Annotation of rpl/lapack/lapack/zlatrd.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZLATRD
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZLATRD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlatrd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlatrd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlatrd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER LDA, LDW, N, NB
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * DOUBLE PRECISION E( * )
! 29: * COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
! 39: *> Hermitian tridiagonal form by a unitary similarity
! 40: *> transformation Q**H * A * Q, and returns the matrices V and W which are
! 41: *> needed to apply the transformation to the unreduced part of A.
! 42: *>
! 43: *> If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
! 44: *> matrix, of which the upper triangle is supplied;
! 45: *> if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
! 46: *> matrix, of which the lower triangle is supplied.
! 47: *>
! 48: *> This is an auxiliary routine called by ZHETRD.
! 49: *> \endverbatim
! 50: *
! 51: * Arguments:
! 52: * ==========
! 53: *
! 54: *> \param[in] UPLO
! 55: *> \verbatim
! 56: *> UPLO is CHARACTER*1
! 57: *> Specifies whether the upper or lower triangular part of the
! 58: *> Hermitian matrix A is stored:
! 59: *> = 'U': Upper triangular
! 60: *> = 'L': Lower triangular
! 61: *> \endverbatim
! 62: *>
! 63: *> \param[in] N
! 64: *> \verbatim
! 65: *> N is INTEGER
! 66: *> The order of the matrix A.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in] NB
! 70: *> \verbatim
! 71: *> NB is INTEGER
! 72: *> The number of rows and columns to be reduced.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in,out] A
! 76: *> \verbatim
! 77: *> A is COMPLEX*16 array, dimension (LDA,N)
! 78: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 79: *> n-by-n upper triangular part of A contains the upper
! 80: *> triangular part of the matrix A, and the strictly lower
! 81: *> triangular part of A is not referenced. If UPLO = 'L', the
! 82: *> leading n-by-n lower triangular part of A contains the lower
! 83: *> triangular part of the matrix A, and the strictly upper
! 84: *> triangular part of A is not referenced.
! 85: *> On exit:
! 86: *> if UPLO = 'U', the last NB columns have been reduced to
! 87: *> tridiagonal form, with the diagonal elements overwriting
! 88: *> the diagonal elements of A; the elements above the diagonal
! 89: *> with the array TAU, represent the unitary matrix Q as a
! 90: *> product of elementary reflectors;
! 91: *> if UPLO = 'L', the first NB columns have been reduced to
! 92: *> tridiagonal form, with the diagonal elements overwriting
! 93: *> the diagonal elements of A; the elements below the diagonal
! 94: *> with the array TAU, represent the unitary matrix Q as a
! 95: *> product of elementary reflectors.
! 96: *> See Further Details.
! 97: *> \endverbatim
! 98: *>
! 99: *> \param[in] LDA
! 100: *> \verbatim
! 101: *> LDA is INTEGER
! 102: *> The leading dimension of the array A. LDA >= max(1,N).
! 103: *> \endverbatim
! 104: *>
! 105: *> \param[out] E
! 106: *> \verbatim
! 107: *> E is DOUBLE PRECISION array, dimension (N-1)
! 108: *> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
! 109: *> elements of the last NB columns of the reduced matrix;
! 110: *> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
! 111: *> the first NB columns of the reduced matrix.
! 112: *> \endverbatim
! 113: *>
! 114: *> \param[out] TAU
! 115: *> \verbatim
! 116: *> TAU is COMPLEX*16 array, dimension (N-1)
! 117: *> The scalar factors of the elementary reflectors, stored in
! 118: *> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
! 119: *> See Further Details.
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[out] W
! 123: *> \verbatim
! 124: *> W is COMPLEX*16 array, dimension (LDW,NB)
! 125: *> The n-by-nb matrix W required to update the unreduced part
! 126: *> of A.
! 127: *> \endverbatim
! 128: *>
! 129: *> \param[in] LDW
! 130: *> \verbatim
! 131: *> LDW is INTEGER
! 132: *> The leading dimension of the array W. LDW >= max(1,N).
! 133: *> \endverbatim
! 134: *
! 135: * Authors:
! 136: * ========
! 137: *
! 138: *> \author Univ. of Tennessee
! 139: *> \author Univ. of California Berkeley
! 140: *> \author Univ. of Colorado Denver
! 141: *> \author NAG Ltd.
! 142: *
! 143: *> \date November 2011
! 144: *
! 145: *> \ingroup complex16OTHERauxiliary
! 146: *
! 147: *> \par Further Details:
! 148: * =====================
! 149: *>
! 150: *> \verbatim
! 151: *>
! 152: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
! 153: *> reflectors
! 154: *>
! 155: *> Q = H(n) H(n-1) . . . H(n-nb+1).
! 156: *>
! 157: *> Each H(i) has the form
! 158: *>
! 159: *> H(i) = I - tau * v * v**H
! 160: *>
! 161: *> where tau is a complex scalar, and v is a complex vector with
! 162: *> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
! 163: *> and tau in TAU(i-1).
! 164: *>
! 165: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
! 166: *> reflectors
! 167: *>
! 168: *> Q = H(1) H(2) . . . H(nb).
! 169: *>
! 170: *> Each H(i) has the form
! 171: *>
! 172: *> H(i) = I - tau * v * v**H
! 173: *>
! 174: *> where tau is a complex scalar, and v is a complex vector with
! 175: *> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
! 176: *> and tau in TAU(i).
