001:       SUBROUTINE SSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
002: *
003: *  -- LAPACK routine (version 3.2) --
004: *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
005: *     November 2006
006: *
007: *     .. Scalar Arguments ..
008:       CHARACTER          UPLO
009:       INTEGER            INFO, LDA, LWORK, N
010: *     ..
011: *     .. Array Arguments ..
012:       INTEGER            IPIV( * )
013:       REAL               A( LDA, * ), WORK( * )
014: *     ..
015: *
016: *  Purpose
017: *  =======
018: *
019: *  SSYTRF computes the factorization of a real symmetric matrix A using
020: *  the Bunch-Kaufman diagonal pivoting method.  The form of the
021: *  factorization is
022: *
023: *     A = U*D*U**T  or  A = L*D*L**T
024: *
025: *  where U (or L) is a product of permutation and unit upper (lower)
026: *  triangular matrices, and D is symmetric and block diagonal with 
027: *  1-by-1 and 2-by-2 diagonal blocks.
028: *
029: *  This is the blocked version of the algorithm, calling Level 3 BLAS.
030: *
031: *  Arguments
032: *  =========
033: *
034: *  UPLO    (input) CHARACTER*1
035: *          = 'U':  Upper triangle of A is stored;
036: *          = 'L':  Lower triangle of A is stored.
037: *
038: *  N       (input) INTEGER
039: *          The order of the matrix A.  N >= 0.
040: *
041: *  A       (input/output) REAL array, dimension (LDA,N)
042: *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
043: *          N-by-N upper triangular part of A contains the upper
044: *          triangular part of the matrix A, and the strictly lower
045: *          triangular part of A is not referenced.  If UPLO = 'L', the
046: *          leading N-by-N lower triangular part of A contains the lower
047: *          triangular part of the matrix A, and the strictly upper
048: *          triangular part of A is not referenced.
049: *
050: *          On exit, the block diagonal matrix D and the multipliers used
051: *          to obtain the factor U or L (see below for further details).
052: *
053: *  LDA     (input) INTEGER
054: *          The leading dimension of the array A.  LDA >= max(1,N).
055: *
056: *  IPIV    (output) INTEGER array, dimension (N)
057: *          Details of the interchanges and the block structure of D.
058: *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
059: *          interchanged and D(k,k) is a 1-by-1 diagonal block.
060: *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
061: *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
062: *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
063: *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
064: *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
065: *
066: *  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
067: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
068: *
069: *  LWORK   (input) INTEGER
070: *          The length of WORK.  LWORK >=1.  For best performance
071: *          LWORK >= N*NB, where NB is the block size returned by ILAENV.
072: *
073: *          If LWORK = -1, then a workspace query is assumed; the routine
074: *          only calculates the optimal size of the WORK array, returns
075: *          this value as the first entry of the WORK array, and no error
076: *          message related to LWORK is issued by XERBLA.
077: *
078: *  INFO    (output) INTEGER
079: *          = 0:  successful exit
080: *          < 0:  if INFO = -i, the i-th argument had an illegal value
081: *          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
082: *                has been completed, but the block diagonal matrix D is
083: *                exactly singular, and division by zero will occur if it
084: *                is used to solve a system of equations.
085: *
086: *  Further Details
087: *  ===============
088: *
089: *  If UPLO = 'U', then A = U*D*U', where
090: *     U = P(n)*U(n)* ... *P(k)U(k)* ...,
091: *  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
092: *  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
093: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
094: *  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
095: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
096: *
097: *             (   I    v    0   )   k-s
098: *     U(k) =  (   0    I    0   )   s
099: *             (   0    0    I   )   n-k
100: *                k-s   s   n-k
101: *
102: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
103: *  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
104: *  and A(k,k), and v overwrites A(1:k-2,k-1:k).
105: *
106: *  If UPLO = 'L', then A = L*D*L', where
107: *     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
108: *  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
109: *  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
110: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
111: *  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
112: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
113: *
114: *             (   I    0     0   )  k-1
115: *     L(k) =  (   0    I     0   )  s
116: *             (   0    v     I   )  n-k-s+1
117: *                k-1   s  n-k-s+1
118: *
119: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
120: *  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
121: *  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
122: *
123: *  =====================================================================
124: *
125: *     .. Local Scalars ..
126:       LOGICAL            LQUERY, UPPER
127:       INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
128: *     ..
129: *     .. External Functions ..
130:       LOGICAL            LSAME
131:       INTEGER            ILAENV
132:       EXTERNAL           LSAME, ILAENV
133: *     ..
134: *     .. External Subroutines ..
135:       EXTERNAL           SLASYF, SSYTF2, XERBLA
136: *     ..
