14 typedef long long BLASLONG;
15 typedef unsigned long long BLASULONG;
17 typedef long BLASLONG;
18 typedef unsigned long BLASULONG;
22 typedef BLASLONG blasint;
24 #define blasabs(x) llabs(x)
26 #define blasabs(x) labs(x)
30 #define blasabs(x) abs(x)
33 typedef blasint integer;
35 typedef unsigned int uinteger;
36 typedef char *address;
37 typedef short int shortint;
39 typedef double doublereal;
40 typedef struct { real r, i; } complex;
41 typedef struct { doublereal r, i; } doublecomplex;
43 static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
44 static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
45 static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
46 static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
48 static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
49 static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
50 static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
51 static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
53 #define pCf(z) (*_pCf(z))
54 #define pCd(z) (*_pCd(z))
56 typedef short int shortlogical;
57 typedef char logical1;
58 typedef char integer1;
63 /* Extern is for use with -E */
74 /*external read, write*/
83 /*internal read, write*/
113 /*rewind, backspace, endfile*/
125 ftnint *inex; /*parameters in standard's order*/
151 union Multitype { /* for multiple entry points */
162 typedef union Multitype Multitype;
164 struct Vardesc { /* for Namelist */
170 typedef struct Vardesc Vardesc;
177 typedef struct Namelist Namelist;
179 #define abs(x) ((x) >= 0 ? (x) : -(x))
180 #define dabs(x) (fabs(x))
181 #define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
182 #define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
183 #define dmin(a,b) (f2cmin(a,b))
184 #define dmax(a,b) (f2cmax(a,b))
185 #define bit_test(a,b) ((a) >> (b) & 1)
186 #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
187 #define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
189 #define abort_() { sig_die("Fortran abort routine called", 1); }
190 #define c_abs(z) (cabsf(Cf(z)))
191 #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
193 #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]);}
194 #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]);}
196 #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
197 #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
199 #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
200 #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
201 #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
202 //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
203 #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
204 #define d_abs(x) (fabs(*(x)))
205 #define d_acos(x) (acos(*(x)))
206 #define d_asin(x) (asin(*(x)))
207 #define d_atan(x) (atan(*(x)))
208 #define d_atn2(x, y) (atan2(*(x),*(y)))
209 #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
210 #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
211 #define d_cos(x) (cos(*(x)))
212 #define d_cosh(x) (cosh(*(x)))
213 #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
214 #define d_exp(x) (exp(*(x)))
215 #define d_imag(z) (cimag(Cd(z)))
216 #define r_imag(z) (cimagf(Cf(z)))
217 #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
218 #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
219 #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
220 #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
221 #define d_log(x) (log(*(x)))
222 #define d_mod(x, y) (fmod(*(x), *(y)))
223 #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
224 #define d_nint(x) u_nint(*(x))
225 #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
226 #define d_sign(a,b) u_sign(*(a),*(b))
227 #define r_sign(a,b) u_sign(*(a),*(b))
228 #define d_sin(x) (sin(*(x)))
229 #define d_sinh(x) (sinh(*(x)))
230 #define d_sqrt(x) (sqrt(*(x)))
231 #define d_tan(x) (tan(*(x)))
232 #define d_tanh(x) (tanh(*(x)))
233 #define i_abs(x) abs(*(x))
234 #define i_dnnt(x) ((integer)u_nint(*(x)))
235 #define i_len(s, n) (n)
236 #define i_nint(x) ((integer)u_nint(*(x)))
237 #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
238 #define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
239 #define pow_si(B,E) spow_ui(*(B),*(E))
240 #define pow_ri(B,E) spow_ui(*(B),*(E))
241 #define pow_di(B,E) dpow_ui(*(B),*(E))
242 #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
243 #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
244 #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
245 #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++ = ' '; }
246 #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
247 #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]; }
248 #define sig_die(s, kill) { exit(1); }
249 #define s_stop(s, n) {exit(0);}
250 static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
251 #define z_abs(z) (cabs(Cd(z)))
252 #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
253 #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
254 #define myexit_() break;
255 #define mycycle() continue;
256 #define myceiling(w) {ceil(w)}
257 #define myhuge(w) {HUGE_VAL}
258 //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
259 #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
