LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ ssytrf_rook()

 subroutine ssytrf_rook ( character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO )

SSYTRF_ROOK

Purpose:
``` SSYTRF_ROOK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
The form of the factorization is

A = U*D*U**T  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.```
Further Details:
```  If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    v    0   )   k-s
U(k) =  (   0    I    0   )   s
(   0    0    I   )   n-k
k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

(   I    0     0   )  k-1
L(k) =  (   0    I     0   )  s
(   0    v     I   )  n-k-s+1
k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).```
Contributors:
```   June 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 207 of file ssytrf_rook.f.

208*
209* -- LAPACK computational routine --
210* -- LAPACK is a software package provided by Univ. of Tennessee, --
211* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
212*
213* .. Scalar Arguments ..
214 CHARACTER UPLO
215 INTEGER INFO, LDA, LWORK, N
216* ..
217* .. Array Arguments ..
218 INTEGER IPIV( * )
219 REAL A( LDA, * ), WORK( * )
220* ..
221*
222* =====================================================================
223*
224* .. Local Scalars ..
225 LOGICAL LQUERY, UPPER
226 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
227* ..
228* .. External Functions ..
229 LOGICAL LSAME
230 INTEGER ILAENV
231 EXTERNAL lsame, ilaenv
232* ..
233* .. External Subroutines ..
235* ..
236* .. Intrinsic Functions ..
237 INTRINSIC max
238* ..
239* .. Executable Statements ..
240*
241* Test the input parameters.
242*
243 info = 0
244 upper = lsame( uplo, 'U' )
245 lquery = ( lwork.EQ.-1 )
246 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
247 info = -1
248 ELSE IF( n.LT.0 ) THEN
249 info = -2
250 ELSE IF( lda.LT.max( 1, n ) ) THEN
251 info = -4
252 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
253 info = -7
254 END IF
255*
256 IF( info.EQ.0 ) THEN
257*
258* Determine the block size
259*
260 nb = ilaenv( 1, 'SSYTRF_ROOK', uplo, n, -1, -1, -1 )
261 lwkopt = max( 1, n*nb )
262 work( 1 ) = lwkopt
263 END IF
264*
265 IF( info.NE.0 ) THEN
266 CALL xerbla( 'SSYTRF_ROOK', -info )
267 RETURN
268 ELSE IF( lquery ) THEN
269 RETURN
270 END IF
271*
272 nbmin = 2
273 ldwork = n
274 IF( nb.GT.1 .AND. nb.LT.n ) THEN
275 iws = ldwork*nb
276 IF( lwork.LT.iws ) THEN
277 nb = max( lwork / ldwork, 1 )
278 nbmin = max( 2, ilaenv( 2, 'SSYTRF_ROOK',
279 \$ uplo, n, -1, -1, -1 ) )
280 END IF
281 ELSE
282 iws = 1
283 END IF
284 IF( nb.LT.nbmin )
285 \$ nb = n
286*
287 IF( upper ) THEN
288*
289* Factorize A as U*D*U**T using the upper triangle of A
290*
291* K is the main loop index, decreasing from N to 1 in steps of
292* KB, where KB is the number of columns factorized by SLASYF_ROOK;
293* KB is either NB or NB-1, or K for the last block
294*
295 k = n
296 10 CONTINUE
297*
298* If K < 1, exit from loop
299*
300 IF( k.LT.1 )
301 \$ GO TO 40
302*
303 IF( k.GT.nb ) THEN
304*
305* Factorize columns k-kb+1:k of A and use blocked code to
306* update columns 1:k-kb
307*
308 CALL slasyf_rook( uplo, k, nb, kb, a, lda,
309 \$ ipiv, work, ldwork, iinfo )
310 ELSE
311*
312* Use unblocked code to factorize columns 1:k of A
313*
314 CALL ssytf2_rook( uplo, k, a, lda, ipiv, iinfo )
315 kb = k
316 END IF
317*
318* Set INFO on the first occurrence of a zero pivot
319*
320 IF( info.EQ.0 .AND. iinfo.GT.0 )
321 \$ info = iinfo
322*
323* No need to adjust IPIV
324*
325* Decrease K and return to the start of the main loop
326*
327 k = k - kb
328 GO TO 10
329*
330 ELSE
331*
332* Factorize A as L*D*L**T using the lower triangle of A
333*
334* K is the main loop index, increasing from 1 to N in steps of
335* KB, where KB is the number of columns factorized by SLASYF_ROOK;
336* KB is either NB or NB-1, or N-K+1 for the last block
337*
338 k = 1
339 20 CONTINUE
340*
341* If K > N, exit from loop
342*
343 IF( k.GT.n )
344 \$ GO TO 40
345*
346 IF( k.LE.n-nb ) THEN
347*
348* Factorize columns k:k+kb-1 of A and use blocked code to
349* update columns k+kb:n
350*
351 CALL slasyf_rook( uplo, n-k+1, nb, kb, a( k, k ), lda,
352 \$ ipiv( k ), work, ldwork, iinfo )
353 ELSE
354*
355* Use unblocked code to factorize columns k:n of A
356*
357 CALL ssytf2_rook( uplo, n-k+1, a( k, k ), lda, ipiv( k ),
358 \$ iinfo )
359 kb = n - k + 1
360 END IF
361*
362* Set INFO on the first occurrence of a zero pivot
363*
364 IF( info.EQ.0 .AND. iinfo.GT.0 )
365 \$ info = iinfo + k - 1
366*
368*
369 DO 30 j = k, k + kb - 1
370 IF( ipiv( j ).GT.0 ) THEN
371 ipiv( j ) = ipiv( j ) + k - 1
372 ELSE
373 ipiv( j ) = ipiv( j ) - k + 1
374 END IF
375 30 CONTINUE
376*
377* Increase K and return to the start of the main loop
378*
379 k = k + kb
380 GO TO 20
381*
382 END IF
383*
384 40 CONTINUE
385 work( 1 ) = lwkopt
386 RETURN
387*
388* End of SSYTRF_ROOK
389*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ssytf2_rook(UPLO, N, A, LDA, IPIV, INFO)
SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-...
Definition: ssytf2_rook.f:194
subroutine slasyf_rook(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufm...
Definition: slasyf_rook.f:184
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