Annotation of rpl/lapack/lapack/dsytrs_3.f, revision 1.4
1.1 bertrand 1: *> \brief \b DSYTRS_3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYTRS_3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrs_3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrs_3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrs_3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDA, LDB, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), E( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *> DSYTRS_3 solves a system of linear equations A * X = B with a real
39: *> symmetric matrix A using the factorization computed
40: *> by DSYTRF_RK or DSYTRF_BK:
41: *>
42: *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
43: *>
44: *> where U (or L) is unit upper (or lower) triangular matrix,
45: *> U**T (or L**T) is the transpose of U (or L), P is a permutation
46: *> matrix, P**T is the transpose of P, and D is symmetric and block
47: *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
48: *>
49: *> This algorithm is using Level 3 BLAS.
50: *> \endverbatim
51: *
52: * Arguments:
53: * ==========
54: *
55: *> \param[in] UPLO
56: *> \verbatim
57: *> UPLO is CHARACTER*1
58: *> Specifies whether the details of the factorization are
59: *> stored as an upper or lower triangular matrix:
60: *> = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
61: *> = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
62: *> \endverbatim
63: *>
64: *> \param[in] N
65: *> \verbatim
66: *> N is INTEGER
67: *> The order of the matrix A. N >= 0.
68: *> \endverbatim
69: *>
70: *> \param[in] NRHS
71: *> \verbatim
72: *> NRHS is INTEGER
73: *> The number of right hand sides, i.e., the number of columns
74: *> of the matrix B. NRHS >= 0.
75: *> \endverbatim
76: *>
77: *> \param[in] A
78: *> \verbatim
79: *> A is DOUBLE PRECISION array, dimension (LDA,N)
80: *> Diagonal of the block diagonal matrix D and factors U or L
81: *> as computed by DSYTRF_RK and DSYTRF_BK:
82: *> a) ONLY diagonal elements of the symmetric block diagonal
83: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
84: *> (superdiagonal (or subdiagonal) elements of D
85: *> should be provided on entry in array E), and
86: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
87: *> If UPLO = 'L': factor L in the subdiagonal part of A.
88: *> \endverbatim
89: *>
90: *> \param[in] LDA
91: *> \verbatim
92: *> LDA is INTEGER
93: *> The leading dimension of the array A. LDA >= max(1,N).
94: *> \endverbatim
95: *>
96: *> \param[in] E
97: *> \verbatim
98: *> E is DOUBLE PRECISION array, dimension (N)
99: *> On entry, contains the superdiagonal (or subdiagonal)
100: *> elements of the symmetric block diagonal matrix D
101: *> with 1-by-1 or 2-by-2 diagonal blocks, where
1.3 bertrand 102: *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
1.1 bertrand 103: *> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
104: *>
105: *> NOTE: For 1-by-1 diagonal block D(k), where
106: *> 1 <= k <= N, the element E(k) is not referenced in both
107: *> UPLO = 'U' or UPLO = 'L' cases.
108: *> \endverbatim
109: *>
110: *> \param[in] IPIV
111: *> \verbatim
112: *> IPIV is INTEGER array, dimension (N)
113: *> Details of the interchanges and the block structure of D
114: *> as determined by DSYTRF_RK or DSYTRF_BK.
115: *> \endverbatim
116: *>
117: *> \param[in,out] B
118: *> \verbatim
119: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
120: *> On entry, the right hand side matrix B.
121: *> On exit, the solution matrix X.
122: *> \endverbatim
123: *>
124: *> \param[in] LDB
125: *> \verbatim
126: *> LDB is INTEGER
127: *> The leading dimension of the array B. LDB >= max(1,N).
128: *> \endverbatim
129: *>
130: *> \param[out] INFO
131: *> \verbatim
132: *> INFO is INTEGER
133: *> = 0: successful exit
134: *> < 0: if INFO = -i, the i-th argument had an illegal value
135: *> \endverbatim
136: *
137: * Authors:
138: * ========
139: *
140: *> \author Univ. of Tennessee
141: *> \author Univ. of California Berkeley
142: *> \author Univ. of Colorado Denver
143: *> \author NAG Ltd.
144: *
1.3 bertrand 145: *> \date June 2017
1.1 bertrand 146: *
147: *> \ingroup doubleSYcomputational
148: *
149: *> \par Contributors:
150: * ==================
151: *>
152: *> \verbatim
153: *>
1.3 bertrand 154: *> June 2017, Igor Kozachenko,
1.1 bertrand 155: *> Computer Science Division,
156: *> University of California, Berkeley
157: *>
158: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
159: *> School of Mathematics,
160: *> University of Manchester
161: *>
162: *> \endverbatim
163: *
164: * =====================================================================
165: SUBROUTINE DSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
166: $ INFO )
167: *
1.3 bertrand 168: * -- LAPACK computational routine (version 3.7.1) --
1.1 bertrand 169: * -- LAPACK is a software package provided by Univ. of Tennessee, --
170: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.3 bertrand 171: * June 2017
1.1 bertrand 172: *
173: * .. Scalar Arguments ..
174: CHARACTER UPLO
175: INTEGER INFO, LDA, LDB, N, NRHS
176: * ..
177: * .. Array Arguments ..
178: INTEGER IPIV( * )
179: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), E( * )
180: * ..
181: *
182: * =====================================================================
183: *
184: * .. Parameters ..
185: DOUBLE PRECISION ONE
186: PARAMETER ( ONE = 1.0D+0 )
187: * ..
188: * .. Local Scalars ..
189: LOGICAL UPPER
190: INTEGER I, J, K, KP
191: DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
192: * ..
193: * .. External Functions ..
194: LOGICAL LSAME
195: EXTERNAL LSAME
196: * ..
197: * .. External Subroutines ..