261 /* procedure parameter types for -A and -C++ */
263 #define F2C_proc_par_types 1
265 typedef logical (*L_fp)(...);
267 typedef logical (*L_fp)();
270 static float spow_ui(float x, integer n) {
271 float pow=1.0; unsigned long int u;
273 if(n < 0) n = -n, x = 1/x;
282 static double dpow_ui(double x, integer n) {
283 double pow=1.0; unsigned long int u;
285 if(n < 0) n = -n, x = 1/x;
295 static _Fcomplex cpow_ui(complex x, integer n) {
296 complex pow={1.0,0.0}; unsigned long int u;
298 if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
300 if(u & 01) pow.r *= x.r, pow.i *= x.i;
301 if(u >>= 1) x.r *= x.r, x.i *= x.i;
305 _Fcomplex p={pow.r, pow.i};
309 static _Complex float cpow_ui(_Complex float x, integer n) {
310 _Complex float pow=1.0; unsigned long int u;
312 if(n < 0) n = -n, x = 1/x;
323 static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
324 _Dcomplex pow={1.0,0.0}; unsigned long int u;
326 if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
328 if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
329 if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
333 _Dcomplex p = {pow._Val[0], pow._Val[1]};
337 static _Complex double zpow_ui(_Complex double x, integer n) {
338 _Complex double pow=1.0; unsigned long int u;
340 if(n < 0) n = -n, x = 1/x;
350 static integer pow_ii(integer x, integer n) {
351 integer pow; unsigned long int u;
353 if (n == 0 || x == 1) pow = 1;
354 else if (x != -1) pow = x == 0 ? 1/x : 0;
357 if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
367 static integer dmaxloc_(double *w, integer s, integer e, integer *n)
369 double m; integer i, mi;
370 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
371 if (w[i-1]>m) mi=i ,m=w[i-1];
374 static integer smaxloc_(float *w, integer s, integer e, integer *n)
376 float m; integer i, mi;
377 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
378 if (w[i-1]>m) mi=i ,m=w[i-1];
381 static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
382 integer n = *n_, incx = *incx_, incy = *incy_, i;
384 _Fcomplex zdotc = {0.0, 0.0};
385 if (incx == 1 && incy == 1) {
386 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
387 zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
388 zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
391 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
392 zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
393 zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
399 _Complex float zdotc = 0.0;
400 if (incx == 1 && incy == 1) {
401 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
402 zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
405 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
406 zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
412 static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
413 integer n = *n_, incx = *incx_, incy = *incy_, i;
415 _Dcomplex zdotc = {0.0, 0.0};
416 if (incx == 1 && incy == 1) {
417 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
418 zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
419 zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
422 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
423 zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
424 zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
430 _Complex double zdotc = 0.0;
431 if (incx == 1 && incy == 1) {
432 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
433 zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
436 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
437 zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
443 static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
444 integer n = *n_, incx = *incx_, incy = *incy_, i;
446 _Fcomplex zdotc = {0.0, 0.0};
447 if (incx == 1 && incy == 1) {
448 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
449 zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
450 zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
453 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
454 zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
455 zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
461 _Complex float zdotc = 0.0;
462 if (incx == 1 && incy == 1) {
463 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
464 zdotc += Cf(&x[i]) * Cf(&y[i]);
467 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
468 zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
474 static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
475 integer n = *n_, incx = *incx_, incy = *incy_, i;
477 _Dcomplex zdotc = {0.0, 0.0};
478 if (incx == 1 && incy == 1) {
479 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
480 zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
481 zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
484 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
485 zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
486 zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
492 _Complex double zdotc = 0.0;
493 if (incx == 1 && incy == 1) {
494 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
495 zdotc += Cd(&x[i]) * Cd(&y[i]);
498 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
499 zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
505 /* -- translated by f2c (version 20000121).
