Annotation of rpl/lapack/lapack/dsytrd.f, revision 1.2
1.1 bertrand 1: SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, LDA, LWORK, N
11: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
14: $ WORK( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DSYTRD reduces a real symmetric matrix A to real symmetric
21: * tridiagonal form T by an orthogonal similarity transformation:
22: * Q**T * A * Q = T.
23: *
24: * Arguments
25: * =========
26: *
27: * UPLO (input) CHARACTER*1
28: * = 'U': Upper triangle of A is stored;
29: * = 'L': Lower triangle of A is stored.
30: *
31: * N (input) INTEGER
32: * The order of the matrix A. N >= 0.
33: *
34: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
35: * On entry, the symmetric matrix A. If UPLO = 'U', the leading
36: * N-by-N upper triangular part of A contains the upper
37: * triangular part of the matrix A, and the strictly lower
38: * triangular part of A is not referenced. If UPLO = 'L', the
39: * leading N-by-N lower triangular part of A contains the lower
40: * triangular part of the matrix A, and the strictly upper
41: * triangular part of A is not referenced.
42: * On exit, if UPLO = 'U', the diagonal and first superdiagonal
43: * of A are overwritten by the corresponding elements of the
44: * tridiagonal matrix T, and the elements above the first
45: * superdiagonal, with the array TAU, represent the orthogonal
46: * matrix Q as a product of elementary reflectors; if UPLO
47: * = 'L', the diagonal and first subdiagonal of A are over-
48: * written by the corresponding elements of the tridiagonal
49: * matrix T, and the elements below the first subdiagonal, with
50: * the array TAU, represent the orthogonal matrix Q as a product
51: * of elementary reflectors. See Further Details.
52: *
53: * LDA (input) INTEGER
54: * The leading dimension of the array A. LDA >= max(1,N).
55: *
56: * D (output) DOUBLE PRECISION array, dimension (N)
57: * The diagonal elements of the tridiagonal matrix T:
58: * D(i) = A(i,i).
59: *
60: * E (output) DOUBLE PRECISION array, dimension (N-1)
61: * The off-diagonal elements of the tridiagonal matrix T:
62: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
63: *
64: * TAU (output) DOUBLE PRECISION array, dimension (N-1)
65: * The scalar factors of the elementary reflectors (see Further
66: * Details).
67: *
68: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
69: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
70: *
71: * LWORK (input) INTEGER
72: * The dimension of the array WORK. LWORK >= 1.
73: * For optimum performance LWORK >= N*NB, where NB is the
74: * optimal blocksize.
75: *
76: * If LWORK = -1, then a workspace query is assumed; the routine
77: * only calculates the optimal size of the WORK array, returns
78: * this value as the first entry of the WORK array, and no error
79: * message related to LWORK is issued by XERBLA.
80: *
81: * INFO (output) INTEGER
82: * = 0: successful exit
83: * < 0: if INFO = -i, the i-th argument had an illegal value
84: *
85: * Further Details
86: * ===============
87: *
88: * If UPLO = 'U', the matrix Q is represented as a product of elementary
89: * reflectors
90: *
91: * Q = H(n-1) . . . H(2) H(1).
92: *
93: * Each H(i) has the form
94: *
95: * H(i) = I - tau * v * v'
96: *
97: * where tau is a real scalar, and v is a real vector with
98: * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
99: * A(1:i-1,i+1), and tau in TAU(i).
100: *
101: * If UPLO = 'L', the matrix Q is represented as a product of elementary
102: * reflectors
103: *
104: * Q = H(1) H(2) . . . H(n-1).
105: *
106: * Each H(i) has the form
107: *
108: * H(i) = I - tau * v * v'
109: *
110: * where tau is a real scalar, and v is a real vector with
111: * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
112: * and tau in TAU(i).
113: *
114: * The contents of A on exit are illustrated by the following examples
115: * with n = 5:
116: *
117: * if UPLO = 'U': if UPLO = 'L':
118: *
119: * ( d e v2 v3 v4 ) ( d )
120: * ( d e v3 v4 ) ( e d )
121: * ( d e v4 ) ( v1 e d )
122: * ( d e ) ( v1 v2 e d )
123: * ( d ) ( v1 v2 v3 e d )
124: *
125: * where d and e denote diagonal and off-diagonal elements of T, and vi
126: * denotes an element of the vector defining H(i).
127: *
128: * =====================================================================
129: *
130: * .. Parameters ..
131: DOUBLE PRECISION ONE
132: PARAMETER ( ONE = 1.0D+0 )
133: * ..
134: * .. Local Scalars ..
