1: *> \brief \b DSYTRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, LWORK, N
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * DOUBLE PRECISION A( LDA, * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYTRF computes the factorization of a real symmetric matrix A using
39: *> the Bunch-Kaufman diagonal pivoting method. The form of the
40: *> factorization is
41: *>
42: *> A = U*D*U**T or A = L*D*L**T
43: *>
44: *> where U (or L) is a product of permutation and unit upper (lower)
45: *> triangular matrices, and D is symmetric and block diagonal with
46: *> 1-by-1 and 2-by-2 diagonal blocks.
47: *>
48: *> This is the blocked version of the algorithm, calling Level 3 BLAS.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] UPLO
55: *> \verbatim
56: *> UPLO is CHARACTER*1
57: *> = 'U': Upper triangle of A is stored;
58: *> = 'L': Lower triangle of A is stored.
59: *> \endverbatim
60: *>
61: *> \param[in] N
62: *> \verbatim
63: *> N is INTEGER
64: *> The order of the matrix A. N >= 0.
65: *> \endverbatim
66: *>
67: *> \param[in,out] A
68: *> \verbatim
69: *> A is DOUBLE PRECISION array, dimension (LDA,N)
70: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
71: *> N-by-N upper triangular part of A contains the upper
72: *> triangular part of the matrix A, and the strictly lower
73: *> triangular part of A is not referenced. If UPLO = 'L', the
74: *> leading N-by-N lower triangular part of A contains the lower
75: *> triangular part of the matrix A, and the strictly upper
76: *> triangular part of A is not referenced.
77: *>
78: *> On exit, the block diagonal matrix D and the multipliers used
79: *> to obtain the factor U or L (see below for further details).
80: *> \endverbatim
81: *>
82: *> \param[in] LDA
83: *> \verbatim
84: *> LDA is INTEGER
85: *> The leading dimension of the array A. LDA >= max(1,N).
86: *> \endverbatim
87: *>
88: *> \param[out] IPIV
89: *> \verbatim
90: *> IPIV is INTEGER array, dimension (N)
91: *> Details of the interchanges and the block structure of D.
92: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
93: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
94: *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
95: *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
96: *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
97: *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
98: *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
99: *> \endverbatim
100: *>
101: *> \param[out] WORK
102: *> \verbatim
103: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
104: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
105: *> \endverbatim
106: *>
107: *> \param[in] LWORK
108: *> \verbatim
109: *> LWORK is INTEGER
110: *> The length of WORK. LWORK >=1. For best performance
111: *> LWORK >= N*NB, where NB is the block size returned by ILAENV.
112: *>
113: *> If LWORK = -1, then a workspace query is assumed; the routine
114: *> only calculates the optimal size of the WORK array, returns
115: *> this value as the first entry of the WORK array, and no error
116: *> message related to LWORK is issued by XERBLA.
117: *> \endverbatim
118: *>
119: *> \param[out] INFO
120: *> \verbatim
121: *> INFO is INTEGER
122: *> = 0: successful exit
123: *> < 0: if INFO = -i, the i-th argument had an illegal value
124: *> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
125: *> has been completed, but the block diagonal matrix D is
126: *> exactly singular, and division by zero will occur if it
127: *> is used to solve a system of equations.
128: *> \endverbatim
129: *
130: * Authors:
131: * ========
132: *
133: *> \author Univ. of Tennessee
134: *> \author Univ. of California Berkeley
135: *> \author Univ. of Colorado Denver
136: *> \author NAG Ltd.
137: *
138: *> \date November 2011
139: *
140: *> \ingroup doubleSYcomputational
141: *
142: *> \par Further Details:
143: * =====================
144: *>
145: *> \verbatim
146: *>
147: *> If UPLO = 'U', then A = U*D*U**T, where
148: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
149: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
150: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
151: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
152: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
153: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
154: *>
155: *> ( I v 0 ) k-s
156: *> U(k) = ( 0 I 0 ) s
157: *> ( 0 0 I ) n-k
158: *> k-s s n-k
159: *>
160: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
161: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
162: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
163: *>
164: *> If UPLO = 'L', then A = L*D*L**T, where
165: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
166: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
167: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
168: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
169: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
170: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
171: *>
172: *> ( I 0 0 ) k-1
173: *> L(k) = ( 0 I 0 ) s
174: *> ( 0 v I ) n-k-s+1
175: *> k-1 s n-k-s+1
176: *>
177: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
178: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
179: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
180: *> \endverbatim
181: *>
182: * =====================================================================
183: SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
184: *
185: * -- LAPACK computational routine (version 3.4.0) --
186: * -- LAPACK is a software package provided by Univ. of Tennessee, --
187: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
188: * November 2011
189: *
190: * .. Scalar Arguments ..
191: CHARACTER UPLO
192: INTEGER INFO, LDA, LWORK, N
193: * ..
