LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
ssytf2_rk.f
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1 *> \brief \b SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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7 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, N
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IPIV( * )
29 * REAL A( LDA, * ), E ( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *> SSYTF2_RK computes the factorization of a real symmetric matrix A
38 *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
39 *>
40 *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
41 *>
42 *> where U (or L) is unit upper (or lower) triangular matrix,
43 *> U**T (or L**T) is the transpose of U (or L), P is a permutation
44 *> matrix, P**T is the transpose of P, and D is symmetric and block
45 *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
46 *>
47 *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[in] UPLO
55 *> \verbatim
56 *> UPLO is CHARACTER*1
57 *> Specifies whether the upper or lower triangular part of the
58 *> symmetric matrix A is stored:
59 *> = 'U': Upper triangular
60 *> = 'L': Lower triangular
61 *> \endverbatim
62 *>
63 *> \param[in] N
64 *> \verbatim
65 *> N is INTEGER
66 *> The order of the matrix A. N >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in,out] A
70 *> \verbatim
71 *> A is REAL array, dimension (LDA,N)
72 *> On entry, the symmetric matrix A.
73 *> If UPLO = 'U': the leading N-by-N upper triangular part
74 *> of A contains the upper triangular part of the matrix A,
75 *> and the strictly lower triangular part of A is not
76 *> referenced.
77 *>
78 *> If UPLO = 'L': the leading N-by-N lower triangular part
79 *> of A contains the lower triangular part of the matrix A,
80 *> and the strictly upper triangular part of A is not
81 *> referenced.
82 *>
83 *> On exit, contains:
84 *> a) ONLY diagonal elements of the symmetric block diagonal
85 *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
86 *> (superdiagonal (or subdiagonal) elements of D
87 *> are stored on exit in array E), and
88 *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
89 *> If UPLO = 'L': factor L in the subdiagonal part of A.
90 *> \endverbatim
91 *>
92 *> \param[in] LDA
93 *> \verbatim
94 *> LDA is INTEGER
95 *> The leading dimension of the array A. LDA >= max(1,N).
96 *> \endverbatim
97 *>
98 *> \param[out] E
99 *> \verbatim
100 *> E is REAL array, dimension (N)
101 *> On exit, contains the superdiagonal (or subdiagonal)
102 *> elements of the symmetric block diagonal matrix D
103 *> with 1-by-1 or 2-by-2 diagonal blocks, where
104 *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
105 *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
106 *>
107 *> NOTE: For 1-by-1 diagonal block D(k), where
108 *> 1 <= k <= N, the element E(k) is set to 0 in both
109 *> UPLO = 'U' or UPLO = 'L' cases.
110 *> \endverbatim
111 *>
112 *> \param[out] IPIV
113 *> \verbatim
114 *> IPIV is INTEGER array, dimension (N)
115 *> IPIV describes the permutation matrix P in the factorization
116 *> of matrix A as follows. The absolute value of IPIV(k)
117 *> represents the index of row and column that were
118 *> interchanged with the k-th row and column. The value of UPLO
119 *> describes the order in which the interchanges were applied.
120 *> Also, the sign of IPIV represents the block structure of
121 *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
122 *> diagonal blocks which correspond to 1 or 2 interchanges
124 *> Details section.
125 *>
126 *> If UPLO = 'U',
127 *> ( in factorization order, k decreases from N to 1 ):
128 *> a) A single positive entry IPIV(k) > 0 means:
129 *> D(k,k) is a 1-by-1 diagonal block.
130 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
131 *> interchanged in the matrix A(1:N,1:N);
132 *> If IPIV(k) = k, no interchange occurred.
133 *>
134 *> b) A pair of consecutive negative entries
135 *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
136 *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
137 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
138 *> 1) If -IPIV(k) != k, rows and columns
139 *> k and -IPIV(k) were interchanged
140 *> in the matrix A(1:N,1:N).
141 *> If -IPIV(k) = k, no interchange occurred.
142 *> 2) If -IPIV(k-1) != k-1, rows and columns
143 *> k-1 and -IPIV(k-1) were interchanged
144 *> in the matrix A(1:N,1:N).
145 *> If -IPIV(k-1) = k-1, no interchange occurred.
146 *>
147 *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
148 *>
149 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
150 *>
151 *> If UPLO = 'L',
152 *> ( in factorization order, k increases from 1 to N ):
153 *> a) A single positive entry IPIV(k) > 0 means:
154 *> D(k,k) is a 1-by-1 diagonal block.
155 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
156 *> interchanged in the matrix A(1:N,1:N).
157 *> If IPIV(k) = k, no interchange occurred.
158 *>
159 *> b) A pair of consecutive negative entries
160 *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
161 *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
162 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
163 *> 1) If -IPIV(k) != k, rows and columns
164 *> k and -IPIV(k) were interchanged
165 *> in the matrix A(1:N,1:N).
166 *> If -IPIV(k) = k, no interchange occurred.
167 *> 2) If -IPIV(k+1) != k+1, rows and columns
168 *> k-1 and -IPIV(k-1) were interchanged
169 *> in the matrix A(1:N,1:N).
170 *> If -IPIV(k+1) = k+1, no interchange occurred.
171 *>
172 *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
173 *>
174 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
175 *> \endverbatim
176 *>
177 *> \param[out] INFO
178 *> \verbatim
179 *> INFO is INTEGER
180 *> = 0: successful exit
181 *>
182 *> < 0: If INFO = -k, the k-th argument had an illegal value
183 *>
184 *> > 0: If INFO = k, the matrix A is singular, because:
185 *> If UPLO = 'U': column k in the upper
186 *> triangular part of A contains all zeros.
187 *> If UPLO = 'L': column k in the lower
188 *> triangular part of A contains all zeros.
189 *>
190 *> Therefore D(k,k) is exactly zero, and superdiagonal
191 *> elements of column k of U (or subdiagonal elements of
192 *> column k of L ) are all zeros. The factorization has
193 *> been completed, but the block diagonal matrix D is
194 *> exactly singular, and division by zero will occur if
195 *> it is used to solve a system of equations.
196 *>
197 *> NOTE: INFO only stores the first occurrence of
198 *> a singularity, any subsequent occurrence of singularity
199 *> is not stored in INFO even though the factorization
200 *> always completes.
201 *> \endverbatim
202 *
203 * Authors:
204 * ========
205 *
206 *> \author Univ. of Tennessee
207 *> \author Univ. of California Berkeley
208 *> \author Univ. of Colorado Denver
209 *> \author NAG Ltd.
210 *
211 *> \ingroup singleSYcomputational
212 *
213 *> \par Further Details:
214 * =====================
215 *>
216 *> \verbatim
217 *> TODO: put further details
218 *> \endverbatim
219 *
220 *> \par Contributors:
221 * ==================
222 *>
223 *> \verbatim
224 *>
225 *> December 2016, Igor Kozachenko,
226 *> Computer Science Division,
227 *> University of California, Berkeley
228 *>
229 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
230 *> School of Mathematics,
231 *> University of Manchester
232 *>
233 *> 01-01-96 - Based on modifications by
234 *> J. Lewis, Boeing Computer Services Company
235 *> A. Petitet, Computer Science Dept.,
236 *> Univ. of Tenn., Knoxville abd , USA
237 *> \endverbatim
238 *
239 * =====================================================================
240  SUBROUTINE ssytf2_rk( UPLO, N, A, LDA, E, IPIV, INFO )
241 *
242 * -- LAPACK computational routine --
243 * -- LAPACK is a software package provided by Univ. of Tennessee, --
244 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
245 *
246 * .. Scalar Arguments ..
247  CHARACTER UPLO
248  INTEGER INFO, LDA, N
249 * ..
250 * .. Array Arguments ..
251  INTEGER IPIV( * )
252  REAL A( LDA, * ), E( * )
253 * ..
254 *
255 * =====================================================================
256 *
257 * .. Parameters ..
258  REAL ZERO, ONE
259  parameter( zero = 0.0e+0, one = 1.0e+0 )
260  REAL EIGHT, SEVTEN
261  parameter( eight = 8.0e+0, sevten = 17.0e+0 )
262 * ..
263 * .. Local Scalars ..
264  LOGICAL UPPER, DONE
265  INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
266  \$ P, II
267  REAL ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
268  \$ ROWMAX, STEMP, T, WK, WKM1, WKP1, SFMIN
269 * ..
270 * .. External Functions ..
271  LOGICAL LSAME
272  INTEGER ISAMAX
273  REAL SLAMCH
274  EXTERNAL lsame, isamax, slamch
275 * ..
276 * .. External Subroutines ..
277  EXTERNAL sscal, sswap, ssyr, xerbla
278 * ..
279 * .. Intrinsic Functions ..
280  INTRINSIC abs, max, sqrt
281 * ..
282 * .. Executable Statements ..
283 *
284 * Test the input parameters.
285 *
286  info = 0
287  upper = lsame( uplo, 'U' )
288  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
289  info = -1
290  ELSE IF( n.LT.0 ) THEN
291  info = -2
292  ELSE IF( lda.LT.max( 1, n ) ) THEN
293  info = -4
294  END IF
295  IF( info.NE.0 ) THEN
296  CALL xerbla( 'SSYTF2_RK', -info )
297  RETURN
298  END IF
299 *
300 * Initialize ALPHA for use in choosing pivot block size.
301 *
302  alpha = ( one+sqrt( sevten ) ) / eight
303 *
304 * Compute machine safe minimum
305 *
306  sfmin = slamch( 'S' )
307 *
308  IF( upper ) THEN
309 *
310 * Factorize A as U*D*U**T using the upper triangle of A
311 *
312 * Initialize the first entry of array E, where superdiagonal
313 * elements of D are stored
314 *
315  e( 1 ) = zero
316 *
317 * K is the main loop index, decreasing from N to 1 in steps of
318 * 1 or 2
319 *
320  k = n
321  10 CONTINUE
322 *
323 * If K < 1, exit from loop
324 *
325  IF( k.LT.1 )
326  \$ GO TO 34
327  kstep = 1
328  p = k
329 *
330 * Determine rows and columns to be interchanged and whether
331 * a 1-by-1 or 2-by-2 pivot block will be used
332 *
333  absakk = abs( a( k, k ) )
334 *
335 * IMAX is the row-index of the largest off-diagonal element in
336 * column K, and COLMAX is its absolute value.
337 * Determine both COLMAX and IMAX.
338 *
339  IF( k.GT.1 ) THEN
340  imax = isamax( k-1, a( 1, k ), 1 )
341  colmax = abs( a( imax, k ) )
342  ELSE
343  colmax = zero
344  END IF
345 *
346  IF( (max( absakk, colmax ).EQ.zero) ) THEN
347 *
348 * Column K is zero or underflow: set INFO and continue
349 *
350  IF( info.EQ.0 )
351  \$ info = k
352  kp = k
353 *
354 * Set E( K ) to zero
355 *
356  IF( k.GT.1 )
357  \$ e( k ) = zero
358 *
359  ELSE
360 *
361 * Test for interchange
362 *
363 * Equivalent to testing for (used to handle NaN and Inf)
364 * ABSAKK.GE.ALPHA*COLMAX
365 *
366  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
367 *
368 * no interchange,
369 * use 1-by-1 pivot block
370 *
371  kp = k
372  ELSE
373 *
374  done = .false.
375 *
376 * Loop until pivot found
377 *
378  12 CONTINUE
379 *
380 * Begin pivot search loop body
381 *
382 * JMAX is the column-index of the largest off-diagonal
383 * element in row IMAX, and ROWMAX is its absolute value.
384 * Determine both ROWMAX and JMAX.
385 *
386  IF( imax.NE.k ) THEN
387  jmax = imax + isamax( k-imax, a( imax, imax+1 ),
388  \$ lda )
389  rowmax = abs( a( imax, jmax ) )
390  ELSE
391  rowmax = zero
392  END IF
393 *
394  IF( imax.GT.1 ) THEN
395  itemp = isamax( imax-1, a( 1, imax ), 1 )
396  stemp = abs( a( itemp, imax ) )
397  IF( stemp.GT.rowmax ) THEN
398  rowmax = stemp
399  jmax = itemp
400  END IF
401  END IF
402 *
403 * Equivalent to testing for (used to handle NaN and Inf)
404 * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
405 *
406  IF( .NOT.( abs( a( imax, imax ) ).LT.alpha*rowmax ) )
407  \$ THEN
408 *
409 * interchange rows and columns K and IMAX,
410 * use 1-by-1 pivot block
411 *
412  kp = imax
413  done = .true.
