Annotation of rpl/lapack/lapack/dsytrd.f, revision 1.18
1.9 bertrand 1: *> \brief \b DSYTRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download DSYTRD + 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/dsytrd.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
1.15 bertrand 22: *
1.9 bertrand 23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, LWORK, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
29: * $ WORK( * )
30: * ..
1.15 bertrand 31: *
1.9 bertrand 32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYTRD reduces a real symmetric matrix A to real symmetric
39: *> tridiagonal form T by an orthogonal similarity transformation:
40: *> Q**T * A * Q = T.
41: *> \endverbatim
42: *
43: * Arguments:
44: * ==========
45: *
46: *> \param[in] UPLO
47: *> \verbatim
48: *> UPLO is CHARACTER*1
49: *> = 'U': Upper triangle of A is stored;
50: *> = 'L': Lower triangle of A is stored.
51: *> \endverbatim
52: *>
53: *> \param[in] N
54: *> \verbatim
55: *> N is INTEGER
56: *> The order of the matrix A. N >= 0.
57: *> \endverbatim
58: *>
59: *> \param[in,out] A
60: *> \verbatim
61: *> A is DOUBLE PRECISION array, dimension (LDA,N)
62: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
63: *> N-by-N upper triangular part of A contains the upper
64: *> triangular part of the matrix A, and the strictly lower
65: *> triangular part of A is not referenced. If UPLO = 'L', the
66: *> leading N-by-N lower triangular part of A contains the lower
67: *> triangular part of the matrix A, and the strictly upper
68: *> triangular part of A is not referenced.
69: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70: *> of A are overwritten by the corresponding elements of the
71: *> tridiagonal matrix T, and the elements above the first
72: *> superdiagonal, with the array TAU, represent the orthogonal
73: *> matrix Q as a product of elementary reflectors; if UPLO
74: *> = 'L', the diagonal and first subdiagonal of A are over-
75: *> written by the corresponding elements of the tridiagonal
76: *> matrix T, and the elements below the first subdiagonal, with
77: *> the array TAU, represent the orthogonal matrix Q as a product
78: *> of elementary reflectors. See Further Details.
79: *> \endverbatim
80: *>
81: *> \param[in] LDA
82: *> \verbatim
83: *> LDA is INTEGER
84: *> The leading dimension of the array A. LDA >= max(1,N).
85: *> \endverbatim
86: *>
87: *> \param[out] D
88: *> \verbatim
89: *> D is DOUBLE PRECISION array, dimension (N)
90: *> The diagonal elements of the tridiagonal matrix T:
91: *> D(i) = A(i,i).
92: *> \endverbatim
93: *>
94: *> \param[out] E
95: *> \verbatim
96: *> E is DOUBLE PRECISION array, dimension (N-1)
97: *> The off-diagonal elements of the tridiagonal matrix T:
98: *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99: *> \endverbatim
100: *>
101: *> \param[out] TAU
102: *> \verbatim
103: *> TAU is DOUBLE PRECISION array, dimension (N-1)
104: *> The scalar factors of the elementary reflectors (see Further
105: *> Details).
106: *> \endverbatim
107: *>
108: *> \param[out] WORK
109: *> \verbatim
110: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
111: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
112: *> \endverbatim
113: *>
114: *> \param[in] LWORK
115: *> \verbatim
116: *> LWORK is INTEGER
117: *> The dimension of the array WORK. LWORK >= 1.
118: *> For optimum performance LWORK >= N*NB, where NB is the
119: *> optimal blocksize.
120: *>
121: *> If LWORK = -1, then a workspace query is assumed; the routine
122: *> only calculates the optimal size of the WORK array, returns
123: *> this value as the first entry of the WORK array, and no error
124: *> message related to LWORK is issued by XERBLA.
125: *> \endverbatim
126: *>
127: *> \param[out] INFO
128: *> \verbatim
129: *> INFO is INTEGER
130: *> = 0: successful exit
131: *> < 0: if INFO = -i, the i-th argument had an illegal value
132: *> \endverbatim
133: *
134: * Authors:
135: * ========
136: *
1.15 bertrand 137: *> \author Univ. of Tennessee
138: *> \author Univ. of California Berkeley
139: *> \author Univ. of Colorado Denver
140: *> \author NAG Ltd.
1.9 bertrand 141: *
142: *> \ingroup doubleSYcomputational
143: *
144: *> \par Further Details:
145: * =====================
146: *>
147: *> \verbatim
148: *>
149: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
150: *> reflectors
151: *>
152: *> Q = H(n-1) . . . H(2) H(1).
153: *>
154: *> Each H(i) has the form
155: *>
156: *> H(i) = I - tau * v * v**T
157: *>
158: *> where tau is a real scalar, and v is a real vector with
159: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
160: *> A(1:i-1,i+1), and tau in TAU(i).
161: *>
162: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
163: *> reflectors
164: *>
165: *> Q = H(1) H(2) . . . H(n-1).
166: *>
167: *> Each H(i) has the form
168: *>
169: *> H(i) = I - tau * v * v**T
170: *>
171: *> where tau is a real scalar, and v is a real vector with
172: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
173: *> and tau in TAU(i).
174: *>
175: *> The contents of A on exit are illustrated by the following examples
176: *> with n = 5:
177: *>
178: *> if UPLO = 'U': if UPLO = 'L':
179: *>
180: *> ( d e v2 v3 v4 ) ( d )
181: *> ( d e v3 v4 ) ( e d )
182: *> ( d e v4 ) ( v1 e d )
183: *> ( d e ) ( v1 v2 e d )
184: *> ( d ) ( v1 v2 v3 e d )
185: *>
186: *> where d and e denote diagonal and off-diagonal elements of T, and vi
187: *> denotes an element of the vector defining H(i).