506 You must link the resulting object file with the libraries:
507 -lf2c -lm (in that order)
513 /* Table of constant values */
515 static integer c__1 = 1;
517 /* > \brief \b SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bu
518 nch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). */
520 /* =========== DOCUMENTATION =========== */
522 /* Online html documentation available at */
523 /* http://www.netlib.org/lapack/explore-html/ */
526 /* > Download SSYTF2_RK + dependencies */
527 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssytf2_
530 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssytf2_
533 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssytf2_
541 /* SUBROUTINE SSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO ) */
544 /* INTEGER INFO, LDA, N */
545 /* INTEGER IPIV( * ) */
546 /* REAL A( LDA, * ), E ( * ) */
549 /* > \par Purpose: */
553 /* > SSYTF2_RK computes the factorization of a real symmetric matrix A */
554 /* > using the bounded Bunch-Kaufman (rook) diagonal pivoting method: */
556 /* > A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T), */
558 /* > where U (or L) is unit upper (or lower) triangular matrix, */
559 /* > U**T (or L**T) is the transpose of U (or L), P is a permutation */
560 /* > matrix, P**T is the transpose of P, and D is symmetric and block */
561 /* > diagonal with 1-by-1 and 2-by-2 diagonal blocks. */
563 /* > This is the unblocked version of the algorithm, calling Level 2 BLAS. */
564 /* > For more information see Further Details section. */
570 /* > \param[in] UPLO */
572 /* > UPLO is CHARACTER*1 */
573 /* > Specifies whether the upper or lower triangular part of the */
574 /* > symmetric matrix A is stored: */
575 /* > = 'U': Upper triangular */
576 /* > = 'L': Lower triangular */
582 /* > The order of the matrix A. N >= 0. */
585 /* > \param[in,out] A */
587 /* > A is REAL array, dimension (LDA,N) */
588 /* > On entry, the symmetric matrix A. */
589 /* > If UPLO = 'U': the leading N-by-N upper triangular part */
590 /* > of A contains the upper triangular part of the matrix A, */
591 /* > and the strictly lower triangular part of A is not */
594 /* > If UPLO = 'L': the leading N-by-N lower triangular part */
595 /* > of A contains the lower triangular part of the matrix A, */
596 /* > and the strictly upper triangular part of A is not */
599 /* > On exit, contains: */
600 /* > a) ONLY diagonal elements of the symmetric block diagonal */
601 /* > matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); */
602 /* > (superdiagonal (or subdiagonal) elements of D */
603 /* > are stored on exit in array E), and */
604 /* > b) If UPLO = 'U': factor U in the superdiagonal part of A. */
605 /* > If UPLO = 'L': factor L in the subdiagonal part of A. */
608 /* > \param[in] LDA */
610 /* > LDA is INTEGER */
611 /* > The leading dimension of the array A. LDA >= f2cmax(1,N). */
614 /* > \param[out] E */
616 /* > E is REAL array, dimension (N) */
617 /* > On exit, contains the superdiagonal (or subdiagonal) */
618 /* > elements of the symmetric block diagonal matrix D */
619 /* > with 1-by-1 or 2-by-2 diagonal blocks, where */
620 /* > If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; */
621 /* > If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. */
623 /* > NOTE: For 1-by-1 diagonal block D(k), where */
624 /* > 1 <= k <= N, the element E(k) is set to 0 in both */
625 /* > UPLO = 'U' or UPLO = 'L' cases. */
628 /* > \param[out] IPIV */
630 /* > IPIV is INTEGER array, dimension (N) */
631 /* > IPIV describes the permutation matrix P in the factorization */
632 /* > of matrix A as follows. The absolute value of IPIV(k) */
633 /* > represents the index of row and column that were */
634 /* > interchanged with the k-th row and column. The value of UPLO */
635 /* > describes the order in which the interchanges were applied. */
636 /* > Also, the sign of IPIV represents the block structure of */
637 /* > the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 */
638 /* > diagonal blocks which correspond to 1 or 2 interchanges */
639 /* > at each factorization step. For more info see Further */
640 /* > Details section. */
642 /* > If UPLO = 'U', */
643 /* > ( in factorization order, k decreases from N to 1 ): */
644 /* > a) A single positive entry IPIV(k) > 0 means: */
645 /* > D(k,k) is a 1-by-1 diagonal block. */
646 /* > If IPIV(k) != k, rows and columns k and IPIV(k) were */
647 /* > interchanged in the matrix A(1:N,1:N); */
648 /* > If IPIV(k) = k, no interchange occurred. */
650 /* > b) A pair of consecutive negative entries */
651 /* > IPIV(k) < 0 and IPIV(k-1) < 0 means: */
652 /* > D(k-1:k,k-1:k) is a 2-by-2 diagonal block. */
653 /* > (NOTE: negative entries in IPIV appear ONLY in pairs). */
654 /* > 1) If -IPIV(k) != k, rows and columns */
655 /* > k and -IPIV(k) were interchanged */
656 /* > in the matrix A(1:N,1:N). */
657 /* > If -IPIV(k) = k, no interchange occurred. */
658 /* > 2) If -IPIV(k-1) != k-1, rows and columns */
659 /* > k-1 and -IPIV(k-1) were interchanged */
660 /* > in the matrix A(1:N,1:N). */
661 /* > If -IPIV(k-1) = k-1, no interchange occurred. */
663 /* > c) In both cases a) and b), always ABS( IPIV(k) ) <= k. */
665 /* > d) NOTE: Any entry IPIV(k) is always NONZERO on output. */
667 /* > If UPLO = 'L', */
668 /* > ( in factorization order, k increases from 1 to N ): */
669 /* > a) A single positive entry IPIV(k) > 0 means: */
670 /* > D(k,k) is a 1-by-1 diagonal block. */
671 /* > If IPIV(k) != k, rows and columns k and IPIV(k) were */
672 /* > interchanged in the matrix A(1:N,1:N). */
673 /* > If IPIV(k) = k, no interchange occurred. */
675 /* > b) A pair of consecutive negative entries */
676 /* > IPIV(k) < 0 and IPIV(k+1) < 0 means: */
677 /* > D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */
678 /* > (NOTE: negative entries in IPIV appear ONLY in pairs). */
679 /* > 1) If -IPIV(k) != k, rows and columns */
680 /* > k and -IPIV(k) were interchanged */
681 /* > in the matrix A(1:N,1:N). */
682 /* > If -IPIV(k) = k, no interchange occurred. */
683 /* > 2) If -IPIV(k+1) != k+1, rows and columns */
684 /* > k-1 and -IPIV(k-1) were interchanged */
685 /* > in the matrix A(1:N,1:N). */
686 /* > If -IPIV(k+1) = k+1, no interchange occurred. */
688 /* > c) In both cases a) and b), always ABS( IPIV(k) ) >= k. */
690 /* > d) NOTE: Any entry IPIV(k) is always NONZERO on output. */
693 /* > \param[out] INFO */
695 /* > INFO is INTEGER */
696 /* > = 0: successful exit */
698 /* > < 0: If INFO = -k, the k-th argument had an illegal value */
700 /* > > 0: If INFO = k, the matrix A is singular, because: */
701 /* > If UPLO = 'U': column k in the upper */
702 /* > triangular part of A contains all zeros. */
703 /* > If UPLO = 'L': column k in the lower */
704 /* > triangular part of A contains all zeros. */
706 /* > Therefore D(k,k) is exactly zero, and superdiagonal */
707 /* > elements of column k of U (or subdiagonal elements of */
708 /* > column k of L ) are all zeros. The factorization has */
709 /* > been completed, but the block diagonal matrix D is */
710 /* > exactly singular, and division by zero will occur if */
711 /* > it is used to solve a system of equations. */
713 /* > NOTE: INFO only stores the first occurrence of */
714 /* > a singularity, any subsequent occurrence of singularity */