135: LOGICAL LQUERY, UPPER
136: INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
137: $ NBMIN, NX
138: * ..
139: * .. External Subroutines ..
140: EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
141: * ..
142: * .. Intrinsic Functions ..
143: INTRINSIC MAX
144: * ..
145: * .. External Functions ..
146: LOGICAL LSAME
147: INTEGER ILAENV
148: EXTERNAL LSAME, ILAENV
149: * ..
150: * .. Executable Statements ..
151: *
152: * Test the input parameters
153: *
154: INFO = 0
155: UPPER = LSAME( UPLO, 'U' )
156: LQUERY = ( LWORK.EQ.-1 )
157: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
158: INFO = -1
159: ELSE IF( N.LT.0 ) THEN
160: INFO = -2
161: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162: INFO = -4
163: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
164: INFO = -9
165: END IF
166: *
167: IF( INFO.EQ.0 ) THEN
168: *
169: * Determine the block size.
170: *
171: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
172: LWKOPT = N*NB
173: WORK( 1 ) = LWKOPT
174: END IF
175: *
176: IF( INFO.NE.0 ) THEN
177: CALL XERBLA( 'DSYTRD', -INFO )
178: RETURN
179: ELSE IF( LQUERY ) THEN
180: RETURN
181: END IF
182: *
183: * Quick return if possible
184: *
185: IF( N.EQ.0 ) THEN
186: WORK( 1 ) = 1
187: RETURN
188: END IF
189: *
190: NX = N
191: IWS = 1
192: IF( NB.GT.1 .AND. NB.LT.N ) THEN
193: *
194: * Determine when to cross over from blocked to unblocked code
195: * (last block is always handled by unblocked code).
196: *
197: NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
198: IF( NX.LT.N ) THEN
199: *
200: * Determine if workspace is large enough for blocked code.
201: *
202: LDWORK = N
203: IWS = LDWORK*NB
204: IF( LWORK.LT.IWS ) THEN
205: *
206: * Not enough workspace to use optimal NB: determine the
207: * minimum value of NB, and reduce NB or force use of
208: * unblocked code by setting NX = N.
209: *
210: NB = MAX( LWORK / LDWORK, 1 )
211: NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
212: IF( NB.LT.NBMIN )
213: $ NX = N
214: END IF
215: ELSE
216: NX = N
217: END IF
218: ELSE
219: NB = 1
220: END IF
221: *
222: IF( UPPER ) THEN
223: *
224: * Reduce the upper triangle of A.
225: * Columns 1:kk are handled by the unblocked method.
226: *
227: KK = N - ( ( N-NX+NB-1 ) / NB )*NB
228: DO 20 I = N - NB + 1, KK + 1, -NB
229: *
230: * Reduce columns i:i+nb-1 to tridiagonal form and form the
231: * matrix W which is needed to update the unreduced part of
232: * the matrix
233: *
234: CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
235: $ LDWORK )
236: *
237: * Update the unreduced submatrix A(1:i-1,1:i-1), using an
238: * update of the form: A := A - V*W' - W*V'
239: *
240: CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
241: $ LDA, WORK, LDWORK, ONE, A, LDA )
242: *
243: * Copy superdiagonal elements back into A, and diagonal
244: * elements into D
245: *
246: DO 10 J = I, I + NB - 1
247: A( J-1, J ) = E( J-1 )
248: D( J ) = A( J, J )
249: 10 CONTINUE
250: 20 CONTINUE
251: *
252: * Use unblocked code to reduce the last or only block
253: *
254: CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
255: ELSE
256: *
257: * Reduce the lower triangle of A
258: *
259: DO 40 I = 1, N - NX, NB
260: *
261: * Reduce columns i:i+nb-1 to tridiagonal form and form the
262: * matrix W which is needed to update the unreduced part of
263: * the matrix
264: *
265: CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
266: $ TAU( I ), WORK, LDWORK )
267: *
268: * Update the unreduced submatrix A(i+ib:n,i+ib:n), using
269: * an update of the form: A := A - V*W' - W*V'
270: *
271: CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
272: $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
273: $ A( I+NB, I+NB ), LDA )
274: *
275: * Copy subdiagonal elements back into A, and diagonal
276: * elements into D
277: *
278: DO 30 J = I, I + NB - 1
279: A( J+1, J ) = E( J )
280: D( J ) = A( J, J )
281: 30 CONTINUE
282: 40 CONTINUE
283: *
284: * Use unblocked code to reduce the last or only block
285: *
286: CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
287: $ TAU( I ), IINFO )
288: END IF
289: *
290: WORK( 1 ) = LWKOPT
291: RETURN
292: *
293: * End of DSYTRD
294: *
295: END
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