414 *
415 * Equivalent to testing for ROWMAX .EQ. COLMAX,
416 * used to handle NaN and Inf
417 *
418  ELSE IF( ( p.EQ.jmax ).OR.( rowmax.LE.colmax ) ) THEN
419 *
420 * interchange rows and columns K+1 and IMAX,
421 * use 2-by-2 pivot block
422 *
423  kp = imax
424  kstep = 2
425  done = .true.
426  ELSE
427 *
429 *
430  p = imax
431  colmax = rowmax
432  imax = jmax
433  END IF
434 *
435 * End pivot search loop body
436 *
437  IF( .NOT. done ) GOTO 12
438 *
439  END IF
440 *
441 * Swap TWO rows and TWO columns
442 *
443 * First swap
444 *
445  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
446 *
447 * Interchange rows and column K and P in the leading
448 * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
449 *
450  IF( p.GT.1 )
451  \$ CALL sswap( p-1, a( 1, k ), 1, a( 1, p ), 1 )
452  IF( p.LT.(k-1) )
453  \$ CALL sswap( k-p-1, a( p+1, k ), 1, a( p, p+1 ),
454  \$ lda )
455  t = a( k, k )
456  a( k, k ) = a( p, p )
457  a( p, p ) = t
458 *
459 * Convert upper triangle of A into U form by applying
460 * the interchanges in columns k+1:N.
461 *
462  IF( k.LT.n )
463  \$ CALL sswap( n-k, a( k, k+1 ), lda, a( p, k+1 ), lda )
464 *
465  END IF
466 *
467 * Second swap
468 *
469  kk = k - kstep + 1
470  IF( kp.NE.kk ) THEN
471 *
472 * Interchange rows and columns KK and KP in the leading
473 * submatrix A(1:k,1:k)
474 *
475  IF( kp.GT.1 )
476  \$ CALL sswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
477  IF( ( kk.GT.1 ) .AND. ( kp.LT.(kk-1) ) )
478  \$ CALL sswap( kk-kp-1, a( kp+1, kk ), 1, a( kp, kp+1 ),
479  \$ lda )
480  t = a( kk, kk )
481  a( kk, kk ) = a( kp, kp )
482  a( kp, kp ) = t
483  IF( kstep.EQ.2 ) THEN
484  t = a( k-1, k )
485  a( k-1, k ) = a( kp, k )
486  a( kp, k ) = t
487  END IF
488 *
489 * Convert upper triangle of A into U form by applying
490 * the interchanges in columns k+1:N.
491 *
492  IF( k.LT.n )
493  \$ CALL sswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
494  \$ lda )
495 *
496  END IF
497 *
498 * Update the leading submatrix
499 *
500  IF( kstep.EQ.1 ) THEN
501 *
502 * 1-by-1 pivot block D(k): column k now holds
503 *
504 * W(k) = U(k)*D(k)
505 *
506 * where U(k) is the k-th column of U
507 *
508  IF( k.GT.1 ) THEN
509 *
510 * Perform a rank-1 update of A(1:k-1,1:k-1) and
511 * store U(k) in column k
512 *
513  IF( abs( a( k, k ) ).GE.sfmin ) THEN
514 *
515 * Perform a rank-1 update of A(1:k-1,1:k-1) as
516 * A := A - U(k)*D(k)*U(k)**T
517 * = A - W(k)*1/D(k)*W(k)**T
518 *
519  d11 = one / a( k, k )
520  CALL ssyr( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
521 *
522 * Store U(k) in column k
523 *
524  CALL sscal( k-1, d11, a( 1, k ), 1 )
525  ELSE
526 *
527 * Store L(k) in column K
528 *
529  d11 = a( k, k )
530  DO 16 ii = 1, k - 1
531  a( ii, k ) = a( ii, k ) / d11
532  16 CONTINUE
533 *
534 * Perform a rank-1 update of A(k+1:n,k+1:n) as
535 * A := A - U(k)*D(k)*U(k)**T
536 * = A - W(k)*(1/D(k))*W(k)**T
537 * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
538 *
539  CALL ssyr( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
540  END IF
541 *