198: EXTERNAL DSCAL, DSWAP, DTRSM, XERBLA
199: * ..
200: * .. Intrinsic Functions ..
201: INTRINSIC ABS, MAX
202: * ..
203: * .. Executable Statements ..
204: *
205: INFO = 0
206: UPPER = LSAME( UPLO, 'U' )
207: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
208: INFO = -1
209: ELSE IF( N.LT.0 ) THEN
210: INFO = -2
211: ELSE IF( NRHS.LT.0 ) THEN
212: INFO = -3
213: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
214: INFO = -5
215: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
216: INFO = -9
217: END IF
218: IF( INFO.NE.0 ) THEN
219: CALL XERBLA( 'DSYTRS_3', -INFO )
220: RETURN
221: END IF
222: *
223: * Quick return if possible
224: *
225: IF( N.EQ.0 .OR. NRHS.EQ.0 )
226: $ RETURN
227: *
228: IF( UPPER ) THEN
229: *
230: * Begin Upper
231: *
232: * Solve A*X = B, where A = U*D*U**T.
233: *
234: * P**T * B
235: *
236: * Interchange rows K and IPIV(K) of matrix B in the same order
237: * that the formation order of IPIV(I) vector for Upper case.
238: *
239: * (We can do the simple loop over IPIV with decrement -1,
240: * since the ABS value of IPIV( I ) represents the row index
241: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
242: *
243: DO K = N, 1, -1
244: KP = ABS( IPIV( K ) )
245: IF( KP.NE.K ) THEN
246: CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
247: END IF
248: END DO
249: *
250: * Compute (U \P**T * B) -> B [ (U \P**T * B) ]
251: *
252: CALL DTRSM( 'L', 'U', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
253: *
254: * Compute D \ B -> B [ D \ (U \P**T * B) ]
255: *
256: I = N
257: DO WHILE ( I.GE.1 )
258: IF( IPIV( I ).GT.0 ) THEN
259: CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
260: ELSE IF ( I.GT.1 ) THEN
261: AKM1K = E( I )
262: AKM1 = A( I-1, I-1 ) / AKM1K
263: AK = A( I, I ) / AKM1K
264: DENOM = AKM1*AK - ONE
265: DO J = 1, NRHS
266: BKM1 = B( I-1, J ) / AKM1K
267: BK = B( I, J ) / AKM1K
268: B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
269: B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
270: END DO
271: I = I - 1
272: END IF
273: I = I - 1
274: END DO
275: *
276: * Compute (U**T \ B) -> B [ U**T \ (D \ (U \P**T * B) ) ]
277: *
278: CALL DTRSM( 'L', 'U', 'T', 'U', N, NRHS, ONE, A, LDA, B, LDB )
279: *
280: * P * B [ P * (U**T \ (D \ (U \P**T * B) )) ]
281: *
282: * Interchange rows K and IPIV(K) of matrix B in reverse order
283: * from the formation order of IPIV(I) vector for Upper case.
284: *
285: * (We can do the simple loop over IPIV with increment 1,
286: * since the ABS value of IPIV(I) represents the row index
287: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
288: *
289: DO K = 1, N
290: KP = ABS( IPIV( K ) )
291: IF( KP.NE.K ) THEN
292: CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
293: END IF
294: END DO
295: *
296: ELSE
297: *
298: * Begin Lower
299: *
300: * Solve A*X = B, where A = L*D*L**T.
301: *
302: * P**T * B
303: * Interchange rows K and IPIV(K) of matrix B in the same order
304: * that the formation order of IPIV(I) vector for Lower case.
305: *
306: * (We can do the simple loop over IPIV with increment 1,
307: * since the ABS value of IPIV(I) represents the row index
308: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
309: *
310: DO K = 1, N
311: KP = ABS( IPIV( K ) )
312: IF( KP.NE.K ) THEN
313: CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
314: END IF
315: END DO
316: *
317: * Compute (L \P**T * B) -> B [ (L \P**T * B) ]
318: *
319: CALL DTRSM( 'L', 'L', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
320: *
321: * Compute D \ B -> B [ D \ (L \P**T * B) ]
322: *
323: I = 1
324: DO WHILE ( I.LE.N )
325: IF( IPIV( I ).GT.0 ) THEN
326: CALL DSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB )
327: ELSE IF( I.LT.N ) THEN
328: AKM1K = E( I )
329: AKM1 = A( I, I ) / AKM1K
330: AK = A( I+1, I+1 ) / AKM1K
331: DENOM = AKM1*AK - ONE
332: DO J = 1, NRHS
333: BKM1 = B( I, J ) / AKM1K
334: BK = B( I+1, J ) / AKM1K
335: B( I, J ) = ( AK*BKM1-BK ) / DENOM
336: B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
337: END DO
338: I = I + 1
339: END IF
340: I = I + 1
341: END DO
342: *
343: * Compute (L**T \ B) -> B [ L**T \ (D \ (L \P**T * B) ) ]
344: *
345: CALL DTRSM('L', 'L', 'T', 'U', N, NRHS, ONE, A, LDA, B, LDB )
346: *
347: * P * B [ P * (L**T \ (D \ (L \P**T * B) )) ]
348: *
349: * Interchange rows K and IPIV(K) of matrix B in reverse order
350: * from the formation order of IPIV(I) vector for Lower case.
351: *
352: * (We can do the simple loop over IPIV with decrement -1,
353: * since the ABS value of IPIV(I) represents the row index
354: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
355: *
356: DO K = N, 1, -1
357: KP = ABS( IPIV( K ) )
358: IF( KP.NE.K ) THEN
359: CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
360: END IF
361: END DO
362: *
363: * END Lower
364: *
365: END IF
366: *
367: RETURN
368: *
369: * End of DSYTRS_3
370: *
371: END
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