715 /* > is not stored in INFO even though the factorization */
716 /* > always completes. */
722 /* > \author Univ. of Tennessee */
723 /* > \author Univ. of California Berkeley */
724 /* > \author Univ. of Colorado Denver */
725 /* > \author NAG Ltd. */
727 /* > \date December 2016 */
729 /* > \ingroup singleSYcomputational */
731 /* > \par Further Details: */
732 /* ===================== */
735 /* > TODO: put further details */
738 /* > \par Contributors: */
739 /* ================== */
743 /* > December 2016, Igor Kozachenko, */
744 /* > Computer Science Division, */
745 /* > University of California, Berkeley */
747 /* > September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, */
748 /* > School of Mathematics, */
749 /* > University of Manchester */
751 /* > 01-01-96 - Based on modifications by */
752 /* > J. Lewis, Boeing Computer Services Company */
753 /* > A. Petitet, Computer Science Dept., */
754 /* > Univ. of Tenn., Knoxville abd , USA */
757 /* ===================================================================== */
758 /* Subroutine */ int ssytf2_rk_(char *uplo, integer *n, real *a, integer *
759 lda, real *e, integer *ipiv, integer *info)
761 /* System generated locals */
762 integer a_dim1, a_offset, i__1, i__2;
765 /* Local variables */
768 extern /* Subroutine */ int ssyr_(char *, integer *, real *, real *,
769 integer *, real *, integer *);
770 integer i__, j, k, p;
772 extern logical lsame_(char *, char *);
773 extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
775 integer itemp, kstep;
778 extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *,
780 real d11, d12, d21, d22;
783 extern real slamch_(char *);
784 extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
785 extern integer isamax_(integer *, real *, integer *);
786 real colmax, rowmax, wkm1, wkp1;
789 /* -- LAPACK computational routine (version 3.7.0) -- */
790 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
791 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
795 /* ===================================================================== */
798 /* Test the input parameters. */
800 /* Parameter adjustments */
802 a_offset = 1 + a_dim1 * 1;
809 upper = lsame_(uplo, "U");
810 if (! upper && ! lsame_(uplo, "L")) {
814 } else if (*lda < f2cmax(1,*n)) {
819 xerbla_("SSYTF2_RK", &i__1, (ftnlen)9);
823 /* Initialize ALPHA for use in choosing pivot block size. */
825 alpha = (sqrt(17.f) + 1.f) / 8.f;
827 /* Compute machine safe minimum */
829 sfmin = slamch_("S");
833 /* Factorize A as U*D*U**T using the upper triangle of A */
835 /* Initialize the first entry of array E, where superdiagonal */
836 /* elements of D are stored */
840 /* K is the main loop index, decreasing from N to 1 in steps of */
846 /* If K < 1, exit from loop */
854 /* Determine rows and columns to be interchanged and whether */
855 /* a 1-by-1 or 2-by-2 pivot block will be used */
857 absakk = (r__1 = a[k + k * a_dim1], abs(r__1));
859 /* IMAX is the row-index of the largest off-diagonal element in */
860 /* column K, and COLMAX is its absolute value. */
861 /* Determine both COLMAX and IMAX. */
865 imax = isamax_(&i__1, &a[k * a_dim1 + 1], &c__1);
866 colmax = (r__1 = a[imax + k * a_dim1], abs(r__1));
871 if (f2cmax(absakk,colmax) == 0.f) {
873 /* Column K is zero or underflow: set INFO and continue */
880 /* Set E( K ) to zero */
888 /* Test for interchange */
890 /* Equivalent to testing for (used to handle NaN and Inf) */
891 /* ABSAKK.GE.ALPHA*COLMAX */
893 if (! (absakk < alpha * colmax)) {
895 /* no interchange, */
896 /* use 1-by-1 pivot block */
903 /* Loop until pivot found */
907 /* Begin pivot search loop body */