542 * Store the superdiagonal element of D in array E
543 *
544  e( k ) = zero
545 *
546  END IF
547 *
548  ELSE
549 *
550 * 2-by-2 pivot block D(k): columns k and k-1 now hold
551 *
552 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
553 *
554 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
555 * of U
556 *
557 * Perform a rank-2 update of A(1:k-2,1:k-2) as
558 *
559 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
560 * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
561 *
562 * and store L(k) and L(k+1) in columns k and k+1
563 *
564  IF( k.GT.2 ) THEN
565 *
566  d12 = a( k-1, k )
567  d22 = a( k-1, k-1 ) / d12
568  d11 = a( k, k ) / d12
569  t = one / ( d11*d22-one )
570 *
571  DO 30 j = k - 2, 1, -1
572 *
573  wkm1 = t*( d11*a( j, k-1 )-a( j, k ) )
574  wk = t*( d22*a( j, k )-a( j, k-1 ) )
575 *
576  DO 20 i = j, 1, -1
577  a( i, j ) = a( i, j ) - (a( i, k ) / d12 )*wk -
578  \$ ( a( i, k-1 ) / d12 )*wkm1
579  20 CONTINUE
580 *
581 * Store U(k) and U(k-1) in cols k and k-1 for row J
582 *
583  a( j, k ) = wk / d12
584  a( j, k-1 ) = wkm1 / d12
585 *
586  30 CONTINUE
587 *
588  END IF
589 *
590 * Copy superdiagonal elements of D(K) to E(K) and
591 * ZERO out superdiagonal entry of A
592 *
593  e( k ) = a( k-1, k )
594  e( k-1 ) = zero
595  a( k-1, k ) = zero
596 *
597  END IF
598 *
599 * End column K is nonsingular
600 *
601  END IF
602 *
603 * Store details of the interchanges in IPIV
604 *
605  IF( kstep.EQ.1 ) THEN
606  ipiv( k ) = kp
607  ELSE
608  ipiv( k ) = -p
609  ipiv( k-1 ) = -kp
610  END IF
611 *
612 * Decrease K and return to the start of the main loop
613 *
614  k = k - kstep
615  GO TO 10
616 *
617  34 CONTINUE
618 *
619  ELSE
620 *
621 * Factorize A as L*D*L**T using the lower triangle of A
622 *
623 * Initialize the unused last entry of the subdiagonal array E.
624 *
625  e( n ) = zero
626 *
627 * K is the main loop index, increasing from 1 to N in steps of
628 * 1 or 2
629 *
630  k = 1
631  40 CONTINUE
632 *
633 * If K > N, exit from loop
634 *
635  IF( k.GT.n )
636  \$ GO TO 64
637  kstep = 1
638  p = k
639 *
640 * Determine rows and columns to be interchanged and whether
641 * a 1-by-1 or 2-by-2 pivot block will be used
642 *
643  absakk = abs( a( k, k ) )
644 *
645 * IMAX is the row-index of the largest off-diagonal element in
646 * column K, and COLMAX is its absolute value.
647 * Determine both COLMAX and IMAX.
648 *
649  IF( k.LT.n ) THEN
650  imax = k + isamax( n-k, a( k+1, k ), 1 )
651  colmax = abs( a( imax, k ) )
652  ELSE
653  colmax = zero
654  END IF
655 *
656  IF( ( max( absakk, colmax ).EQ.zero ) ) THEN
657 *
658 * Column K is zero or underflow: set INFO and continue
659 *
660  IF( info.EQ.0 )
661  \$ info = k
662  kp = k
663 *
664 * Set E( K ) to zero
665 *
666  IF( k.LT.n )
667  \$ e( k ) = zero
668 *
669  ELSE
670 *
671 * Test for interchange
672 *
673 * Equivalent to testing for (used to handle NaN and Inf)
674 * ABSAKK.GE.ALPHA*COLMAX
675 *
676  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
677 *
678 * no interchange, use 1-by-1 pivot block
679 *
680  kp = k
681 *
682  ELSE
683 *
684  done = .false.
