Annotation of rpl/lapack/lapack/zhetrd.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZHETRD
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZHETRD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetrd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetrd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER INFO, LDA, LWORK, N
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * DOUBLE PRECISION D( * ), E( * )
! 29: * COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZHETRD reduces a complex Hermitian matrix A to real symmetric
! 39: *> tridiagonal form T by a unitary similarity transformation:
! 40: *> Q**H * 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 COMPLEX*16 array, dimension (LDA,N)
! 62: *> On entry, the Hermitian 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 unitary
! 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 unitary 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 COMPLEX*16 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 COMPLEX*16 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: *
! 137: *> \author Univ. of Tennessee
! 138: *> \author Univ. of California Berkeley
! 139: *> \author Univ. of Colorado Denver
! 140: *> \author NAG Ltd.
! 141: *
! 142: *> \date November 2011
! 143: *
! 144: *> \ingroup complex16HEcomputational
! 145: *
! 146: *> \par Further Details:
! 147: * =====================
! 148: *>
! 149: *> \verbatim
! 150: *>
! 151: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
! 152: *> reflectors
! 153: *>
! 154: *> Q = H(n-1) . . . H(2) H(1).
! 155: *>
! 156: *> Each H(i) has the form
! 157: *>
! 158: *> H(i) = I - tau * v * v**H
! 159: *>
! 160: *> where tau is a complex scalar, and v is a complex vector with
! 161: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
! 162: *> A(1:i-1,i+1), and tau in TAU(i).
! 163: *>
! 164: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
! 165: *> reflectors
! 166: *>
! 167: *> Q = H(1) H(2) . . . H(n-1).
! 168: *>
! 169: *> Each H(i) has the form
! 170: *>
! 171: *> H(i) = I - tau * v * v**H
! 172: *>
! 173: *> where tau is a complex scalar, and v is a complex vector with
! 174: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
! 175: *> and tau in TAU(i).
! 176: *>
! 177: *> The contents of A on exit are illustrated by the following examples
! 178: *> with n = 5:
! 179: *>
! 180: *> if UPLO = 'U': if UPLO = 'L':
! 181: *>
! 182: *> ( d e v2 v3 v4 ) ( d )
! 183: *> ( d e v3 v4 ) ( e d )
! 184: *> ( d e v4 ) ( v1 e d )
! 185: *> ( d e ) ( v1 v2 e d )
! 186: *> ( d ) ( v1 v2 v3 e d )
! 187: *>
! 188: *> where d and e denote diagonal and off-diagonal elements of T, and vi
! 189: *> denotes an element of the vector defining H(i).
! 190: *> \endverbatim
! 191: *>
! 192: * =====================================================================
1.1 bertrand 193: SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
194: *
1.9 ! bertrand 195: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 196: * -- LAPACK is a software package provided by Univ. of Tennessee, --
197: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 198: * November 2011
1.1 bertrand 199: *
200: * .. Scalar Arguments ..
201: CHARACTER UPLO
202: INTEGER INFO, LDA, LWORK, N
203: * ..
204: * .. Array Arguments ..
205: DOUBLE PRECISION D( * ), E( * )
206: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
207: * ..
208: *
209: * =====================================================================
210: *
211: * .. Parameters ..
212: DOUBLE PRECISION ONE
213: PARAMETER ( ONE = 1.0D+0 )
214: COMPLEX*16 CONE
215: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
216: * ..
217: * .. Local Scalars ..
218: LOGICAL LQUERY, UPPER
219: INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
220: $ NBMIN, NX
221: * ..
222: * .. External Subroutines ..
223: EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC MAX
227: * ..
228: * .. External Functions ..
229: LOGICAL LSAME
230: INTEGER ILAENV
231: EXTERNAL LSAME, ILAENV
232: * ..
233: * .. Executable Statements ..
