1: *> \brief \b ZHETRS_3
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHETRS_3 + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrs_3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHETRS_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: * COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *> ZHETRS_3 solves a system of linear equations A * X = B with a complex
39: *> Hermitian matrix A using the factorization computed
40: *> by ZHETRF_RK or ZHETRF_BK:
41: *>
42: *> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
43: *>
44: *> where U (or L) is unit upper (or lower) triangular matrix,
45: *> U**H (or L**H) is the conjugate of U (or L), P is a permutation
46: *> matrix, P**T is the transpose of P, and D is Hermitian 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**H)*(P**T);
61: *> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(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 COMPLEX*16 array, dimension (LDA,N)
80: *> Diagonal of the block diagonal matrix D and factors U or L
81: *> as computed by ZHETRF_RK and ZHETRF_BK:
82: *> a) ONLY diagonal elements of the Hermitian 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 COMPLEX*16 array, dimension (N)
99: *> On entry, contains the superdiagonal (or subdiagonal)
100: *> elements of the Hermitian block diagonal matrix D
101: *> with 1-by-1 or 2-by-2 diagonal blocks, where
102: *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not refernced;
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 ZHETRF_RK or ZHETRF_BK.
115: *> \endverbatim
116: *>
117: *> \param[in,out] B
118: *> \verbatim
119: *> B is COMPLEX*16 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: *
145: *> \date December 2016
146: *
147: *> \ingroup complex16HEcomputational
148: *
149: *> \par Contributors:
150: * ==================
151: *>
152: *> \verbatim
153: *>
154: *> December 2016, Igor Kozachenko,
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 ZHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
166: $ INFO )
167: *
168: * -- LAPACK computational routine (version 3.7.0) --
169: * -- LAPACK is a software package provided by Univ. of Tennessee, --
170: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
171: * December 2016
172: *
173: * .. Scalar Arguments ..
174: CHARACTER UPLO
175: INTEGER INFO, LDA, LDB, N, NRHS
176: * ..
177: * .. Array Arguments ..
178: INTEGER IPIV( * )
179: COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * )
180: * ..
181: *
182: * =====================================================================
183: *
184: * .. Parameters ..
185: COMPLEX*16 ONE
186: PARAMETER ( ONE = ( 1.0D+0,0.0D+0 ) )
187: * ..
188: * .. Local Scalars ..
189: LOGICAL UPPER
190: INTEGER I, J, K, KP
191: DOUBLE PRECISION S
192: COMPLEX*16 AK, AKM1, AKM1K, BK, BKM1, DENOM
193: * ..
194: * .. External Functions ..
195: LOGICAL LSAME
196: EXTERNAL LSAME
197: * ..
198: * .. External Subroutines ..
199: EXTERNAL ZDSCAL, ZSWAP, ZTRSM, XERBLA
200: * ..
201: * .. Intrinsic Functions ..
202: INTRINSIC ABS, DBLE, DCONJG, MAX
203: * ..
204: * .. Executable Statements ..
205: *
206: INFO = 0
207: UPPER = LSAME( UPLO, 'U' )
208: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
209: INFO = -1
210: ELSE IF( N.LT.0 ) THEN
211: INFO = -2
212: ELSE IF( NRHS.LT.0 ) THEN
213: INFO = -3
214: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
215: INFO = -5
216: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
217: INFO = -9
218: END IF
219: IF( INFO.NE.0 ) THEN
220: CALL XERBLA( 'ZHETRS_3', -INFO )
221: RETURN
222: END IF
223: *
224: * Quick return if possible
225: *
226: IF( N.EQ.0 .OR. NRHS.EQ.0 )
227: $ RETURN
228: *
229: IF( UPPER ) THEN
230: *
231: * Begin Upper
232: *
233: * Solve A*X = B, where A = U*D*U**H.
234: *
235: * P**T * B
236: *
237: * Interchange rows K and IPIV(K) of matrix B in the same order
238: * that the formation order of IPIV(I) vector for Upper case.
