lsame_(order, "E") && *nsplit > 1) { i__1 = *m - 1; for (je = 1; je <= i__1; ++je) { ie = 0; tmp1 = w[je]; i__2 = *m; for (j = je + 1; j <= i__2; ++j) { if (w[j] < tmp1) { ie = j; tmp1 = w[j]; } /* L140: */ } if (ie != 0) { tmp2 = werr[ie]; itmp1 = iblock[ie]; itmp2 = indexw[ie]; w[ie] = w[je]; werr[ie] = werr[je]; iblock[ie] = iblock[je]; indexw[ie] = indexw[je]; w[je] = tmp1; werr[je] = tmp2; iblock[je] = itmp1; indexw[je] = itmp2; } /* L150: */ } } *info = 0; if (ncnvrg) { ++(*info); } if (toofew) { *info += 2; } return 0; /* End of DLARRD */ } /* dlarrd_ */