234: *
235: * Test the input parameters
236: *
237: INFO = 0
238: UPPER = LSAME( UPLO, 'U' )
239: LQUERY = ( LWORK.EQ.-1 )
240: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
241: INFO = -1
242: ELSE IF( N.LT.0 ) THEN
243: INFO = -2
244: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
245: INFO = -4
246: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
247: INFO = -9
248: END IF
249: *
250: IF( INFO.EQ.0 ) THEN
251: *
252: * Determine the block size.
253: *
254: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
255: LWKOPT = N*NB
256: WORK( 1 ) = LWKOPT
257: END IF
258: *
259: IF( INFO.NE.0 ) THEN
260: CALL XERBLA( 'ZHETRD', -INFO )
261: RETURN
262: ELSE IF( LQUERY ) THEN
263: RETURN
264: END IF
265: *
266: * Quick return if possible
267: *
268: IF( N.EQ.0 ) THEN
269: WORK( 1 ) = 1
270: RETURN
271: END IF
272: *
273: NX = N
274: IWS = 1
275: IF( NB.GT.1 .AND. NB.LT.N ) THEN
276: *
277: * Determine when to cross over from blocked to unblocked code
278: * (last block is always handled by unblocked code).
279: *
280: NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
281: IF( NX.LT.N ) THEN
282: *
283: * Determine if workspace is large enough for blocked code.
284: *
285: LDWORK = N
286: IWS = LDWORK*NB
287: IF( LWORK.LT.IWS ) THEN
288: *
289: * Not enough workspace to use optimal NB: determine the
290: * minimum value of NB, and reduce NB or force use of
291: * unblocked code by setting NX = N.
292: *
293: NB = MAX( LWORK / LDWORK, 1 )
294: NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 )
295: IF( NB.LT.NBMIN )
296: $ NX = N
297: END IF
298: ELSE
299: NX = N
300: END IF
301: ELSE
302: NB = 1
303: END IF
304: *
305: IF( UPPER ) THEN
306: *
307: * Reduce the upper triangle of A.
308: * Columns 1:kk are handled by the unblocked method.
309: *
310: KK = N - ( ( N-NX+NB-1 ) / NB )*NB
311: DO 20 I = N - NB + 1, KK + 1, -NB
312: *
313: * Reduce columns i:i+nb-1 to tridiagonal form and form the
314: * matrix W which is needed to update the unreduced part of
315: * the matrix
316: *
317: CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
318: $ LDWORK )
319: *
320: * Update the unreduced submatrix A(1:i-1,1:i-1), using an
1.8 bertrand 321: * update of the form: A := A - V*W**H - W*V**H
1.1 bertrand 322: *
323: CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
324: $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA )
325: *
326: * Copy superdiagonal elements back into A, and diagonal
327: * elements into D
328: *
329: DO 10 J = I, I + NB - 1
330: A( J-1, J ) = E( J-1 )
331: D( J ) = A( J, J )
332: 10 CONTINUE
333: 20 CONTINUE
334: *
335: * Use unblocked code to reduce the last or only block
336: *
337: CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
338: ELSE
339: *
340: * Reduce the lower triangle of A
341: *
342: DO 40 I = 1, N - NX, NB
343: *
344: * Reduce columns i:i+nb-1 to tridiagonal form and form the
345: * matrix W which is needed to update the unreduced part of
346: * the matrix
347: *
348: CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
349: $ TAU( I ), WORK, LDWORK )
350: *
351: * Update the unreduced submatrix A(i+nb:n,i+nb:n), using
1.8 bertrand 352: * an update of the form: A := A - V*W**H - W*V**H
1.1 bertrand 353: *
354: CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
355: $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
356: $ A( I+NB, I+NB ), LDA )
357: *
358: * Copy subdiagonal elements back into A, and diagonal
359: * elements into D
360: *
361: DO 30 J = I, I + NB - 1
362: A( J+1, J ) = E( J )
363: D( J ) = A( J, J )
364: 30 CONTINUE
365: 40 CONTINUE
366: *
367: * Use unblocked code to reduce the last or only block
368: *
369: CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
370: $ TAU( I ), IINFO )
371: END IF
372: *
373: WORK( 1 ) = LWKOPT
374: RETURN
375: *
376: * End of ZHETRD
377: *
378: END
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