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