909 /* JMAX is the column-index of the largest off-diagonal */
910 /* element in row IMAX, and ROWMAX is its absolute value. */
911 /* Determine both ROWMAX and JMAX. */
915 jmax = imax + isamax_(&i__1, &a[imax + (imax + 1) *
917 rowmax = (r__1 = a[imax + jmax * a_dim1], abs(r__1));
924 itemp = isamax_(&i__1, &a[imax * a_dim1 + 1], &c__1);
925 stemp = (r__1 = a[itemp + imax * a_dim1], abs(r__1));
926 if (stemp > rowmax) {
932 /* Equivalent to testing for (used to handle NaN and Inf) */
933 /* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX */
935 if (! ((r__1 = a[imax + imax * a_dim1], abs(r__1)) < alpha *
938 /* interchange rows and columns K and IMAX, */
939 /* use 1-by-1 pivot block */
944 /* Equivalent to testing for ROWMAX .EQ. COLMAX, */
945 /* used to handle NaN and Inf */
947 } else if (p == jmax || rowmax <= colmax) {
949 /* interchange rows and columns K+1 and IMAX, */
950 /* use 2-by-2 pivot block */
957 /* Pivot NOT found, set variables and repeat */
964 /* End pivot search loop body */
972 /* Swap TWO rows and TWO columns */
976 if (kstep == 2 && p != k) {
978 /* Interchange rows and column K and P in the leading */
979 /* submatrix A(1:k,1:k) if we have a 2-by-2 pivot */
983 sswap_(&i__1, &a[k * a_dim1 + 1], &c__1, &a[p * a_dim1 +
988 sswap_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + (p +
991 t = a[k + k * a_dim1];
992 a[k + k * a_dim1] = a[p + p * a_dim1];
993 a[p + p * a_dim1] = t;
995 /* Convert upper triangle of A into U form by applying */
996 /* the interchanges in columns k+1:N. */
1000 sswap_(&i__1, &a[k + (k + 1) * a_dim1], lda, &a[p + (k +
1011 /* Interchange rows and columns KK and KP in the leading */
1012 /* submatrix A(1:k,1:k) */
1016 sswap_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1
1019 if (kk > 1 && kp < kk - 1) {
1021 sswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (
1022 kp + 1) * a_dim1], lda);
1024 t = a[kk + kk * a_dim1];
1025 a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
1026 a[kp + kp * a_dim1] = t;
1028 t = a[k - 1 + k * a_dim1];
1029 a[k - 1 + k * a_dim1] = a[kp + k * a_dim1];
1030 a[kp + k * a_dim1] = t;
1033 /* Convert upper triangle of A into U form by applying */
1034 /* the interchanges in columns k+1:N. */
1038 sswap_(&i__1, &a[kk + (k + 1) * a_dim1], lda, &a[kp + (k
1039 + 1) * a_dim1], lda);
1044 /* Update the leading submatrix */
1048 /* 1-by-1 pivot block D(k): column k now holds */
1050 /* W(k) = U(k)*D(k) */
1052 /* where U(k) is the k-th column of U */
1056 /* Perform a rank-1 update of A(1:k-1,1:k-1) and */
1057 /* store U(k) in column k */
1059 if ((r__1 = a[k + k * a_dim1], abs(r__1)) >= sfmin) {
1061 /* Perform a rank-1 update of A(1:k-1,1:k-1) as */
1062 /* A := A - U(k)*D(k)*U(k)**T */
1063 /* = A - W(k)*1/D(k)*W(k)**T */
1065 d11 = 1.f / a[k + k * a_dim1];
1068 ssyr_(uplo, &i__1, &r__1, &a[k * a_dim1 + 1], &c__1, &
1071 /* Store U(k) in column k */
1074 sscal_(&i__1, &d11, &a[k * a_dim1 + 1], &c__1);
1077 /* Store L(k) in column K */
1079 d11 = a[k + k * a_dim1];
1081 for (ii = 1; ii <= i__1; ++ii) {
1082 a[ii + k * a_dim1] /= d11;
1086 /* Perform a rank-1 update of A(k+1:n,k+1:n) as */
1087 /* A := A - U(k)*D(k)*U(k)**T */
1088 /* = A - W(k)*(1/D(k))*W(k)**T */
1089 /* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T */
1093 ssyr_(uplo, &i__1, &r__1, &a[k * a_dim1 + 1], &c__1, &
1097 /* Store the superdiagonal element of D in array E */
1105 /* 2-by-2 pivot block D(k): columns k and k-1 now hold */
1107 /* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) */
1109 /* where U(k) and U(k-1) are the k-th and (k-1)-th columns */
1112 /* Perform a rank-2 update of A(1:k-2,1:k-2) as */