! 177: *>
! 178: *> The elements of the vectors v together form the n-by-nb matrix V
! 179: *> which is needed, with W, to apply the transformation to the unreduced
! 180: *> part of the matrix, using a Hermitian rank-2k update of the form:
! 181: *> A := A - V*W**H - W*V**H.
! 182: *>
! 183: *> The contents of A on exit are illustrated by the following examples
! 184: *> with n = 5 and nb = 2:
! 185: *>
! 186: *> if UPLO = 'U': if UPLO = 'L':
! 187: *>
! 188: *> ( a a a v4 v5 ) ( d )
! 189: *> ( a a v4 v5 ) ( 1 d )
! 190: *> ( a 1 v5 ) ( v1 1 a )
! 191: *> ( d 1 ) ( v1 v2 a a )
! 192: *> ( d ) ( v1 v2 a a a )
! 193: *>
! 194: *> where d denotes a diagonal element of the reduced matrix, a denotes
! 195: *> an element of the original matrix that is unchanged, and vi denotes
! 196: *> an element of the vector defining H(i).
! 197: *> \endverbatim
! 198: *>
! 199: * =====================================================================
1.1 bertrand 200: SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
201: *
1.9 ! bertrand 202: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 203: * -- LAPACK is a software package provided by Univ. of Tennessee, --
204: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 205: * November 2011
1.1 bertrand 206: *
207: * .. Scalar Arguments ..
208: CHARACTER UPLO
209: INTEGER LDA, LDW, N, NB
210: * ..
211: * .. Array Arguments ..
212: DOUBLE PRECISION E( * )
213: COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
214: * ..
215: *
216: * =====================================================================
217: *
218: * .. Parameters ..
219: COMPLEX*16 ZERO, ONE, HALF
220: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
221: $ ONE = ( 1.0D+0, 0.0D+0 ),
222: $ HALF = ( 0.5D+0, 0.0D+0 ) )
223: * ..
224: * .. Local Scalars ..
225: INTEGER I, IW
226: COMPLEX*16 ALPHA
227: * ..
228: * .. External Subroutines ..
229: EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL
230: * ..
231: * .. External Functions ..
232: LOGICAL LSAME
233: COMPLEX*16 ZDOTC
234: EXTERNAL LSAME, ZDOTC
235: * ..
236: * .. Intrinsic Functions ..
237: INTRINSIC DBLE, MIN
238: * ..
239: * .. Executable Statements ..
240: *
241: * Quick return if possible
242: *
243: IF( N.LE.0 )
244: $ RETURN
245: *
246: IF( LSAME( UPLO, 'U' ) ) THEN
247: *
248: * Reduce last NB columns of upper triangle
249: *
250: DO 10 I = N, N - NB + 1, -1
251: IW = I - N + NB
252: IF( I.LT.N ) THEN
253: *
254: * Update A(1:i,i)
255: *
256: A( I, I ) = DBLE( A( I, I ) )
257: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
258: CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
259: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
260: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
261: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
262: CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
263: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
264: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
265: A( I, I ) = DBLE( A( I, I ) )
266: END IF
267: IF( I.GT.1 ) THEN
268: *
269: * Generate elementary reflector H(i) to annihilate
270: * A(1:i-2,i)
271: *
272: ALPHA = A( I-1, I )
273: CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )
274: E( I-1 ) = ALPHA
275: A( I-1, I ) = ONE
276: *
277: * Compute W(1:i-1,i)
278: *
279: CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
280: $ ZERO, W( 1, IW ), 1 )
281: IF( I.LT.N ) THEN
282: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
283: $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO,
284: $ W( I+1, IW ), 1 )
285: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
286: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
287: $ W( 1, IW ), 1 )
288: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
289: $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO,
290: $ W( I+1, IW ), 1 )
291: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
292: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
293: $ W( 1, IW ), 1 )
294: END IF
295: CALL ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
296: ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1,
297: $ A( 1, I ), 1 )
298: CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
299: END IF
300: *
301: 10 CONTINUE
302: ELSE
303: *
304: * Reduce first NB columns of lower triangle
305: *
306: DO 20 I = 1, NB
307: *
308: * Update A(i:n,i)
309: *
310: A( I, I ) = DBLE( A( I, I ) )
311: CALL ZLACGV( I-1, W( I, 1 ), LDW )
312: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
313: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
314: CALL ZLACGV( I-1, W( I, 1 ), LDW )
315: CALL ZLACGV( I-1, A( I, 1 ), LDA )
316: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
317: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
318: CALL ZLACGV( I-1, A( I, 1 ), LDA )
319: A( I, I ) = DBLE( A( I, I ) )
320: IF( I.LT.N ) THEN
321: *
322: * Generate elementary reflector H(i) to annihilate
323: * A(i+2:n,i)
324: *
325: ALPHA = A( I+1, I )
326: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,
327: $ TAU( I ) )
328: E( I ) = ALPHA
329: A( I+1, I ) = ONE
330: *
331: * Compute W(i+1:n,i)
332: *
333: CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
334: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
335: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
336: $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO,
337: $ W( 1, I ), 1 )
338: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
339: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
340: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
341: $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
342: $ W( 1, I ), 1 )
343: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
344: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
345: CALL ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
346: ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1,
347: $ A( I+1, I ), 1 )
348: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
349: END IF
350: *
351: 20 CONTINUE
352: END IF
353: *
354: RETURN
355: *
356: * End of ZLATRD
357: *
358: END
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