137: *     .. Intrinsic Functions ..
138:       INTRINSIC          MAX
139: *     ..
140: *     .. Executable Statements ..
141: *
142: *     Test the input parameters.
143: *
144:       INFO = 0
145:       UPPER = LSAME( UPLO, 'U' )
146:       LQUERY = ( LWORK.EQ.-1 )
147:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
148:          INFO = -1
149:       ELSE IF( N.LT.0 ) THEN
150:          INFO = -2
151:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
152:          INFO = -4
153:       ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
154:          INFO = -7
155:       END IF
156: *
157:       IF( INFO.EQ.0 ) THEN
158: *
159: *        Determine the block size
160: *
161:          NB = ILAENV( 1, 'SSYTRF', UPLO, N, -1, -1, -1 )
162:          LWKOPT = N*NB
163:          WORK( 1 ) = LWKOPT
164:       END IF
165: *
166:       IF( INFO.NE.0 ) THEN
167:          CALL XERBLA( 'SSYTRF', -INFO )
168:          RETURN
169:       ELSE IF( LQUERY ) THEN
170:          RETURN
171:       END IF
172: *
173:       NBMIN = 2
174:       LDWORK = N
175:       IF( NB.GT.1 .AND. NB.LT.N ) THEN
176:          IWS = LDWORK*NB
177:          IF( LWORK.LT.IWS ) THEN
178:             NB = MAX( LWORK / LDWORK, 1 )
179:             NBMIN = MAX( 2, ILAENV( 2, 'SSYTRF', UPLO, N, -1, -1, -1 ) )
180:          END IF
181:       ELSE
182:          IWS = 1
183:       END IF
184:       IF( NB.LT.NBMIN )
185:      $   NB = N
186: *
187:       IF( UPPER ) THEN
188: *
189: *        Factorize A as U*D*U' using the upper triangle of A
190: *
191: *        K is the main loop index, decreasing from N to 1 in steps of
192: *        KB, where KB is the number of columns factorized by SLASYF;
193: *        KB is either NB or NB-1, or K for the last block
194: *
195:          K = N
196:    10    CONTINUE
197: *
198: *        If K < 1, exit from loop
199: *
200:          IF( K.LT.1 )
201:      $      GO TO 40
202: *
203:          IF( K.GT.NB ) THEN
204: *
205: *           Factorize columns k-kb+1:k of A and use blocked code to
206: *           update columns 1:k-kb
207: *
208:             CALL SLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
209:      $                   IINFO )
210:          ELSE
211: *
212: *           Use unblocked code to factorize columns 1:k of A
213: *
214:             CALL SSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
215:             KB = K
216:          END IF
217: *
218: *        Set INFO on the first occurrence of a zero pivot
219: *
220:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
221:      $      INFO = IINFO
222: *
223: *        Decrease K and return to the start of the main loop
224: *
225:          K = K - KB
226:          GO TO 10
227: *
228:       ELSE
229: *
230: *        Factorize A as L*D*L' using the lower triangle of A
231: *
232: *        K is the main loop index, increasing from 1 to N in steps of
233: *        KB, where KB is the number of columns factorized by SLASYF;
234: *        KB is either NB or NB-1, or N-K+1 for the last block
235: *
236:          K = 1
237:    20    CONTINUE
238: *
239: *        If K > N, exit from loop
240: *
241:          IF( K.GT.N )
242:      $      GO TO 40
243: *
244:          IF( K.LE.N-NB ) THEN
245: *
246: *           Factorize columns k:k+kb-1 of A and use blocked code to
247: *           update columns k+kb:n
248: *
249:             CALL SLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
250:      $                   WORK, LDWORK, IINFO )
251:          ELSE
252: *
253: *           Use unblocked code to factorize columns k:n of A
254: *
255:             CALL SSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
256:             KB = N - K + 1
257:          END IF
258: *
259: *        Set INFO on the first occurrence of a zero pivot
260: *
261:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
262:      $      INFO = IINFO + K - 1
263: *
264: *        Adjust IPIV
265: *
266:          DO 30 J = K, K + KB - 1
267:             IF( IPIV( J ).GT.0 ) THEN
268:                IPIV( J ) = IPIV( J ) + K - 1
269:             ELSE
270:                IPIV( J ) = IPIV( J ) - K + 1
271:             END IF
272:    30    CONTINUE
273: *
274: *        Increase K and return to the start of the main loop
275: *
276:          K = K + KB
277:          GO TO 20
278: *
279:       END IF
280: *
281:    40 CONTINUE
282:       WORK( 1 ) = LWKOPT
283:       RETURN
284: *
285: *     End of SSYTRF
286: *
287:       END
288: