Annotation of rpl/lapack/lapack/zlahr2.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZLAHR2
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZLAHR2 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahr2.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahr2.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahr2.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * INTEGER K, LDA, LDT, LDY, N, NB
! 25: * ..
! 26: * .. Array Arguments ..
! 27: * COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
! 28: * $ Y( LDY, NB )
! 29: * ..
! 30: *
! 31: *
! 32: *> \par Purpose:
! 33: * =============
! 34: *>
! 35: *> \verbatim
! 36: *>
! 37: *> ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
! 38: *> matrix A so that elements below the k-th subdiagonal are zero. The
! 39: *> reduction is performed by an unitary similarity transformation
! 40: *> Q**H * A * Q. The routine returns the matrices V and T which determine
! 41: *> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
! 42: *>
! 43: *> This is an auxiliary routine called by ZGEHRD.
! 44: *> \endverbatim
! 45: *
! 46: * Arguments:
! 47: * ==========
! 48: *
! 49: *> \param[in] N
! 50: *> \verbatim
! 51: *> N is INTEGER
! 52: *> The order of the matrix A.
! 53: *> \endverbatim
! 54: *>
! 55: *> \param[in] K
! 56: *> \verbatim
! 57: *> K is INTEGER
! 58: *> The offset for the reduction. Elements below the k-th
! 59: *> subdiagonal in the first NB columns are reduced to zero.
! 60: *> K < N.
! 61: *> \endverbatim
! 62: *>
! 63: *> \param[in] NB
! 64: *> \verbatim
! 65: *> NB is INTEGER
! 66: *> The number of columns to be reduced.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in,out] A
! 70: *> \verbatim
! 71: *> A is COMPLEX*16 array, dimension (LDA,N-K+1)
! 72: *> On entry, the n-by-(n-k+1) general matrix A.
! 73: *> On exit, the elements on and above the k-th subdiagonal in
! 74: *> the first NB columns are overwritten with the corresponding
! 75: *> elements of the reduced matrix; the elements below the k-th
! 76: *> subdiagonal, with the array TAU, represent the matrix Q as a
! 77: *> product of elementary reflectors. The other columns of A are
! 78: *> unchanged. 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] TAU
! 88: *> \verbatim
! 89: *> TAU is COMPLEX*16 array, dimension (NB)
! 90: *> The scalar factors of the elementary reflectors. See Further
! 91: *> Details.
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[out] T
! 95: *> \verbatim
! 96: *> T is COMPLEX*16 array, dimension (LDT,NB)
! 97: *> The upper triangular matrix T.
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] LDT
! 101: *> \verbatim
! 102: *> LDT is INTEGER
! 103: *> The leading dimension of the array T. LDT >= NB.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[out] Y
! 107: *> \verbatim
! 108: *> Y is COMPLEX*16 array, dimension (LDY,NB)
! 109: *> The n-by-nb matrix Y.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] LDY
! 113: *> \verbatim
! 114: *> LDY is INTEGER
! 115: *> The leading dimension of the array Y. LDY >= N.
! 116: *> \endverbatim
! 117: *
! 118: * Authors:
! 119: * ========
! 120: *
! 121: *> \author Univ. of Tennessee
! 122: *> \author Univ. of California Berkeley
! 123: *> \author Univ. of Colorado Denver
! 124: *> \author NAG Ltd.
! 125: *
! 126: *> \date November 2011
! 127: *
! 128: *> \ingroup complex16OTHERauxiliary
! 129: *
! 130: *> \par Further Details:
! 131: * =====================
! 132: *>
! 133: *> \verbatim
! 134: *>
! 135: *> The matrix Q is represented as a product of nb elementary reflectors
! 136: *>
! 137: *> Q = H(1) H(2) . . . H(nb).
! 138: *>
! 139: *> Each H(i) has the form
! 140: *>
! 141: *> H(i) = I - tau * v * v**H
! 142: *>
! 143: *> where tau is a complex scalar, and v is a complex vector with
! 144: *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
! 145: *> A(i+k+1:n,i), and tau in TAU(i).
! 146: *>
! 147: *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
! 148: *> V which is needed, with T and Y, to apply the transformation to the
! 149: *> unreduced part of the matrix, using an update of the form:
! 150: *> A := (I - V*T*V**H) * (A - Y*V**H).
! 151: *>
! 152: *> The contents of A on exit are illustrated by the following example
! 153: *> with n = 7, k = 3 and nb = 2:
! 154: *>
! 155: *> ( a a a a a )
! 156: *> ( a a a a a )
! 157: *> ( a a a a a )
! 158: *> ( h h a a a )
! 159: *> ( v1 h a a a )
! 160: *> ( v1 v2 a a a )
! 161: *> ( v1 v2 a a a )
! 162: *>
! 163: *> where a denotes an element of the original matrix A, h denotes a
! 164: *> modified element of the upper Hessenberg matrix H, and vi denotes an
! 165: *> element of the vector defining H(i).
