#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 blasint logical; 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) {float nenn=crealf(_FCmulcc(Cf(b),conjf(Cf(b)))); _Fcomplex zaehl=_FCmulcc(Cf(a),conjf(Cf(b))); pCf(c)=_FCbuild(crealf(zaehl)/nenn,cimagf(zaehl)/nenn);} #define z_div(c, a, b) {double nenn=creal(_Cmulcc(Cd(b),conj(Cd(b)))); _Dcomplex zaehl=_Cmulcc(Cd(a),conj(Cd(b))); pCd(c)=_Cbuild(creal(zaehl)/nenn,cimag(zaehl)/nenn);} #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);} #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++ */ #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 CLARFT forms the triangular factor T of a block reflector H = I - vtvH */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > Download CLARFT + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* Definition: */ /* =========== */ /* SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) */ /* CHARACTER DIRECT, STOREV */ /* INTEGER K, LDT, LDV, N */ /* COMPLEX T( LDT, * ), TAU( * ), V( LDV, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > CLARFT forms the triangular factor T of a complex block reflector H */ /* > of order n, which is defined as a product of k elementary reflectors. */ /* > */ /* > If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; */ /* > */ /* > If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. */ /* > */ /* > If STOREV = 'C', the vector which defines the elementary reflector */ /* > H(i) is stored in the i-th column of the array V, and */ /* > */ /* > H = I - V * T * V**H */ /* > */ /* > If STOREV = 'R', the vector which defines the elementary reflector */ /* > H(i) is stored in the i-th row of the array V, and */ /* > */ /* > H = I - V**H * T * V */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] DIRECT */ /* > \verbatim */ /* > DIRECT is CHARACTER*1 */ /* > Specifies the order in which the elementary reflectors are */ /* > multiplied to form the block reflector: */ /* > = 'F': H = H(1) H(2) . . . H(k) (Forward) */ /* > = 'B': H = H(k) . . . H(2) H(1) (Backward) */ /* > \endverbatim */ /* > */ /* > \param[in] STOREV */ /* > \verbatim */ /* > STOREV is CHARACTER*1 */ /* > Specifies how the vectors which define the elementary */ /* > reflectors are stored (see also Further Details): */ /* > = 'C': columnwise */ /* > = 'R': rowwise */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the block reflector H. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] K */ /* > \verbatim */ /* > K is INTEGER */ /* > The order of the triangular factor T (= the number of */ /* > elementary reflectors). K >= 1. */ /* > \endverbatim */ /* > */ /* > \param[in] V */ /* > \verbatim */ /* > V is COMPLEX array, dimension */ /* > (LDV,K) if STOREV = 'C' */ /* > (LDV,N) if STOREV = 'R' */ /* > The matrix V. See further details. */ /* > \endverbatim */ /* > */ /* > \param[in] LDV */ /* > \verbatim */ /* > LDV is INTEGER */ /* > The leading dimension of the array V. */ /* > If STOREV = 'C', LDV >= f2cmax(1,N); if STOREV = 'R', LDV >= K. */ /* > \endverbatim */ /* > */ /* > \param[in] TAU */ /* > \verbatim */ /* > TAU is COMPLEX array, dimension (K) */ /* > TAU(i) must contain the scalar factor of the elementary */ /* > reflector H(i). */ /* > \endverbatim */ /* > */ /* > \param[out] T */ /* > \verbatim */ /* > T is COMPLEX array, dimension (LDT,K) */ /* > The k by k triangular factor T of the block reflector. */ /* > If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is */ /* > lower triangular. The rest of the array is not used. */ /* > \endverbatim */ /* > */ /* > \param[in] LDT */ /* > \verbatim */ /* > LDT is INTEGER */ /* > The leading dimension of the array T. LDT >= K. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \ingroup larft */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > The shape of the matrix V and the storage of the vectors which define */ /* > the H(i) is best illustrated by the following example with n = 5 and */ /* > k = 3. The elements equal to 1 are not stored. */ /* > */ /* > DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': */ /* > */ /* > V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) */ /* > ( v1 1 ) ( 1 v2 v2 v2 ) */ /* > ( v1 v2 1 ) ( 1 v3 v3 ) */ /* > ( v1 v2 v3 ) */ /* > ( v1 v2 v3 ) */ /* > */ /* > DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': */ /* > */ /* > V = ( v1 v2 v3 ) V = ( v1 v1 1 ) */ /* > ( v1 v2 v3 ) ( v2 v2 v2 1 ) */ /* > ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) */ /* > ( 1 v3 ) */ /* > ( 1 ) */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int clarft_(char *direct, char *storev, integer *n, integer * k, complex *v, integer *ldv, complex *tau, complex *t, integer *ldt) { /* System generated locals */ address a__1[2]; integer t_dim1, t_offset, v_dim1, v_offset, i__1, i__2[2], i__3, i__4; complex q__1; char ch__1[2]; /* Local variables */ integer i__, j, l; logical lq, ql, qr; integer nx; extern /* Subroutine */ int clarft_lvl2__(char *, char *, integer *, integer *, complex *, integer *, complex *, complex *, integer *); logical dirf, colv; extern /* Subroutine */ int cgemm_(char *, char *, integer *, integer *, integer *, complex *, complex *, integer *, complex *, integer *, complex *, complex *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int ctrmm_(char *, char *, char *, char *, integer *, integer *, complex *, complex *, integer *, complex *, integer *), clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); /* -- LAPACK auxiliary routine -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* The general scheme used is inspired by the approach inside DGEQRT3 */ /* which was (at the time of writing this code): */ /* Based on the algorithm of Elmroth and Gustavson, */ /* IBM J. Res. Develop. Vol 44 No. 4 July 2000. */ /* Quick return if possible */ /* Parameter adjustments */ v_dim1 = *ldv; v_offset = 1 + v_dim1; v -= v_offset; --tau; t_dim1 = *ldt; t_offset = 1 + t_dim1; t -= t_offset; /* Function Body */ if (*n == 0 || *k == 0) { return 0; } /* Base case */ if (*n == 1 || *k == 1) { i__1 = t_dim1 + 1; t[i__1].r = tau[1].r, t[i__1].i = tau[1].i; return 0; } /* Determine when to cross over into the level 2 based implementation */ /* Writing concatenation */ i__2[0] = 1, a__1[0] = direct; i__2[1] = 1, a__1[1] = storev; s_cat(ch__1, a__1, i__2, &c__2, (ftnlen)2); nx = ilaenv_(&c__3, "CLARFT", ch__1, n, k, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)2); if (*k < nx) { clarft_lvl2__(direct, storev, n, k, &v[v_offset], ldv, &tau[1], &t[ t_offset], ldt); return 0; } /* Beginning of executable statements */ l = *k / 2; /* Determine what kind of Q we need to compute */ /* We assume that if the user doesn't provide 'F' for DIRECT, */ /* then they meant to provide 'B' and if they don't provide */ /* 'C' for STOREV, then they meant to provide 'R' */ dirf = lsame_(direct, "F"); colv = lsame_(storev, "C"); /* QR happens when we have forward direction in column storage */ qr = dirf && colv; /* LQ happens when we have forward direction in row storage */ lq = dirf && ! colv; /* QL happens when we have backward direction in column storage */ ql = ! dirf && colv; /* The last case is RQ. Due to how we structured this, if the */ /* above 3 are false, then RQ must be true, so we never store */ /* this */ /* RQ happens when we have backward direction in row storage */ /* RQ = (.NOT.DIRF).AND.(.NOT.COLV) */ if (qr) { /* Break V apart into 6 components */ /* V = |---------------| */ /* |V_{1,1} 0 | */ /* |V_{2,1} V_{2,2}| */ /* |V_{3,1} V_{3,2}| */ /* |---------------| */ /* V_{1,1}\in\C^{l,l} unit lower triangular */ /* V_{2,1}\in\C^{k-l,l} rectangular */ /* V_{3,1}\in\C^{n-k,l} rectangular */ /* V_{2,2}\in\C^{k-l,k-l} unit lower triangular */ /* V_{3,2}\in\C^{n-k,k-l} rectangular */ /* We will construct the T matrix */ /* T = |---------------| */ /* |T_{1,1} T_{1,2}| */ /* |0 T_{2,2}| */ /* |---------------| */ /* T is the triangular factor obtained from block reflectors. */ /* To motivate the structure, assume we have already computed T_{1,1} */ /* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2 */ /* T_{1,1}\in\C^{l, l} upper triangular */ /* T_{2,2}\in\C^{k-l, k-l} upper triangular */ /* T_{1,2}\in\C^{l, k-l} rectangular */ /* Where l = floor(k/2) */ /* Then, consider the product: */ /* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2') */ /* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2' */ /* Define T{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2} */ /* Then, we can define the matrix V as */ /* V = |-------| */ /* |V_1 V_2| */ /* |-------| */ /* So, our product is equivalent to the matrix product */ /* I - V*T*V' */ /* This means, we can compute T_{1,1} and T_{2,2}, then use this information */ /* to compute T_{1,2} */ /* Compute T_{1,1} recursively */ clarft_(direct, storev, n, &l, &v[v_offset], ldv, &tau[1], &t[ t_offset], ldt); /* Compute T_{2,2} recursively */ i__1 = *n - l; i__3 = *k - l; clarft_(direct, storev, &i__1, &i__3, &v[l + 1 + (l + 1) * v_dim1], ldv, &tau[l + 1], &t[l + 1 + (l + 1) * t_dim1], ldt); /* Compute T_{1,2} */ /* T_{1,2} = V_{2,1}' */ i__1 = l; for (j = 1; j <= i__1; ++j) { i__3 = *k - l; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = j + (l + i__) * t_dim1; r_cnjg(&q__1, &v[l + i__ + j * v_dim1]); t[i__4].r = q__1.r, t[i__4].i = q__1.i; } } /* T_{1,2} = T_{1,2}*V_{2,2} */ i__1 = *k - l; ctrmm_("Right", "Lower", "No transpose", "Unit", &l, &i__1, &c_b1, &v[ l + 1 + (l + 1) * v_dim1], ldv, &t[(l + 1) * t_dim1 + 1], ldt); /* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2} */ /* Note: We assume K <= N, and GEMM will do nothing if N=K */ i__1 = *k - l; i__3 = *n - *k; cgemm_("Conjugate", "No transpose", &l, &i__1, &i__3, &c_b1, &v[*k + 1 + v_dim1], ldv, &v[*k + 1 + (l + 1) * v_dim1], ldv, &c_b1, & t[(l + 1) * t_dim1 + 1], ldt); /* At this point, we have that T_{1,2} = V_1'*V_2 */ /* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2} */ /* respectively. */ /* T_{1,2} = -T_{1,1}*T_{1,2} */ i__1 = *k - l; ctrmm_("Left", "Upper", "No transpose", "Non-unit", &l, &i__1, &c_b3, &t[t_offset], ldt, &t[(l + 1) * t_dim1 + 1], ldt); /* T_{1,2} = T_{1,2}*T_{2,2} */ i__1 = *k - l; ctrmm_("Right", "Upper", "No transpose", "Non-unit", &l, &i__1, &c_b1, &t[l + 1 + (l + 1) * t_dim1], ldt, &t[(l + 1) * t_dim1 + 1], ldt); } else if (lq) { /* Break V apart into 6 components */ /* V = |----------------------| */ /* |V_{1,1} V_{1,2} V{1,3}| */ /* |0 V_{2,2} V{2,3}| */ /* |----------------------| */ /* V_{1,1}\in\C^{l,l} unit upper triangular */ /* V_{1,2}\in\C^{l,k-l} rectangular */ /* V_{1,3}\in\C^{l,n-k} rectangular */ /* V_{2,2}\in\C^{k-l,k-l} unit upper triangular */ /* V_{2,3}\in\C^{k-l,n-k} rectangular */ /* Where l = floor(k/2) */ /* We will construct the T matrix */ /* T = |---------------| */ /* |T_{1,1} T_{1,2}| */ /* |0 T_{2,2}| */ /* |---------------| */ /* T is the triangular factor obtained from block reflectors. */ /* To motivate the structure, assume we have already computed T_{1,1} */ /* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2 */ /* T_{1,1}\in\C^{l, l} upper triangular */ /* T_{2,2}\in\C^{k-l, k-l} upper triangular */ /* T_{1,2}\in\C^{l, k-l} rectangular */ /* Then, consider the product: */ /* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2) */ /* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2 */ /* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2} */ /* Then, we can define the matrix V as */ /* V = |---| */ /* |V_1| */ /* |V_2| */ /* |---| */ /* So, our product is equivalent to the matrix product */ /* I - V'*T*V */ /* This means, we can compute T_{1,1} and T_{2,2}, then use this information */ /* to compute T_{1,2} */ /* Compute T_{1,1} recursively */ clarft_(direct, storev, n, &l, &v[v_offset], ldv, &tau[1], &t[ t_offset], ldt); /* Compute T_{2,2} recursively */ i__1 = *n - l; i__3 = *k - l; clarft_(direct, storev, &i__1, &i__3, &v[l + 1 + (l + 1) * v_dim1], ldv, &tau[l + 1], &t[l + 1 + (l + 1) * t_dim1], ldt); /* Compute T_{1,2} */ /* T_{1,2} = V_{1,2} */ i__1 = *k - l; clacpy_("All", &l, &i__1, &v[(l + 1) * v_dim1 + 1], ldv, &t[(l + 1) * t_dim1 + 1], ldt); /* T_{1,2} = T_{1,2}*V_{2,2}' */ i__1 = *k - l; ctrmm_("Right", "Upper", "Conjugate", "Unit", &l, &i__1, &c_b1, &v[l + 1 + (l + 1) * v_dim1], ldv, &t[(l + 1) * t_dim1 + 1], ldt); /* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2} */ /* Note: We assume K <= N, and GEMM will do nothing if N=K */ i__1 = *k - l; i__3 = *n - *k; cgemm_("No transpose", "Conjugate", &l, &i__1, &i__3, &c_b1, &v[(*k + 1) * v_dim1 + 1], ldv, &v[l + 1 + (*k + 1) * v_dim1], ldv, & c_b1, &t[(l + 1) * t_dim1 + 1], ldt); /* At this point, we have that T_{1,2} = V_1*V_2' */ /* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2} */ /* respectively. */ /* T_{1,2} = -T_{1,1}*T_{1,2} */ i__1 = *k - l; ctrmm_("Left", "Upper", "No transpose", "Non-unit", &l, &i__1, &c_b3, &t[t_offset], ldt, &t[(l + 1) * t_dim1 + 1], ldt); /* T_{1,2} = T_{1,2}*T_{2,2} */ i__1 = *k - l; ctrmm_("Right", "Upper", "No transpose", "Non-unit", &l, &i__1, &c_b1, &t[l + 1 + (l + 1) * t_dim1], ldt, &t[(l + 1) * t_dim1 + 1], ldt); } else if (ql) { /* Break V apart into 6 components */ /* V = |---------------| */ /* |V_{1,1} V_{1,2}| */ /* |V_{2,1} V_{2,2}| */ /* |0 V_{3,2}| */ /* |---------------| */ /* V_{1,1}\in\C^{n-k,k-l} rectangular */ /* V_{2,1}\in\C^{k-l,k-l} unit upper triangular */ /* V_{1,2}\in\C^{n-k,l} rectangular */ /* V_{2,2}\in\C^{k-l,l} rectangular */ /* V_{3,2}\in\C^{l,l} unit upper triangular */ /* We will construct the T matrix */ /* T = |---------------| */ /* |T_{1,1} 0 | */ /* |T_{2,1} T_{2,2}| */ /* |---------------| */ /* T is the triangular factor obtained from block reflectors. */ /* To motivate the structure, assume we have already computed T_{1,1} */ /* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2 */ /* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular */ /* T_{2,2}\in\C^{l, l} non-unit lower triangular */ /* T_{2,1}\in\C^{k-l, l} rectangular */ /* Where l = floor(k/2) */ /* Then, consider the product: */ /* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1') */ /* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1' */ /* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1} */ /* Then, we can define the matrix V as */ /* V = |-------| */ /* |V_1 V_2| */ /* |-------| */ /* So, our product is equivalent to the matrix product */ /* I - V*T*V' */ /* This means, we can compute T_{1,1} and T_{2,2}, then use this information */ /* to compute T_{2,1} */ /* Compute T_{1,1} recursively */ i__1 = *n - l; i__3 = *k - l; clarft_(direct, storev, &i__1, &i__3, &v[v_offset], ldv, &tau[1], &t[ t_offset], ldt); /* Compute T_{2,2} recursively */ clarft_(direct, storev, n, &l, &v[(*k - l + 1) * v_dim1 + 1], ldv, & tau[*k - l + 1], &t[*k - l + 1 + (*k - l + 1) * t_dim1], ldt); /* Compute T_{2,1} */ /* T_{2,1} = V_{2,2}' */ i__1 = *k - l; for (j = 1; j <= i__1; ++j) { i__3 = l; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = *k - l + i__ + j * t_dim1; r_cnjg(&q__1, &v[*n - *k + j + (*k - l + i__) * v_dim1]); t[i__4].r = q__1.r, t[i__4].i = q__1.i; } } /* T_{2,1} = T_{2,1}*V_{2,1} */ i__1 = *k - l; ctrmm_("Right", "Upper", "No transpose", "Unit", &l, &i__1, &c_b1, &v[ *n - *k + 1 + v_dim1], ldv, &t[*k - l + 1 + t_dim1], ldt); /* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1} */ /* Note: We assume K <= N, and GEMM will do nothing if N=K */ i__1 = *k - l; i__3 = *n - *k; cgemm_("Conjugate", "No transpose", &l, &i__1, &i__3, &c_b1, &v[(*k - l + 1) * v_dim1 + 1], ldv, &v[v_offset], ldv, &c_b1, &t[*k - l + 1 + t_dim1], ldt); /* At this point, we have that T_{2,1} = V_2'*V_1 */ /* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1} */ /* respectively. */ /* T_{2,1} = -T_{2,2}*T_{2,1} */ i__1 = *k - l; ctrmm_("Left", "Lower", "No transpose", "Non-unit", &l, &i__1, &c_b3, &t[*k - l + 1 + (*k - l + 1) * t_dim1], ldt, &t[*k - l + 1 + t_dim1], ldt); /* T_{2,1} = T_{2,1}*T_{1,1} */ i__1 = *k - l; ctrmm_("Right", "Lower", "No transpose", "Non-unit", &l, &i__1, &c_b1, &t[t_offset], ldt, &t[*k - l + 1 + t_dim1], ldt); } else { /* Else means RQ case */ /* Break V apart into 6 components */ /* V = |-----------------------| */ /* |V_{1,1} V_{1,2} 0 | */ /* |V_{2,1} V_{2,2} V_{2,3}| */ /* |-----------------------| */ /* V_{1,1}\in\C^{k-l,n-k} rectangular */ /* V_{1,2}\in\C^{k-l,k-l} unit lower triangular */ /* V_{2,1}\in\C^{l,n-k} rectangular */ /* V_{2,2}\in\C^{l,k-l} rectangular */ /* V_{2,3}\in\C^{l,l} unit lower triangular */ /* We will construct the T matrix */ /* T = |---------------| */ /* |T_{1,1} 0 | */ /* |T_{2,1} T_{2,2}| */ /* |---------------| */ /* T is the triangular factor obtained from block reflectors. */ /* To motivate the structure, assume we have already computed T_{1,1} */ /* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2 */ /* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular */ /* T_{2,2}\in\C^{l, l} non-unit lower triangular */ /* T_{2,1}\in\C^{k-l, l} rectangular */ /* Where l = floor(k/2) */ /* Then, consider the product: */ /* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1) */ /* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1 */ /* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1} */ /* Then, we can define the matrix V as */ /* V = |---| */ /* |V_1| */ /* |V_2| */ /* |---| */ /* So, our product is equivalent to the matrix product */ /* I - V'*T*V */ /* This means, we can compute T_{1,1} and T_{2,2}, then use this information */ /* to compute T_{2,1} */ /* Compute T_{1,1} recursively */ i__1 = *n - l; i__3 = *k - l; clarft_(direct, storev, &i__1, &i__3, &v[v_offset], ldv, &tau[1], &t[ t_offset], ldt); /* Compute T_{2,2} recursively */ clarft_(direct, storev, n, &l, &v[*k - l + 1 + v_dim1], ldv, &tau[*k - l + 1], &t[*k - l + 1 + (*k - l + 1) * t_dim1], ldt); /* Compute T_{2,1} */ /* T_{2,1} = V_{2,2} */ i__1 = *k - l; clacpy_("All", &l, &i__1, &v[*k - l + 1 + (*n - *k + 1) * v_dim1], ldv, &t[*k - l + 1 + t_dim1], ldt); /* T_{2,1} = T_{2,1}*V_{1,2}' */ i__1 = *k - l; ctrmm_("Right", "Lower", "Conjugate", "Unit", &l, &i__1, &c_b1, &v[(* n - *k + 1) * v_dim1 + 1], ldv, &t[*k - l + 1 + t_dim1], ldt); /* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1} */ /* Note: We assume K <= N, and GEMM will do nothing if N=K */ i__1 = *k - l; i__3 = *n - *k; cgemm_("No transpose", "Conjugate", &l, &i__1, &i__3, &c_b1, &v[*k - l + 1 + v_dim1], ldv, &v[v_offset], ldv, &c_b1, &t[*k - l + 1 + t_dim1], ldt); /* At this point, we have that T_{2,1} = V_2*V_1' */ /* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1} */ /* respectively. */ /* T_{2,1} = -T_{2,2}*T_{2,1} */ i__1 = *k - l; ctrmm_("Left", "Lower", "No tranpose", "Non-unit", &l, &i__1, &c_b3, & t[*k - l + 1 + (*k - l + 1) * t_dim1], ldt, &t[*k - l + 1 + t_dim1], ldt); /* T_{2,1} = T_{2,1}*T_{1,1} */ i__1 = *k - l; ctrmm_("Right", "Lower", "No tranpose", "Non-unit", &l, &i__1, &c_b1, &t[t_offset], ldt, &t[*k - l + 1 + t_dim1], ldt); } return 0; } /* clarft_ */