194: * .. Array Arguments ..
195: INTEGER IPIV( * )
196: DOUBLE PRECISION A( LDA, * ), WORK( * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Local Scalars ..
202: LOGICAL LQUERY, UPPER
203: INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
204: * ..
205: * .. External Functions ..
206: LOGICAL LSAME
207: INTEGER ILAENV
208: EXTERNAL LSAME, ILAENV
209: * ..
210: * .. External Subroutines ..
211: EXTERNAL DLASYF, DSYTF2, XERBLA
212: * ..
213: * .. Intrinsic Functions ..
214: INTRINSIC MAX
215: * ..
216: * .. Executable Statements ..
217: *
218: * Test the input parameters.
219: *
220: INFO = 0
221: UPPER = LSAME( UPLO, 'U' )
222: LQUERY = ( LWORK.EQ.-1 )
223: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
224: INFO = -1
225: ELSE IF( N.LT.0 ) THEN
226: INFO = -2
227: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
228: INFO = -4
229: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
230: INFO = -7
231: END IF
232: *
233: IF( INFO.EQ.0 ) THEN
234: *
235: * Determine the block size
236: *
237: NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
238: LWKOPT = N*NB
239: WORK( 1 ) = LWKOPT
240: END IF
241: *
242: IF( INFO.NE.0 ) THEN
243: CALL XERBLA( 'DSYTRF', -INFO )
244: RETURN
245: ELSE IF( LQUERY ) THEN
246: RETURN
247: END IF
248: *
249: NBMIN = 2
250: LDWORK = N
251: IF( NB.GT.1 .AND. NB.LT.N ) THEN
252: IWS = LDWORK*NB
253: IF( LWORK.LT.IWS ) THEN
254: NB = MAX( LWORK / LDWORK, 1 )
255: NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1, -1 ) )
256: END IF
257: ELSE
258: IWS = 1
259: END IF
260: IF( NB.LT.NBMIN )
261: $ NB = N
262: *
263: IF( UPPER ) THEN
264: *
265: * Factorize A as U*D*U**T using the upper triangle of A
266: *
267: * K is the main loop index, decreasing from N to 1 in steps of
268: * KB, where KB is the number of columns factorized by DLASYF;
269: * KB is either NB or NB-1, or K for the last block
270: *
271: K = N
272: 10 CONTINUE
273: *
274: * If K < 1, exit from loop
275: *
276: IF( K.LT.1 )
277: $ GO TO 40
278: *
279: IF( K.GT.NB ) THEN
280: *
281: * Factorize columns k-kb+1:k of A and use blocked code to
282: * update columns 1:k-kb
283: *
284: CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
285: $ IINFO )
286: ELSE
287: *
288: * Use unblocked code to factorize columns 1:k of A
289: *
290: CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
291: KB = K
292: END IF
293: *
294: * Set INFO on the first occurrence of a zero pivot
295: *
296: IF( INFO.EQ.0 .AND. IINFO.GT.0 )
297: $ INFO = IINFO
298: *
299: * Decrease K and return to the start of the main loop
300: *
301: K = K - KB
302: GO TO 10
303: *
304: ELSE
305: *
306: * Factorize A as L*D*L**T using the lower triangle of A
307: *
308: * K is the main loop index, increasing from 1 to N in steps of
309: * KB, where KB is the number of columns factorized by DLASYF;
310: * KB is either NB or NB-1, or N-K+1 for the last block
311: *
312: K = 1
313: 20 CONTINUE
314: *
315: * If K > N, exit from loop
316: *
317: IF( K.GT.N )
318: $ GO TO 40
319: *
320: IF( K.LE.N-NB ) THEN
321: *
322: * Factorize columns k:k+kb-1 of A and use blocked code to
323: * update columns k+kb:n
324: *
325: CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
326: $ WORK, LDWORK, IINFO )
327: ELSE
328: *
329: * Use unblocked code to factorize columns k:n of A
330: *
331: CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
332: KB = N - K + 1
333: END IF
334: *
335: * Set INFO on the first occurrence of a zero pivot
336: *
337: IF( INFO.EQ.0 .AND. IINFO.GT.0 )
338: $ INFO = IINFO + K - 1
339: *
340: * Adjust IPIV
341: *
342: DO 30 J = K, K + KB - 1
343: IF( IPIV( J ).GT.0 ) THEN
344: IPIV( J ) = IPIV( J ) + K - 1
345: ELSE
346: IPIV( J ) = IPIV( J ) - K + 1
347: END IF
348: 30 CONTINUE
349: *
350: * Increase K and return to the start of the main loop
351: *
352: K = K + KB
353: GO TO 20
354: *
355: END IF
356: *
357: 40 CONTINUE
358: WORK( 1 ) = LWKOPT
359: RETURN
360: *
361: * End of DSYTRF
362: *
363: END
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