#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 DGELQ */ /* Definition: */ /* =========== */ /* SUBROUTINE DGELQ( M, N, A, LDA, T, TSIZE, WORK, LWORK, */ /* INFO ) */ /* INTEGER INFO, LDA, M, N, TSIZE, LWORK */ /* DOUBLE PRECISION A( LDA, * ), T( * ), WORK( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DGELQ computes an LQ factorization of a real M-by-N matrix A: */ /* > */ /* > A = ( L 0 ) * Q */ /* > */ /* > where: */ /* > */ /* > Q is a N-by-N orthogonal matrix; */ /* > L is a lower-triangular M-by-M matrix; */ /* > 0 is a M-by-(N-M) zero matrix, if M < N. */ /* > */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] M */ /* > \verbatim */ /* > M is INTEGER */ /* > The number of rows of the matrix A. M >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The number of columns of the matrix A. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] A */ /* > \verbatim */ /* > A is DOUBLE PRECISION array, dimension (LDA,N) */ /* > On entry, the M-by-N matrix A. */ /* > On exit, the elements on and below the diagonal of the array */ /* > contain the M-by-f2cmin(M,N) lower trapezoidal matrix L */ /* > (L is lower triangular if M <= N); */ /* > the elements above the diagonal are used to store part of the */ /* > data structure to represent Q. */ /* > \endverbatim */ /* > */ /* > \param[in] LDA */ /* > \verbatim */ /* > LDA is INTEGER */ /* > The leading dimension of the array A. LDA >= f2cmax(1,M). */ /* > \endverbatim */ /* > */ /* > \param[out] T */ /* > \verbatim */ /* > T is DOUBLE PRECISION array, dimension (MAX(5,TSIZE)) */ /* > On exit, if INFO = 0, T(1) returns optimal (or either minimal */ /* > or optimal, if query is assumed) TSIZE. See TSIZE for details. */ /* > Remaining T contains part of the data structure used to represent Q. */ /* > If one wants to apply or construct Q, then one needs to keep T */ /* > (in addition to A) and pass it to further subroutines. */ /* > \endverbatim */ /* > */ /* > \param[in] TSIZE */ /* > \verbatim */ /* > TSIZE is INTEGER */ /* > If TSIZE >= 5, the dimension of the array T. */ /* > If TSIZE = -1 or -2, then a workspace query is assumed. The routine */ /* > only calculates the sizes of the T and WORK arrays, returns these */ /* > values as the first entries of the T and WORK arrays, and no error */ /* > message related to T or WORK is issued by XERBLA. */ /* > If TSIZE = -1, the routine calculates optimal size of T for the */ /* > optimum performance and returns this value in T(1). */ /* > If TSIZE = -2, the routine calculates minimal size of T and */ /* > returns this value in T(1). */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */ /* > On exit, if INFO = 0, WORK(1) contains optimal (or either minimal */ /* > or optimal, if query was assumed) LWORK. */ /* > See LWORK for details. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > The dimension of the array WORK. */ /* > If LWORK = -1 or -2, then a workspace query is assumed. The routine */ /* > only calculates the sizes of the T and WORK arrays, returns these */ /* > values as the first entries of the T and WORK arrays, and no error */ /* > message related to T or WORK is issued by XERBLA. */ /* > If LWORK = -1, the routine calculates optimal size of WORK for the */ /* > optimal performance and returns this value in WORK(1). */ /* > If LWORK = -2, the routine calculates minimal size of WORK and */ /* > returns this value in WORK(1). */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > < 0: if INFO = -i, the i-th argument had an illegal value */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \par Further Details */ /* ==================== */ /* > */ /* > \verbatim */ /* > */ /* > The goal of the interface is to give maximum freedom to the developers for */ /* > creating any LQ factorization algorithm they wish. The triangular */ /* > (trapezoidal) L has to be stored in the lower part of A. The lower part of A */ /* > and the array T can be used to store any relevant information for applying or */ /* > constructing the Q factor. The WORK array can safely be discarded after exit. */ /* > */ /* > Caution: One should not expect the sizes of T and WORK to be the same from one */ /* > LAPACK implementation to the other, or even from one execution to the other. */ /* > A workspace query (for T and WORK) is needed at each execution. However, */ /* > for a given execution, the size of T and WORK are fixed and will not change */ /* > from one query to the next. */ /* > */ /* > \endverbatim */ /* > */ /* > \par