1: *> \brief \b DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYGS2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygs2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygs2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygs2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, ITYPE, LDA, LDB, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), B( LDB, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DSYGS2 reduces a real symmetric-definite generalized eigenproblem
38: *> to standard form.
39: *>
40: *> If ITYPE = 1, the problem is A*x = lambda*B*x,
41: *> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
42: *>
43: *> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
44: *> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
45: *>
46: *> B must have been previously factorized as U**T *U or L*L**T by DPOTRF.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] ITYPE
53: *> \verbatim
54: *> ITYPE is INTEGER
55: *> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
56: *> = 2 or 3: compute U*A*U**T or L**T *A*L.
57: *> \endverbatim
58: *>
59: *> \param[in] UPLO
60: *> \verbatim
61: *> UPLO is CHARACTER*1
62: *> Specifies whether the upper or lower triangular part of the
63: *> symmetric matrix A is stored, and how B has been factorized.
64: *> = 'U': Upper triangular
65: *> = 'L': Lower triangular
66: *> \endverbatim
67: *>
68: *> \param[in] N
69: *> \verbatim
70: *> N is INTEGER
71: *> The order of the matrices A and B. N >= 0.
72: *> \endverbatim
73: *>
74: *> \param[in,out] A
75: *> \verbatim
76: *> A is DOUBLE PRECISION array, dimension (LDA,N)
77: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
78: *> n by n upper triangular part of A contains the upper
79: *> triangular part of the matrix A, and the strictly lower
80: *> triangular part of A is not referenced. If UPLO = 'L', the
81: *> leading n by n lower triangular part of A contains the lower
82: *> triangular part of the matrix A, and the strictly upper
83: *> triangular part of A is not referenced.
84: *>
85: *> On exit, if INFO = 0, the transformed matrix, stored in the
86: *> same format as A.
87: *> \endverbatim
88: *>
89: *> \param[in] LDA
90: *> \verbatim
91: *> LDA is INTEGER
92: *> The leading dimension of the array A. LDA >= max(1,N).
93: *> \endverbatim
94: *>
95: *> \param[in] B
96: *> \verbatim
97: *> B is DOUBLE PRECISION array, dimension (LDB,N)
98: *> The triangular factor from the Cholesky factorization of B,
99: *> as returned by DPOTRF.
100: *> \endverbatim
101: *>
102: *> \param[in] LDB
103: *> \verbatim
104: *> LDB is INTEGER
105: *> The leading dimension of the array B. LDB >= max(1,N).
106: *> \endverbatim
107: *>
108: *> \param[out] INFO
109: *> \verbatim
110: *> INFO is INTEGER
111: *> = 0: successful exit.
112: *> < 0: if INFO = -i, the i-th argument had an illegal value.
113: *> \endverbatim
114: *
115: * Authors:
116: * ========
117: *
118: *> \author Univ. of Tennessee
119: *> \author Univ. of California Berkeley
120: *> \author Univ. of Colorado Denver
121: *> \author NAG Ltd.
122: *
123: *> \ingroup doubleSYcomputational
124: *
125: * =====================================================================
126: SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
127: *
128: * -- LAPACK computational routine --
129: * -- LAPACK is a software package provided by Univ. of Tennessee, --
130: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
131: *
132: * .. Scalar Arguments ..
133: CHARACTER UPLO
134: INTEGER INFO, ITYPE, LDA, LDB, N
135: * ..
136: * .. Array Arguments ..
137: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
138: * ..
139: *
140: * =====================================================================
141: *
142: * .. Parameters ..
143: DOUBLE PRECISION ONE, HALF
144: PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
145: * ..
146: * .. Local Scalars ..
147: LOGICAL UPPER
148: INTEGER K
149: DOUBLE PRECISION AKK, BKK, CT
150: * ..
151: * .. External Subroutines ..
152: EXTERNAL DAXPY, DSCAL, DSYR2, DTRMV, DTRSV, XERBLA
153: * ..
154: * .. Intrinsic Functions ..
155: INTRINSIC MAX
156: * ..
157: * .. External Functions ..
158: LOGICAL LSAME
159: EXTERNAL LSAME
160: * ..
161: * .. Executable Statements ..
162: *
163: * Test the input parameters.
164: *
165: INFO = 0
166: UPPER = LSAME( UPLO, 'U' )
167: IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
168: INFO = -1
169: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
170: INFO = -2
171: ELSE IF( N.LT.0 ) THEN
172: INFO = -3
173: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
174: INFO = -5
175: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
176: INFO = -7
177: END IF
178: IF( INFO.NE.0 ) THEN
179: CALL XERBLA( 'DSYGS2', -INFO )
180: RETURN
181: END IF
182: *
183: IF( ITYPE.EQ.1 ) THEN
184: IF( UPPER ) THEN
185: *
186: * Compute inv(U**T)*A*inv(U)
187: *
188: DO 10 K = 1, N
189: *
190: * Update the upper triangle of A(k:n,k:n)
191: *
192: AKK = A( K, K )
193: BKK = B( K, K )
194: AKK = AKK / BKK**2
195: A( K, K ) = AKK
196: IF( K.LT.N ) THEN
197: CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
198: CT = -HALF*AKK
199: CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
200: $ LDA )
201: CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
202: $ B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
203: CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
204: $ LDA )
205: CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
206: $ B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
207: END IF
208: 10 CONTINUE
209: ELSE
210: *
211: * Compute inv(L)*A*inv(L**T)
212: *
213: DO 20 K = 1, N
214: *
215: * Update the lower triangle of A(k:n,k:n)
216: *
217: AKK = A( K, K )
218: BKK = B( K, K )
219: AKK = AKK / BKK**2
220: A( K, K ) = AKK
221: IF( K.LT.N ) THEN
222: CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
223: CT = -HALF*AKK
224: CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
225: CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
226: $ B( K+1, K ), 1, A( K+1, K+1 ), LDA )
227: CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
228: CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
229: $ B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
230: END IF
231: 20 CONTINUE
232: END IF
233: ELSE
234: IF( UPPER ) THEN
235: *
236: * Compute U*A*U**T
237: *
238: DO 30 K = 1, N
239: *
240: * Update the upper triangle of A(1:k,1:k)
241: *
242: AKK = A( K, K )
243: BKK = B( K, K )
244: CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
245: $ LDB, A( 1, K ), 1 )
246: CT = HALF*AKK
247: CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
248: CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ), 1,
249: $ A, LDA )
250: CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
251: CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
252: A( K, K ) = AKK*BKK**2
253: 30 CONTINUE
254: ELSE
255: *
256: * Compute L**T *A*L
257: *
258: DO 40 K = 1, N
259: *
260: * Update the lower triangle of A(1:k,1:k)
261: *
262: AKK = A( K, K )
263: BKK = B( K, K )
264: CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B, LDB,
265: $ A( K, 1 ), LDA )
266: CT = HALF*AKK
267: CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
268: CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
269: $ LDB, A, LDA )
270: CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
271: CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
272: A( K, K ) = AKK*BKK**2
273: 40 CONTINUE
274: END IF
275: END IF
276: RETURN
277: *
278: * End of DSYGS2
279: *
280: END
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