239: *
240: * (We can do the simple loop over IPIV with decrement -1,
241: * since the ABS value of IPIV(I) represents the row index
242: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
243: *
244: DO K = N, 1, -1
245: KP = ABS( IPIV( K ) )
246: IF( KP.NE.K ) THEN
247: CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
248: END IF
249: END DO
250: *
251: * Compute (U \P**T * B) -> B [ (U \P**T * B) ]
252: *
253: CALL ZTRSM( 'L', 'U', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
254: *
255: * Compute D \ B -> B [ D \ (U \P**T * B) ]
256: *
257: I = N
258: DO WHILE ( I.GE.1 )
259: IF( IPIV( I ).GT.0 ) THEN
260: S = DBLE( ONE ) / DBLE( A( I, I ) )
261: CALL ZDSCAL( NRHS, S, B( I, 1 ), LDB )
262: ELSE IF ( I.GT.1 ) THEN
263: AKM1K = E( I )
264: AKM1 = A( I-1, I-1 ) / AKM1K
265: AK = A( I, I ) / DCONJG( AKM1K )
266: DENOM = AKM1*AK - ONE
267: DO J = 1, NRHS
268: BKM1 = B( I-1, J ) / AKM1K
269: BK = B( I, J ) / DCONJG( AKM1K )
270: B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
271: B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
272: END DO
273: I = I - 1
274: END IF
275: I = I - 1
276: END DO
277: *
278: * Compute (U**H \ B) -> B [ U**H \ (D \ (U \P**T * B) ) ]
279: *
280: CALL ZTRSM( 'L', 'U', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
281: *
282: * P * B [ P * (U**H \ (D \ (U \P**T * B) )) ]
283: *
284: * Interchange rows K and IPIV(K) of matrix B in reverse order
285: * from the formation order of IPIV(I) vector for Upper case.
286: *
287: * (We can do the simple loop over IPIV with increment 1,
288: * since the ABS value of IPIV(I) represents the row index
289: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
290: *
291: DO K = 1, N, 1
292: KP = ABS( IPIV( K ) )
293: IF( KP.NE.K ) THEN
294: CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
295: END IF
296: END DO
297: *
298: ELSE
299: *
300: * Begin Lower
301: *
302: * Solve A*X = B, where A = L*D*L**H.
303: *
304: * P**T * B
305: * Interchange rows K and IPIV(K) of matrix B in the same order
306: * that the formation order of IPIV(I) vector for Lower case.
307: *
308: * (We can do the simple loop over IPIV with increment 1,
309: * since the ABS value of IPIV(I) represents the row index
310: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
311: *
312: DO K = 1, N, 1
313: KP = ABS( IPIV( K ) )
314: IF( KP.NE.K ) THEN
315: CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
316: END IF
317: END DO
318: *
319: * Compute (L \P**T * B) -> B [ (L \P**T * B) ]
320: *
321: CALL ZTRSM( 'L', 'L', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
322: *
323: * Compute D \ B -> B [ D \ (L \P**T * B) ]
324: *
325: I = 1
326: DO WHILE ( I.LE.N )
327: IF( IPIV( I ).GT.0 ) THEN
328: S = DBLE( ONE ) / DBLE( A( I, I ) )
329: CALL ZDSCAL( NRHS, S, B( I, 1 ), LDB )
330: ELSE IF( I.LT.N ) THEN
331: AKM1K = E( I )
332: AKM1 = A( I, I ) / DCONJG( AKM1K )
333: AK = A( I+1, I+1 ) / AKM1K
334: DENOM = AKM1*AK - ONE
335: DO J = 1, NRHS
336: BKM1 = B( I, J ) / DCONJG( AKM1K )
337: BK = B( I+1, J ) / AKM1K
338: B( I, J ) = ( AK*BKM1-BK ) / DENOM
339: B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
340: END DO
341: I = I + 1
342: END IF
343: I = I + 1
344: END DO
345: *
346: * Compute (L**H \ B) -> B [ L**H \ (D \ (L \P**T * B) ) ]
347: *
348: CALL ZTRSM('L', 'L', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
349: *
350: * P * B [ P * (L**H \ (D \ (L \P**T * B) )) ]
351: *
352: * Interchange rows K and IPIV(K) of matrix B in reverse order
353: * from the formation order of IPIV(I) vector for Lower case.
354: *
355: * (We can do the simple loop over IPIV with decrement -1,
356: * since the ABS value of IPIV(I) represents the row index
357: * of the interchange with row i in both 1x1 and 2x2 pivot cases)
358: *
359: DO K = N, 1, -1
360: KP = ABS( IPIV( K ) )
361: IF( KP.NE.K ) THEN
362: CALL ZSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
363: END IF
364: END DO
365: *
366: * END Lower
367: *
368: END IF
369: *
370: RETURN
371: *
372: * End of ZHETRS_3
373: *
374: END
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