188: *> \endverbatim
189: *>
190: * =====================================================================
1.1 bertrand 191: SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
192: *
1.18 ! bertrand 193: * -- LAPACK computational routine --
1.1 bertrand 194: * -- LAPACK is a software package provided by Univ. of Tennessee, --
195: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196: *
197: * .. Scalar Arguments ..
198: CHARACTER UPLO
199: INTEGER INFO, LDA, LWORK, N
200: * ..
201: * .. Array Arguments ..
202: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
203: $ WORK( * )
204: * ..
205: *
206: * =====================================================================
207: *
208: * .. Parameters ..
209: DOUBLE PRECISION ONE
210: PARAMETER ( ONE = 1.0D+0 )
211: * ..
212: * .. Local Scalars ..
213: LOGICAL LQUERY, UPPER
214: INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
215: $ NBMIN, NX
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA
219: * ..
220: * .. Intrinsic Functions ..
221: INTRINSIC MAX
222: * ..
223: * .. External Functions ..
224: LOGICAL LSAME
225: INTEGER ILAENV
226: EXTERNAL LSAME, ILAENV
227: * ..
228: * .. Executable Statements ..
229: *
230: * Test the input parameters
231: *
232: INFO = 0
233: UPPER = LSAME( UPLO, 'U' )
234: LQUERY = ( LWORK.EQ.-1 )
235: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
236: INFO = -1
237: ELSE IF( N.LT.0 ) THEN
238: INFO = -2
239: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
240: INFO = -4
241: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
242: INFO = -9
243: END IF
244: *
245: IF( INFO.EQ.0 ) THEN
246: *
247: * Determine the block size.
248: *
249: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
250: LWKOPT = N*NB
251: WORK( 1 ) = LWKOPT
252: END IF
253: *
254: IF( INFO.NE.0 ) THEN
255: CALL XERBLA( 'DSYTRD', -INFO )
256: RETURN
257: ELSE IF( LQUERY ) THEN
258: RETURN
259: END IF
260: *
261: * Quick return if possible
262: *
263: IF( N.EQ.0 ) THEN
264: WORK( 1 ) = 1
265: RETURN
266: END IF
267: *
268: NX = N
269: IWS = 1
270: IF( NB.GT.1 .AND. NB.LT.N ) THEN
271: *
272: * Determine when to cross over from blocked to unblocked code
273: * (last block is always handled by unblocked code).
274: *
275: NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) )
276: IF( NX.LT.N ) THEN
277: *
278: * Determine if workspace is large enough for blocked code.
279: *
280: LDWORK = N
281: IWS = LDWORK*NB
282: IF( LWORK.LT.IWS ) THEN
283: *
284: * Not enough workspace to use optimal NB: determine the
285: * minimum value of NB, and reduce NB or force use of
286: * unblocked code by setting NX = N.
287: *
288: NB = MAX( LWORK / LDWORK, 1 )
289: NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 )
290: IF( NB.LT.NBMIN )
291: $ NX = N
292: END IF
293: ELSE
294: NX = N
295: END IF
296: ELSE
297: NB = 1
298: END IF
299: *
300: IF( UPPER ) THEN
301: *
302: * Reduce the upper triangle of A.
303: * Columns 1:kk are handled by the unblocked method.
304: *
305: KK = N - ( ( N-NX+NB-1 ) / NB )*NB
306: DO 20 I = N - NB + 1, KK + 1, -NB
307: *
308: * Reduce columns i:i+nb-1 to tridiagonal form and form the
309: * matrix W which is needed to update the unreduced part of
310: * the matrix
311: *
312: CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
313: $ LDWORK )
314: *
315: * Update the unreduced submatrix A(1:i-1,1:i-1), using an
1.8 bertrand 316: * update of the form: A := A - V*W**T - W*V**T
1.1 bertrand 317: *
318: CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
319: $ LDA, WORK, LDWORK, ONE, A, LDA )
320: *
321: * Copy superdiagonal elements back into A, and diagonal
322: * elements into D
323: *
324: DO 10 J = I, I + NB - 1
325: A( J-1, J ) = E( J-1 )
326: D( J ) = A( J, J )
327: 10 CONTINUE
328: 20 CONTINUE
329: *
330: * Use unblocked code to reduce the last or only block
331: *
332: CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
333: ELSE
334: *
335: * Reduce the lower triangle of A
336: *
337: DO 40 I = 1, N - NX, NB
338: *
339: * Reduce columns i:i+nb-1 to tridiagonal form and form the
340: * matrix W which is needed to update the unreduced part of
341: * the matrix
342: *
343: CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
344: $ TAU( I ), WORK, LDWORK )
345: *
346: * Update the unreduced submatrix A(i+ib:n,i+ib:n), using
1.8 bertrand 347: * an update of the form: A := A - V*W**T - W*V**T
1.1 bertrand 348: *
349: CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
350: $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
351: $ A( I+NB, I+NB ), LDA )
352: *
353: * Copy subdiagonal elements back into A, and diagonal
354: * elements into D
355: *
356: DO 30 J = I, I + NB - 1
357: A( J+1, J ) = E( J )
358: D( J ) = A( J, J )
359: 30 CONTINUE
360: 40 CONTINUE
361: *
362: * Use unblocked code to reduce the last or only block
363: *
364: CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
365: $ TAU( I ), IINFO )
366: END IF
367: *
368: WORK( 1 ) = LWKOPT
369: RETURN
370: *
371: * End of DSYTRD
372: *
373: END
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