685 *
686 * Loop until pivot found
687 *
688  42 CONTINUE
689 *
690 * Begin pivot search loop body
691 *
692 * JMAX is the column-index of the largest off-diagonal
693 * element in row IMAX, and ROWMAX is its absolute value.
694 * Determine both ROWMAX and JMAX.
695 *
696  IF( imax.NE.k ) THEN
697  jmax = k - 1 + isamax( imax-k, a( imax, k ), lda )
698  rowmax = abs( a( imax, jmax ) )
699  ELSE
700  rowmax = zero
701  END IF
702 *
703  IF( imax.LT.n ) THEN
704  itemp = imax + isamax( n-imax, a( imax+1, imax ),
705  \$ 1 )
706  stemp = abs( a( itemp, imax ) )
707  IF( stemp.GT.rowmax ) THEN
708  rowmax = stemp
709  jmax = itemp
710  END IF
711  END IF
712 *
713 * Equivalent to testing for (used to handle NaN and Inf)
714 * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
715 *
716  IF( .NOT.( abs( a( imax, imax ) ).LT.alpha*rowmax ) )
717  \$ THEN
718 *
719 * interchange rows and columns K and IMAX,
720 * use 1-by-1 pivot block
721 *
722  kp = imax
723  done = .true.
724 *
725 * Equivalent to testing for ROWMAX .EQ. COLMAX,
726 * used to handle NaN and Inf
727 *
728  ELSE IF( ( p.EQ.jmax ).OR.( rowmax.LE.colmax ) ) THEN
729 *
730 * interchange rows and columns K+1 and IMAX,
731 * use 2-by-2 pivot block
732 *
733  kp = imax
734  kstep = 2
735  done = .true.
736  ELSE
737 *
739 *
740  p = imax
741  colmax = rowmax
742  imax = jmax
743  END IF
744 *
745 * End pivot search loop body
746 *
747  IF( .NOT. done ) GOTO 42
748 *
749  END IF
750 *
751 * Swap TWO rows and TWO columns
752 *
753 * First swap
754 *
755  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
756 *
757 * Interchange rows and column K and P in the trailing
758 * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
759 *
760  IF( p.LT.n )
761  \$ CALL sswap( n-p, a( p+1, k ), 1, a( p+1, p ), 1 )
762  IF( p.GT.(k+1) )
763  \$ CALL sswap( p-k-1, a( k+1, k ), 1, a( p, k+1 ), lda )
764  t = a( k, k )
765  a( k, k ) = a( p, p )
766  a( p, p ) = t
767 *
768 * Convert lower triangle of A into L form by applying
769 * the interchanges in columns 1:k-1.
770 *
771  IF ( k.GT.1 )
772  \$ CALL sswap( k-1, a( k, 1 ), lda, a( p, 1 ), lda )
773 *
774  END IF
775 *
776 * Second swap
777 *
778  kk = k + kstep - 1
779  IF( kp.NE.kk ) THEN
780 *
781 * Interchange rows and columns KK and KP in the trailing
782 * submatrix A(k:n,k:n)
783 *
784  IF( kp.LT.n )
785  \$ CALL sswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
786  IF( ( kk.LT.n ) .AND. ( kp.GT.(kk+1) ) )
787  \$ CALL sswap( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
788  \$ lda )
789  t = a( kk, kk )
790  a( kk, kk ) = a( kp, kp )
791  a( kp, kp ) = t
792  IF( kstep.EQ.2 ) THEN
793  t = a( k+1, k )
794  a( k+1, k ) = a( kp, k )
795  a( kp, k ) = t
796  END IF
797 *
798 * Convert lower triangle of A into L form by applying
799 * the interchanges in columns 1:k-1.
800 *
801  IF ( k.GT.1 )
802  \$ CALL sswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
803 *
804  END IF
805 *
806 * Update the trailing submatrix
807 *
808  IF( kstep.EQ.1 ) THEN
809 *
810 * 1-by-1 pivot block D(k): column k now holds
811 *
812 * W(k) = L(k)*D(k)
813 *
814 * where L(k) is the k-th column of L
815 *
816  IF( k.LT.n ) THEN
817 *
818 * Perform a rank-1 update of A(k+1:n,k+1:n) and
819 * store L(k) in column k
820 *
821  IF( abs( a( k, k ) ).GE.sfmin ) THEN
822 *
823 * Perform a rank-1 update of A(k+1:n,k+1:n) as
824 * A := A - L(k)*D(k)*L(k)**T
825 * = A - W(k)*(1/D(k))*W(k)**T
826 *
827  d11 = one / a( k, k )
828  CALL ssyr( uplo, n-k, -d11, a( k+1, k ), 1,
829  \$ a( k+1, k+1 ), lda )
830 *
831 * Store L(k) in column k
832 *
833  CALL sscal( n-k, d11, a( k+1, k ), 1 )
834  ELSE
835 *
836 * Store L(k) in column k
837 *
838  d11 = a( k, k )
839  DO 46 ii = k + 1, n
840  a( ii, k ) = a( ii, k ) / d11
841  46 CONTINUE
842 *
843 * Perform a rank-1 update of A(k+1:n,k+1:n) as
844 * A := A - L(k)*D(k)*L(k)**T
845 * = A - W(k)*(1/D(k))*W(k)**T
846 * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
847 *
848  CALL ssyr( uplo, n-k, -d11, a( k+1, k ), 1,
849  \$ a( k+1, k+1 ), lda )
850  END IF
851 *
852 * Store the subdiagonal element of D in array E
853 *
854  e( k ) = zero
855 *
856  END IF
857 *
858  ELSE
859 *
860 * 2-by-2 pivot block D(k): columns k and k+1 now hold
861 *
862 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
863 *
864 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
865 * of L
866 *
867 *
868 * Perform a rank-2 update of A(k+2:n,k+2:n) as
869 *
870 * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
871 * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
872 *
873 * and store L(k) and L(k+1) in columns k and k+1
874 *
875  IF( k.LT.n-1 ) THEN
876 *
877  d21 = a( k+1, k )
878  d11 = a( k+1, k+1 ) / d21
879  d22 = a( k, k ) / d21
880  t = one / ( d11*d22-one )
881 *
882  DO 60 j = k + 2, n
883 *
884 * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
885 *
886  wk = t*( d11*a( j, k )-a( j, k+1 ) )
887  wkp1 = t*( d22*a( j, k+1 )-a( j, k ) )
888 *
889 * Perform a rank-2 update of A(k+2:n,k+2:n)
890 *
891  DO 50 i = j, n
892  a( i, j ) = a( i, j ) - ( a( i, k ) / d21 )*wk -
893  \$ ( a( i, k+1 ) / d21 )*wkp1
894  50 CONTINUE
895 *
896 * Store L(k) and L(k+1) in cols k and k+1 for row J
897 *
898  a( j, k ) = wk / d21
899  a( j, k+1 ) = wkp1 / d21
900 *
901  60 CONTINUE
902 *
903  END IF
904 *
905 * Copy subdiagonal elements of D(K) to E(K) and
906 * ZERO out subdiagonal entry of A
907 *
908  e( k ) = a( k+1, k )
909  e( k+1 ) = zero
910  a( k+1, k ) = zero
911 *
912  END IF
913 *
914 * End column K is nonsingular
915 *
916  END IF
917 *
918 * Store details of the interchanges in IPIV
919 *
920  IF( kstep.EQ.1 ) THEN
921  ipiv( k ) = kp
922  ELSE
923  ipiv( k ) = -p
924  ipiv( k+1 ) = -kp
925  END IF
926 *
927 * Increase K and return to the start of the main loop
928 *
929  k = k + kstep
930  GO TO 40
931 *
932  64 CONTINUE
933 *
934  END IF
935 *
936  RETURN
937 *
938 * End of SSYTF2_RK
939 *
940  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine ssyr(UPLO, N, ALPHA, X, INCX, A, LDA)
SSYR
Definition: ssyr.f:132
subroutine ssytf2_rk(UPLO, N, A, LDA, E, IPIV, INFO)
SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Ka...
Definition: ssytf2_rk.f:241