LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ ssytrf()

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

SSYTRF

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Purpose:
``` SSYTRF computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method.  The form of the
factorization is

A = U**T*D*U  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 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 UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and 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**T*D*U, 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).```

Definition at line 181 of file ssytrf.f.

182 *
183 * -- LAPACK computational routine --
184 * -- LAPACK is a software package provided by Univ. of Tennessee, --
185 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186 *
187 * .. Scalar Arguments ..
188  CHARACTER UPLO
189  INTEGER INFO, LDA, LWORK, N
190 * ..
191 * .. Array Arguments ..
192  INTEGER IPIV( * )
193  REAL A( LDA, * ), WORK( * )
194 * ..
195 *
196 * =====================================================================
197 *
198 * .. Local Scalars ..
199  LOGICAL LQUERY, UPPER
200  INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
201 * ..
202 * .. External Functions ..
203  LOGICAL LSAME
204  INTEGER ILAENV
205  EXTERNAL lsame, ilaenv
206 * ..
207 * .. External Subroutines ..
208  EXTERNAL slasyf, ssytf2, xerbla
209 * ..
210 * .. Intrinsic Functions ..
211  INTRINSIC max
212 * ..
213 * .. Executable Statements ..
214 *
215 * Test the input parameters.
216 *
217  info = 0
218  upper = lsame( uplo, 'U' )
219  lquery = ( lwork.EQ.-1 )
220  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
221  info = -1
222  ELSE IF( n.LT.0 ) THEN
223  info = -2
224  ELSE IF( lda.LT.max( 1, n ) ) THEN
225  info = -4
226  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
227  info = -7
228  END IF
229 *
230  IF( info.EQ.0 ) THEN
231 *
232 * Determine the block size
233 *
234  nb = ilaenv( 1, 'SSYTRF', uplo, n, -1, -1, -1 )
235  lwkopt = n*nb
236  work( 1 ) = lwkopt
237  END IF
238 *
239  IF( info.NE.0 ) THEN
240  CALL xerbla( 'SSYTRF', -info )
241  RETURN
242  ELSE IF( lquery ) THEN
243  RETURN
244  END IF
245 *
246  nbmin = 2
247  ldwork = n
248  IF( nb.GT.1 .AND. nb.LT.n ) THEN
249  iws = ldwork*nb
250  IF( lwork.LT.iws ) THEN
251  nb = max( lwork / ldwork, 1 )
252  nbmin = max( 2, ilaenv( 2, 'SSYTRF', uplo, n, -1, -1, -1 ) )
253  END IF
254  ELSE
255  iws = 1
256  END IF
257  IF( nb.LT.nbmin )
258  \$ nb = n
259 *
260  IF( upper ) THEN
261 *
262 * Factorize A as U**T*D*U using the upper triangle of A
263 *
264 * K is the main loop index, decreasing from N to 1 in steps of
265 * KB, where KB is the number of columns factorized by SLASYF;
266 * KB is either NB or NB-1, or K for the last block
267 *
268  k = n
269  10 CONTINUE
270 *
271 * If K < 1, exit from loop
272 *
273  IF( k.LT.1 )
274  \$ GO TO 40
275 *
276  IF( k.GT.nb ) THEN
277 *
278 * Factorize columns k-kb+1:k of A and use blocked code to
279 * update columns 1:k-kb
280 *
281  CALL slasyf( uplo, k, nb, kb, a, lda, ipiv, work, ldwork,
282  \$ iinfo )
283  ELSE
284 *
285 * Use unblocked code to factorize columns 1:k of A
286 *
287  CALL ssytf2( uplo, k, a, lda, ipiv, iinfo )
288  kb = k
289  END IF
290 *
291 * Set INFO on the first occurrence of a zero pivot
292 *
293  IF( info.EQ.0 .AND. iinfo.GT.0 )
294  \$ info = iinfo
295 *
296 * Decrease K and return to the start of the main loop
297 *
298  k = k - kb
299  GO TO 10
300 *
301  ELSE
302 *
303 * Factorize A as L*D*L**T using the lower triangle of A
304 *
305 * K is the main loop index, increasing from 1 to N in steps of
306 * KB, where KB is the number of columns factorized by SLASYF;
307 * KB is either NB or NB-1, or N-K+1 for the last block
308 *
309  k = 1
310  20 CONTINUE
311 *
312 * If K > N, exit from loop
313 *
314  IF( k.GT.n )
315  \$ GO TO 40
316 *
317  IF( k.LE.n-nb ) THEN
318 *
319 * Factorize columns k:k+kb-1 of A and use blocked code to
320 * update columns k+kb:n
321 *
322  CALL slasyf( uplo, n-k+1, nb, kb, a( k, k ), lda, ipiv( k ),
323  \$ work, ldwork, iinfo )
324  ELSE
325 *
326 * Use unblocked code to factorize columns k:n of A
327 *
328  CALL ssytf2( uplo, n-k+1, a( k, k ), lda, ipiv( k ), iinfo )
329  kb = n - k + 1
330  END IF
331 *
332 * Set INFO on the first occurrence of a zero pivot
333 *
334  IF( info.EQ.0 .AND. iinfo.GT.0 )
335  \$ info = iinfo + k - 1
336 *
337 * Adjust IPIV
338 *
339  DO 30 j = k, k + kb - 1
340  IF( ipiv( j ).GT.0 ) THEN
341  ipiv( j ) = ipiv( j ) + k - 1
342  ELSE
343  ipiv( j ) = ipiv( j ) - k + 1
344  END IF
345  30 CONTINUE
346 *
347 * Increase K and return to the start of the main loop
348 *
349  k = k + kb
350  GO TO 20
351 *
352  END IF
353 *
354  40 CONTINUE
355  work( 1 ) = lwkopt
356  RETURN
357 *
358 * End of SSYTRF
359 *
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 slasyf(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal p...
Definition: slasyf.f:176
subroutine ssytf2(UPLO, N, A, LDA, IPIV, INFO)
SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting ...
Definition: ssytf2.f:195
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