Further Details particular to this LAPACK implementation: */ /* ============================================================== */ /* > */ /* > \verbatim */ /* > */ /* > These details are particular for this LAPACK implementation. Users should not */ /* > take them for granted. These details may change in the future, and are not likely */ /* > true for another LAPACK implementation. These details are relevant if one wants */ /* > to try to understand the code. They are not part of the interface. */ /* > */ /* > In this version, */ /* > */ /* > T(2): row block size (MB) */ /* > T(3): column block size (NB) */ /* > T(6:TSIZE): data structure needed for Q, computed by */ /* > DLASWLQ or DGELQT */ /* > */ /* > Depending on the matrix dimensions M and N, and row and column */ /* > block sizes MB and NB returned by ILAENV, DGELQ will use either */ /* > DLASWLQ (if the matrix is short-and-wide) or DGELQT to compute */ /* > the LQ factorization. */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int dgelq_(integer *m, integer *n, doublereal *a, integer * lda, doublereal *t, integer *tsize, doublereal *work, integer *lwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, i__1, i__2; /* Local variables */ logical mint, minw; integer lwmin, lwreq, lwopt, mb, nb, nblcks; extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int dgelqt_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *); logical lminws, lquery; integer mintsz; extern /* Subroutine */ int dlaswlq_(integer *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); /* -- LAPACK computational routine (version 3.9.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. -- */ /* November 2019 */ /* ===================================================================== */ /* Test the input arguments */ /* Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; --t; --work; /* Function Body */ *info = 0; lquery = *tsize == -1 || *tsize == -2 || *lwork == -1 || *lwork == -2; mint = FALSE_; minw = FALSE_; if (*tsize == -2 || *lwork == -2) { if (*tsize != -1) { mint = TRUE_; } if (*lwork != -1) { minw = TRUE_; } } /* Determine the block size */ if (f2cmin(*m,*n) > 0) { mb = ilaenv_(&c__1, "DGELQ ", " ", m, n, &c__1, &c_n1, (ftnlen)6, ( ftnlen)1); nb = ilaenv_(&c__1, "DGELQ ", " ", m, n, &c__2, &c_n1, (ftnlen)6, ( ftnlen)1); } else { mb = 1; nb = *n; } if (mb > f2cmin(*m,*n) || mb < 1) { mb = 1; } if (nb > *n || nb <= *m) { nb = *n; } mintsz = *m + 5; if (nb > *m && *n > *m) { if ((*n - *m) % (nb - *m) == 0) { nblcks = (*n - *m) / (nb - *m); } else { nblcks = (*n - *m) / (nb - *m) + 1; } } else { nblcks = 1; } /* Determine if the workspace size satisfies minimal size */ if (*n <= *m || nb <= *m || nb >= *n) { lwmin = f2cmax(1,*n); /* Computing MAX */ i__1 = 1, i__2 = mb * *n; lwopt = f2cmax(i__1,i__2); } else { lwmin = f2cmax(1,*m); /* Computing MAX */ i__1 = 1, i__2 = mb * *m; lwopt = f2cmax(i__1,i__2); } lminws = FALSE_; /* Computing MAX */ i__1 = 1, i__2 = mb * *m * nblcks + 5; if ((*tsize < f2cmax(i__1,i__2) || *lwork < lwopt) && *lwork >= lwmin && * tsize >= mintsz && ! lquery) { /* Computing MAX */ i__1 = 1, i__2 = mb * *m * nblcks + 5; if (*tsize < f2cmax(i__1,i__2)) { lminws = TRUE_; mb = 1; nb = *n; } if (*lwork < lwopt) { lminws = TRUE_; mb = 1; } } if (*n <= *m || nb <= *m || nb >= *n) { /* Computing MAX */ i__1 = 1, i__2 = mb * *n; lwreq = f2cmax(i__1,i__2); } else { /* Computing MAX */ i__1 = 1, i__2 = mb * *m; lwreq = f2cmax(i__1,i__2); } if (*m < 0) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*lda < f2cmax(1,*m)) { *info = -4; } else /* if(complicated condition) */ { /* Computing MAX */ i__1 = 1, i__2 = mb * *m * nblcks + 5; if (*tsize < f2cmax(i__1,i__2) && ! lquery && ! lminws) { *info = -6; } else if (*lwork < lwreq && ! lquery && ! lminws) { *info = -8; } } if (*info == 0) { if (mint) { t[1] = (doublereal) mintsz; } else { t[1] = (doublereal) (mb * *m * nblcks + 5); } t[2] = (doublereal) mb; t[3] = (doublereal) nb; if (minw) { work[1] = (doublereal) lwmin; } else { work[1] = (doublereal) lwreq; } } if (*info != 0) { i__1 = -(*info); xerbla_("DGELQ", &i__1, (ftnlen)5); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (f2cmin(*m,*n) == 0) { return 0; } /* The LQ Decomposition */ if (*n <= *m || nb <= *m || nb >= *n) { dgelqt_(m, n, &mb, &a[a_offset], lda, &t[6], &mb, &work[1], info); } else { dlaswlq_(m, n, &mb, &nb, &a[a_offset], lda, &t[6], &mb, &work[1], lwork, info); } work[1] = (doublereal) lwreq; return 0; /* End of DGELQ */ } /* dgelq_ */