1114 /* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T */
1115 /* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T */
1117 /* and store L(k) and L(k+1) in columns k and k+1 */
1121 d12 = a[k - 1 + k * a_dim1];
1122 d22 = a[k - 1 + (k - 1) * a_dim1] / d12;
1123 d11 = a[k + k * a_dim1] / d12;
1124 t = 1.f / (d11 * d22 - 1.f);
1126 for (j = k - 2; j >= 1; --j) {
1128 wkm1 = t * (d11 * a[j + (k - 1) * a_dim1] - a[j + k *
1130 wk = t * (d22 * a[j + k * a_dim1] - a[j + (k - 1) *
1133 for (i__ = j; i__ >= 1; --i__) {
1134 a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
1135 + k * a_dim1] / d12 * wk - a[i__ + (k - 1)
1136 * a_dim1] / d12 * wkm1;
1140 /* Store U(k) and U(k-1) in cols k and k-1 for row J */
1142 a[j + k * a_dim1] = wk / d12;
1143 a[j + (k - 1) * a_dim1] = wkm1 / d12;
1150 /* Copy superdiagonal elements of D(K) to E(K) and */
1151 /* ZERO out superdiagonal entry of A */
1153 e[k] = a[k - 1 + k * a_dim1];
1155 a[k - 1 + k * a_dim1] = 0.f;
1159 /* End column K is nonsingular */
1163 /* Store details of the interchanges in IPIV */
1172 /* Decrease K and return to the start of the main loop */
1182 /* Factorize A as L*D*L**T using the lower triangle of A */
1184 /* Initialize the unused last entry of the subdiagonal array E. */
1188 /* K is the main loop index, increasing from 1 to N in steps of */
1194 /* If K > N, exit from loop */
1202 /* Determine rows and columns to be interchanged and whether */
1203 /* a 1-by-1 or 2-by-2 pivot block will be used */
1205 absakk = (r__1 = a[k + k * a_dim1], abs(r__1));
1207 /* IMAX is the row-index of the largest off-diagonal element in */
1208 /* column K, and COLMAX is its absolute value. */
1209 /* Determine both COLMAX and IMAX. */
1213 imax = k + isamax_(&i__1, &a[k + 1 + k * a_dim1], &c__1);
1214 colmax = (r__1 = a[imax + k * a_dim1], abs(r__1));
1219 if (f2cmax(absakk,colmax) == 0.f) {
1221 /* Column K is zero or underflow: set INFO and continue */
1228 /* Set E( K ) to zero */
1236 /* Test for interchange */
1238 /* Equivalent to testing for (used to handle NaN and Inf) */
1239 /* ABSAKK.GE.ALPHA*COLMAX */
1241 if (! (absakk < alpha * colmax)) {
1243 /* no interchange, use 1-by-1 pivot block */
1251 /* Loop until pivot found */
1255 /* Begin pivot search loop body */
1257 /* JMAX is the column-index of the largest off-diagonal */
1258 /* element in row IMAX, and ROWMAX is its absolute value. */
1259 /* Determine both ROWMAX and JMAX. */
1263 jmax = k - 1 + isamax_(&i__1, &a[imax + k * a_dim1], lda);
1264 rowmax = (r__1 = a[imax + jmax * a_dim1], abs(r__1));
1271 itemp = imax + isamax_(&i__1, &a[imax + 1 + imax * a_dim1]
1273 stemp = (r__1 = a[itemp + imax * a_dim1], abs(r__1));
1274 if (stemp > rowmax) {
1280 /* Equivalent to testing for (used to handle NaN and Inf) */
1281 /* ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX */
1283 if (! ((r__1 = a[imax + imax * a_dim1], abs(r__1)) < alpha *
1286 /* interchange rows and columns K and IMAX, */
1287 /* use 1-by-1 pivot block */
1292 /* Equivalent to testing for ROWMAX .EQ. COLMAX, */
1293 /* used to handle NaN and Inf */
1295 } else if (p == jmax || rowmax <= colmax) {
1297 /* interchange rows and columns K+1 and IMAX, */
1298 /* use 2-by-2 pivot block */
1305 /* Pivot NOT found, set variables and repeat */
1312 /* End pivot search loop body */
1320 /* Swap TWO rows and TWO columns */
1324 if (kstep == 2 && p != k) {
1326 /* Interchange rows and column K and P in the trailing */
1327 /* submatrix A(k:n,k:n) if we have a 2-by-2 pivot */
1331 sswap_(&i__1, &a[p + 1 + k * a_dim1], &c__1, &a[p + 1 + p
1336 sswap_(&i__1, &a[k + 1 + k * a_dim1], &c__1, &a[p + (k +
1339 t = a[k + k * a_dim1];
