Annotation of rpl/lapack/lapack/dlahrd.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DLAHRD
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLAHRD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahrd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahrd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahrd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLAHRD( 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: * DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
! 28: * $ Y( LDY, NB )
! 29: * ..
! 30: *
! 31: *
! 32: *> \par Purpose:
! 33: * =============
! 34: *>
! 35: *> \verbatim
! 36: *>
! 37: *> DLAHRD reduces the first NB columns of a real 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 orthogonal similarity transformation
! 40: *> Q**T * A * Q. The routine returns the matrices V and T which determine
! 41: *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
! 42: *>
! 43: *> This is an OBSOLETE auxiliary routine.
! 44: *> This routine will be 'deprecated' in a future release.
! 45: *> Please use the new routine DLAHR2 instead.
! 46: *> \endverbatim
! 47: *
! 48: * Arguments:
! 49: * ==========
! 50: *
! 51: *> \param[in] N
! 52: *> \verbatim
! 53: *> N is INTEGER
! 54: *> The order of the matrix A.
! 55: *> \endverbatim
! 56: *>
! 57: *> \param[in] K
! 58: *> \verbatim
! 59: *> K is INTEGER
! 60: *> The offset for the reduction. Elements below the k-th
! 61: *> subdiagonal in the first NB columns are reduced to zero.
! 62: *> \endverbatim
! 63: *>
! 64: *> \param[in] NB
! 65: *> \verbatim
! 66: *> NB is INTEGER
! 67: *> The number of columns to be reduced.
! 68: *> \endverbatim
! 69: *>
! 70: *> \param[in,out] A
! 71: *> \verbatim
! 72: *> A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
! 73: *> On entry, the n-by-(n-k+1) general matrix A.
! 74: *> On exit, the elements on and above the k-th subdiagonal in
! 75: *> the first NB columns are overwritten with the corresponding
! 76: *> elements of the reduced matrix; the elements below the k-th
! 77: *> subdiagonal, with the array TAU, represent the matrix Q as a
! 78: *> product of elementary reflectors. The other columns of A are
! 79: *> unchanged. See Further Details.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[in] LDA
! 83: *> \verbatim
! 84: *> LDA is INTEGER
! 85: *> The leading dimension of the array A. LDA >= max(1,N).
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[out] TAU
! 89: *> \verbatim
! 90: *> TAU is DOUBLE PRECISION array, dimension (NB)
! 91: *> The scalar factors of the elementary reflectors. See Further
! 92: *> Details.
! 93: *> \endverbatim
! 94: *>
! 95: *> \param[out] T
! 96: *> \verbatim
! 97: *> T is DOUBLE PRECISION array, dimension (LDT,NB)
! 98: *> The upper triangular matrix T.
! 99: *> \endverbatim
! 100: *>
! 101: *> \param[in] LDT
! 102: *> \verbatim
! 103: *> LDT is INTEGER
! 104: *> The leading dimension of the array T. LDT >= NB.
! 105: *> \endverbatim
! 106: *>
! 107: *> \param[out] Y
! 108: *> \verbatim
! 109: *> Y is DOUBLE PRECISION array, dimension (LDY,NB)
! 110: *> The n-by-nb matrix Y.
! 111: *> \endverbatim
! 112: *>
! 113: *> \param[in] LDY
! 114: *> \verbatim
! 115: *> LDY is INTEGER
! 116: *> The leading dimension of the array Y. LDY >= N.
! 117: *> \endverbatim
! 118: *
! 119: * Authors:
! 120: * ========
! 121: *
! 122: *> \author Univ. of Tennessee
! 123: *> \author Univ. of California Berkeley
! 124: *> \author Univ. of Colorado Denver
! 125: *> \author NAG Ltd.
! 126: *
! 127: *> \date November 2011
! 128: *
! 129: *> \ingroup doubleOTHERauxiliary
! 130: *
! 131: *> \par Further Details:
! 132: * =====================
! 133: *>
! 134: *> \verbatim
! 135: *>
! 136: *> The matrix Q is represented as a product of nb elementary reflectors
! 137: *>
! 138: *> Q = H(1) H(2) . . . H(nb).