! 166: *>
! 167: *> This subroutine is a slight modification of LAPACK-3.0's DLAHRD
! 168: *> incorporating improvements proposed by Quintana-Orti and Van de
! 169: *> Gejin. Note that the entries of A(1:K,2:NB) differ from those
! 170: *> returned by the original LAPACK-3.0's DLAHRD routine. (This
! 171: *> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
! 172: *> \endverbatim
! 173: *
! 174: *> \par References:
! 175: * ================
! 176: *>
! 177: *> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
! 178: *> performance of reduction to Hessenberg form," ACM Transactions on
! 179: *> Mathematical Software, 32(2):180-194, June 2006.
! 180: *>
! 181: * =====================================================================
1.1 bertrand 182: SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
183: *
1.9 ! bertrand 184: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 185: * -- LAPACK is a software package provided by Univ. of Tennessee, --
186: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 187: * November 2011
1.1 bertrand 188: *
189: * .. Scalar Arguments ..
190: INTEGER K, LDA, LDT, LDY, N, NB
191: * ..
192: * .. Array Arguments ..
193: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
194: $ Y( LDY, NB )
195: * ..
196: *
197: * =====================================================================
198: *
199: * .. Parameters ..
200: COMPLEX*16 ZERO, ONE
201: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
202: $ ONE = ( 1.0D+0, 0.0D+0 ) )
203: * ..
204: * .. Local Scalars ..
205: INTEGER I
206: COMPLEX*16 EI
207: * ..
208: * .. External Subroutines ..
209: EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZGEMV, ZLACPY,
210: $ ZLARFG, ZSCAL, ZTRMM, ZTRMV, ZLACGV
211: * ..
212: * .. Intrinsic Functions ..
213: INTRINSIC MIN
214: * ..
215: * .. Executable Statements ..
216: *
217: * Quick return if possible
218: *
219: IF( N.LE.1 )
220: $ RETURN
221: *
222: DO 10 I = 1, NB
223: IF( I.GT.1 ) THEN
224: *
225: * Update A(K+1:N,I)
226: *
1.8 bertrand 227: * Update I-th column of A - Y * V**H
1.1 bertrand 228: *
229: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
230: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
231: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
232: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
233: *
1.8 bertrand 234: * Apply I - V * T**H * V**H to this column (call it b) from the
1.1 bertrand 235: * left, using the last column of T as workspace
236: *
237: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
238: * ( V2 ) ( b2 )
239: *
240: * where V1 is unit lower triangular
241: *
1.8 bertrand 242: * w := V1**H * b1
1.1 bertrand 243: *
244: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
245: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
246: $ I-1, A( K+1, 1 ),
247: $ LDA, T( 1, NB ), 1 )
248: *
1.8 bertrand 249: * w := w + V2**H * b2
1.1 bertrand 250: *
251: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
252: $ ONE, A( K+I, 1 ),
253: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
254: *
1.8 bertrand 255: * w := T**H * w
1.1 bertrand 256: *
257: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
258: $ I-1, T, LDT,
259: $ T( 1, NB ), 1 )
260: *
261: * b2 := b2 - V2*w
262: *
263: CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
264: $ A( K+I, 1 ),
265: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
266: *
267: * b1 := b1 - V1*w
268: *
269: CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
270: $ 'UNIT', I-1,
271: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
272: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
273: *
274: A( K+I-1, I-1 ) = EI
275: END IF
276: *
277: * Generate the elementary reflector H(I) to annihilate
278: * A(K+I+1:N,I)
279: *
280: CALL ZLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
281: $ TAU( I ) )
282: EI = A( K+I, I )
283: A( K+I, I ) = ONE
284: *
285: * Compute Y(K+1:N,I)
286: *
287: CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
288: $ ONE, A( K+1, I+1 ),
289: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
290: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
291: $ ONE, A( K+I, 1 ), LDA,
292: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
293: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
294: $ Y( K+1, 1 ), LDY,
295: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
296: CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
297: *
298: * Compute T(1:I,I)
299: *
300: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
301: CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
302: $ I-1, T, LDT,
303: $ T( 1, I ), 1 )
304: T( I, I ) = TAU( I )
305: *
306: 10 CONTINUE
307: A( K+NB, NB ) = EI
308: *
309: * Compute Y(1:K,1:NB)
310: *
311: CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
312: CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
313: $ 'UNIT', K, NB,
314: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
315: IF( N.GT.K+NB )
316: $ CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
317: $ NB, N-K-NB, ONE,
318: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
319: $ LDY )
320: CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
321: $ 'NON-UNIT', K, NB,
322: $ ONE, T, LDT, Y, LDY )
323: *
324: RETURN
325: *
326: * End of ZLAHR2
327: *
328: END
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