1340 a[k + k * a_dim1] = a[p + p * a_dim1];
1341 a[p + p * a_dim1] = t;
1343 /* Convert lower triangle of A into L form by applying */
1344 /* the interchanges in columns 1:k-1. */
1348 sswap_(&i__1, &a[k + a_dim1], lda, &a[p + a_dim1], lda);
1358 /* Interchange rows and columns KK and KP in the trailing */
1359 /* submatrix A(k:n,k:n) */
1363 sswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1
1364 + kp * a_dim1], &c__1);
1366 if (kk < *n && kp > kk + 1) {
1368 sswap_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + (
1369 kk + 1) * a_dim1], lda);
1371 t = a[kk + kk * a_dim1];
1372 a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
1373 a[kp + kp * a_dim1] = t;
1375 t = a[k + 1 + k * a_dim1];
1376 a[k + 1 + k * a_dim1] = a[kp + k * a_dim1];
1377 a[kp + k * a_dim1] = t;
1380 /* Convert lower triangle of A into L form by applying */
1381 /* the interchanges in columns 1:k-1. */
1385 sswap_(&i__1, &a[kk + a_dim1], lda, &a[kp + a_dim1], lda);
1390 /* Update the trailing submatrix */
1394 /* 1-by-1 pivot block D(k): column k now holds */
1396 /* W(k) = L(k)*D(k) */
1398 /* where L(k) is the k-th column of L */
1402 /* Perform a rank-1 update of A(k+1:n,k+1:n) and */
1403 /* store L(k) in column k */
1405 if ((r__1 = a[k + k * a_dim1], abs(r__1)) >= sfmin) {
1407 /* Perform a rank-1 update of A(k+1:n,k+1:n) as */
1408 /* A := A - L(k)*D(k)*L(k)**T */
1409 /* = A - W(k)*(1/D(k))*W(k)**T */
1411 d11 = 1.f / a[k + k * a_dim1];
1414 ssyr_(uplo, &i__1, &r__1, &a[k + 1 + k * a_dim1], &
1415 c__1, &a[k + 1 + (k + 1) * a_dim1], lda);
1417 /* Store L(k) in column k */
1420 sscal_(&i__1, &d11, &a[k + 1 + k * a_dim1], &c__1);
1423 /* Store L(k) in column k */
1425 d11 = a[k + k * a_dim1];
1427 for (ii = k + 1; ii <= i__1; ++ii) {
1428 a[ii + k * a_dim1] /= d11;
1432 /* Perform a rank-1 update of A(k+1:n,k+1:n) as */
1433 /* A := A - L(k)*D(k)*L(k)**T */
1434 /* = A - W(k)*(1/D(k))*W(k)**T */
1435 /* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T */
1439 ssyr_(uplo, &i__1, &r__1, &a[k + 1 + k * a_dim1], &
1440 c__1, &a[k + 1 + (k + 1) * a_dim1], lda);
1443 /* Store the subdiagonal element of D in array E */
1451 /* 2-by-2 pivot block D(k): columns k and k+1 now hold */
1453 /* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) */
1455 /* where L(k) and L(k+1) are the k-th and (k+1)-th columns */
1459 /* Perform a rank-2 update of A(k+2:n,k+2:n) as */
1461 /* A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T */
1462 /* = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T */
1464 /* and store L(k) and L(k+1) in columns k and k+1 */
1468 d21 = a[k + 1 + k * a_dim1];
1469 d11 = a[k + 1 + (k + 1) * a_dim1] / d21;
1470 d22 = a[k + k * a_dim1] / d21;
1471 t = 1.f / (d11 * d22 - 1.f);
1474 for (j = k + 2; j <= i__1; ++j) {
1476 /* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J */
1478 wk = t * (d11 * a[j + k * a_dim1] - a[j + (k + 1) *
1480 wkp1 = t * (d22 * a[j + (k + 1) * a_dim1] - a[j + k *
1483 /* Perform a rank-2 update of A(k+2:n,k+2:n) */
1486 for (i__ = j; i__ <= i__2; ++i__) {
1487 a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
1488 + k * a_dim1] / d21 * wk - a[i__ + (k + 1)
1489 * a_dim1] / d21 * wkp1;
1493 /* Store L(k) and L(k+1) in cols k and k+1 for row J */
1495 a[j + k * a_dim1] = wk / d21;
1496 a[j + (k + 1) * a_dim1] = wkp1 / d21;
1503 /* Copy subdiagonal elements of D(K) to E(K) and */
1504 /* ZERO out subdiagonal entry of A */
1506 e[k] = a[k + 1 + k * a_dim1];
1508 a[k + 1 + k * a_dim1] = 0.f;
1512 /* End column K is nonsingular */
1516 /* Store details of the interchanges in IPIV */
1525 /* Increase K and return to the start of the main loop */
1537 /* End of SSYTF2_RK */