! 139: *>
! 140: *> Each H(i) has the form
! 141: *>
! 142: *> H(i) = I - tau * v * v**T
! 143: *>
! 144: *> where tau is a real scalar, and v is a real vector with
! 145: *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
! 146: *> A(i+k+1:n,i), and tau in TAU(i).
! 147: *>
! 148: *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
! 149: *> V which is needed, with T and Y, to apply the transformation to the
! 150: *> unreduced part of the matrix, using an update of the form:
! 151: *> A := (I - V*T*V**T) * (A - Y*V**T).
! 152: *>
! 153: *> The contents of A on exit are illustrated by the following example
! 154: *> with n = 7, k = 3 and nb = 2:
! 155: *>
! 156: *> ( a h a a a )
! 157: *> ( a h a a a )
! 158: *> ( a h a a a )
! 159: *> ( h h a a a )
! 160: *> ( v1 h a a a )
! 161: *> ( v1 v2 a a a )
! 162: *> ( v1 v2 a a a )
! 163: *>
! 164: *> where a denotes an element of the original matrix A, h denotes a
! 165: *> modified element of the upper Hessenberg matrix H, and vi denotes an
! 166: *> element of the vector defining H(i).
! 167: *> \endverbatim
! 168: *>
! 169: * =====================================================================
1.1 bertrand 170: SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
171: *
1.9 ! bertrand 172: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 175: * November 2011
1.1 bertrand 176: *
177: * .. Scalar Arguments ..
178: INTEGER K, LDA, LDT, LDY, N, NB
179: * ..
180: * .. Array Arguments ..
181: DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
182: $ Y( LDY, NB )
183: * ..
184: *
185: * =====================================================================
186: *
187: * .. Parameters ..
188: DOUBLE PRECISION ZERO, ONE
189: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
190: * ..
191: * .. Local Scalars ..
192: INTEGER I
193: DOUBLE PRECISION EI
194: * ..
195: * .. External Subroutines ..
196: EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV
197: * ..
198: * .. Intrinsic Functions ..
199: INTRINSIC MIN
200: * ..
201: * .. Executable Statements ..
202: *
203: * Quick return if possible
204: *
205: IF( N.LE.1 )
206: $ RETURN
207: *
208: DO 10 I = 1, NB
209: IF( I.GT.1 ) THEN
210: *
211: * Update A(1:n,i)
212: *
1.8 bertrand 213: * Compute i-th column of A - Y * V**T
1.1 bertrand 214: *
215: CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
216: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
217: *
1.8 bertrand 218: * Apply I - V * T**T * V**T to this column (call it b) from the
1.1 bertrand 219: * left, using the last column of T as workspace
220: *
221: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
222: * ( V2 ) ( b2 )
223: *
224: * where V1 is unit lower triangular
225: *
1.8 bertrand 226: * w := V1**T * b1
1.1 bertrand 227: *
228: CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
229: CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ),
230: $ LDA, T( 1, NB ), 1 )
231: *
1.8 bertrand 232: * w := w + V2**T *b2
1.1 bertrand 233: *
234: CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ),
235: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
236: *
1.8 bertrand 237: * w := T**T *w
1.1 bertrand 238: *
239: CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT,
240: $ T( 1, NB ), 1 )
241: *
242: * b2 := b2 - V2*w
243: *
244: CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
245: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
246: *
247: * b1 := b1 - V1*w
248: *
249: CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1,
250: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
251: CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
252: *
253: A( K+I-1, I-1 ) = EI
254: END IF
255: *
256: * Generate the elementary reflector H(i) to annihilate
257: * A(k+i+1:n,i)
258: *
259: CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
260: $ TAU( I ) )
261: EI = A( K+I, I )
262: A( K+I, I ) = ONE
263: *
264: * Compute Y(1:n,i)
265: *
266: CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
267: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
268: CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA,
269: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
270: CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
271: $ ONE, Y( 1, I ), 1 )
272: CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 )
273: *
274: * Compute T(1:i,i)
275: *
276: CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
277: CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
278: $ T( 1, I ), 1 )
279: T( I, I ) = TAU( I )
280: *
281: 10 CONTINUE
282: A( K+NB, NB ) = EI
283: *
284: RETURN
285: *
286: * End of DLAHRD
287: *
288: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to dlahrd.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack