Annotation of rpl/lapack/lapack/dlaqps.f, revision 1.10
1.10 ! bertrand 1: *> \brief \b DLAQPS
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLAQPS + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqps.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqps.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqps.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
! 22: * VN2, AUXV, F, LDF )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER KB, LDA, LDF, M, N, NB, OFFSET
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * INTEGER JPVT( * )
! 29: * DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
! 30: * $ VN1( * ), VN2( * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *>
! 39: *> DLAQPS computes a step of QR factorization with column pivoting
! 40: *> of a real M-by-N matrix A by using Blas-3. It tries to factorize
! 41: *> NB columns from A starting from the row OFFSET+1, and updates all
! 42: *> of the matrix with Blas-3 xGEMM.
! 43: *>
! 44: *> In some cases, due to catastrophic cancellations, it cannot
! 45: *> factorize NB columns. Hence, the actual number of factorized
! 46: *> columns is returned in KB.
! 47: *>
! 48: *> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
! 49: *> \endverbatim
! 50: *
! 51: * Arguments:
! 52: * ==========
! 53: *
! 54: *> \param[in] M
! 55: *> \verbatim
! 56: *> M is INTEGER
! 57: *> The number of rows of the matrix A. M >= 0.
! 58: *> \endverbatim
! 59: *>
! 60: *> \param[in] N
! 61: *> \verbatim
! 62: *> N is INTEGER
! 63: *> The number of columns of the matrix A. N >= 0
! 64: *> \endverbatim
! 65: *>
! 66: *> \param[in] OFFSET
! 67: *> \verbatim
! 68: *> OFFSET is INTEGER
! 69: *> The number of rows of A that have been factorized in
! 70: *> previous steps.
! 71: *> \endverbatim
! 72: *>
! 73: *> \param[in] NB
! 74: *> \verbatim
! 75: *> NB is INTEGER
! 76: *> The number of columns to factorize.
! 77: *> \endverbatim
! 78: *>
! 79: *> \param[out] KB
! 80: *> \verbatim
! 81: *> KB is INTEGER
! 82: *> The number of columns actually factorized.
! 83: *> \endverbatim
! 84: *>
! 85: *> \param[in,out] A
! 86: *> \verbatim
! 87: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 88: *> On entry, the M-by-N matrix A.
! 89: *> On exit, block A(OFFSET+1:M,1:KB) is the triangular
! 90: *> factor obtained and block A(1:OFFSET,1:N) has been
! 91: *> accordingly pivoted, but no factorized.
! 92: *> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
! 93: *> been updated.
! 94: *> \endverbatim
! 95: *>
! 96: *> \param[in] LDA
! 97: *> \verbatim
! 98: *> LDA is INTEGER
! 99: *> The leading dimension of the array A. LDA >= max(1,M).
! 100: *> \endverbatim
! 101: *>
! 102: *> \param[in,out] JPVT
! 103: *> \verbatim
! 104: *> JPVT is INTEGER array, dimension (N)
! 105: *> JPVT(I) = K <==> Column K of the full matrix A has been
! 106: *> permuted into position I in AP.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] TAU
! 110: *> \verbatim
! 111: *> TAU is DOUBLE PRECISION array, dimension (KB)
! 112: *> The scalar factors of the elementary reflectors.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[in,out] VN1
! 116: *> \verbatim
! 117: *> VN1 is DOUBLE PRECISION array, dimension (N)
! 118: *> The vector with the partial column norms.
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in,out] VN2
! 122: *> \verbatim
! 123: *> VN2 is DOUBLE PRECISION array, dimension (N)
! 124: *> The vector with the exact column norms.
! 125: *> \endverbatim
! 126: *>
! 127: *> \param[in,out] AUXV
! 128: *> \verbatim
! 129: *> AUXV is DOUBLE PRECISION array, dimension (NB)
! 130: *> Auxiliar vector.
! 131: *> \endverbatim
! 132: *>
! 133: *> \param[in,out] F
! 134: *> \verbatim
! 135: *> F is DOUBLE PRECISION array, dimension (LDF,NB)
! 136: *> Matrix F**T = L*Y**T*A.
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[in] LDF
! 140: *> \verbatim
! 141: *> LDF is INTEGER
! 142: *> The leading dimension of the array F. LDF >= max(1,N).
! 143: *> \endverbatim
! 144: *
! 145: * Authors:
! 146: * ========
! 147: *
! 148: *> \author Univ. of Tennessee
! 149: *> \author Univ. of California Berkeley
! 150: *> \author Univ. of Colorado Denver
! 151: *> \author NAG Ltd.
! 152: *
! 153: *> \date November 2011
! 154: *
! 155: *> \ingroup doubleOTHERauxiliary
! 156: *
! 157: *> \par Contributors:
! 158: * ==================
! 159: *>
! 160: *> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
! 161: *> X. Sun, Computer Science Dept., Duke University, USA
! 162: *> \n
! 163: *> Partial column norm updating strategy modified on April 2011
! 164: *> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
! 165: *> University of Zagreb, Croatia.
! 166: *
! 167: *> \par References:
! 168: * ================
! 169: *>
! 170: *> LAPACK Working Note 176
! 171: *
! 172: *> \htmlonly
! 173: *> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
! 174: *> \endhtmlonly
! 175: *
! 176: * =====================================================================
1.1 bertrand 177: SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
178: $ VN2, AUXV, F, LDF )
179: *
1.10 ! bertrand 180: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 181: * -- LAPACK is a software package provided by Univ. of Tennessee, --
182: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 ! bertrand 183: * November 2011
1.1 bertrand 184: *
185: * .. Scalar Arguments ..
186: INTEGER KB, LDA, LDF, M, N, NB, OFFSET
187: * ..
188: * .. Array Arguments ..
189: INTEGER JPVT( * )
190: DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
191: $ VN1( * ), VN2( * )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: DOUBLE PRECISION ZERO, ONE
198: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
199: * ..
200: * .. Local Scalars ..
201: INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
202: DOUBLE PRECISION AKK, TEMP, TEMP2, TOL3Z
203: * ..
204: * .. External Subroutines ..
1.5 bertrand 205: EXTERNAL DGEMM, DGEMV, DLARFG, DSWAP
1.1 bertrand 206: * ..
207: * .. Intrinsic Functions ..
208: INTRINSIC ABS, DBLE, MAX, MIN, NINT, SQRT
209: * ..
210: * .. External Functions ..
211: INTEGER IDAMAX
212: DOUBLE PRECISION DLAMCH, DNRM2
213: EXTERNAL IDAMAX, DLAMCH, DNRM2
214: * ..
215: * .. Executable Statements ..
216: *
217: LASTRK = MIN( M, N+OFFSET )
218: LSTICC = 0
219: K = 0
220: TOL3Z = SQRT(DLAMCH('Epsilon'))
221: *
222: * Beginning of while loop.
223: *
224: 10 CONTINUE
225: IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
226: K = K + 1
227: RK = OFFSET + K
228: *
229: * Determine ith pivot column and swap if necessary
230: *
231: PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
232: IF( PVT.NE.K ) THEN
233: CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
234: CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
235: ITEMP = JPVT( PVT )
236: JPVT( PVT ) = JPVT( K )
237: JPVT( K ) = ITEMP
238: VN1( PVT ) = VN1( K )
239: VN2( PVT ) = VN2( K )
240: END IF
241: *
242: * Apply previous Householder reflectors to column K:
1.9 bertrand 243: * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**T.
1.1 bertrand 244: *
245: IF( K.GT.1 ) THEN
246: CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ),
247: $ LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
248: END IF
249: *
250: * Generate elementary reflector H(k).
251: *
252: IF( RK.LT.M ) THEN
1.5 bertrand 253: CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
1.1 bertrand 254: ELSE
1.5 bertrand 255: CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
1.1 bertrand 256: END IF
257: *
258: AKK = A( RK, K )
259: A( RK, K ) = ONE
260: *
261: * Compute Kth column of F:
262: *
1.9 bertrand 263: * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**T*A(RK:M,K).
1.1 bertrand 264: *
265: IF( K.LT.N ) THEN
266: CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
267: $ A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
268: $ F( K+1, K ), 1 )
269: END IF
270: *
271: * Padding F(1:K,K) with zeros.
272: *
273: DO 20 J = 1, K
274: F( J, K ) = ZERO
275: 20 CONTINUE
276: *
277: * Incremental updating of F:
1.9 bertrand 278: * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**T
1.1 bertrand 279: * *A(RK:M,K).
280: *
281: IF( K.GT.1 ) THEN
282: CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ),
283: $ LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
284: *
285: CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
286: $ AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
287: END IF
288: *
289: * Update the current row of A:
1.9 bertrand 290: * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**T.
1.1 bertrand 291: *
292: IF( K.LT.N ) THEN
293: CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF,
294: $ A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
295: END IF
296: *
297: * Update partial column norms.
298: *
299: IF( RK.LT.LASTRK ) THEN
300: DO 30 J = K + 1, N
301: IF( VN1( J ).NE.ZERO ) THEN
302: *
303: * NOTE: The following 4 lines follow from the analysis in
304: * Lapack Working Note 176.
305: *
306: TEMP = ABS( A( RK, J ) ) / VN1( J )
307: TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
308: TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
309: IF( TEMP2 .LE. TOL3Z ) THEN
310: VN2( J ) = DBLE( LSTICC )
311: LSTICC = J
312: ELSE
313: VN1( J ) = VN1( J )*SQRT( TEMP )
314: END IF
315: END IF
316: 30 CONTINUE
317: END IF
318: *
319: A( RK, K ) = AKK
320: *
321: * End of while loop.
322: *
323: GO TO 10
324: END IF
325: KB = K
326: RK = OFFSET + KB
327: *
328: * Apply the block reflector to the rest of the matrix:
329: * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
1.9 bertrand 330: * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**T.
1.1 bertrand 331: *
332: IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
333: CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE,
334: $ A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
335: $ A( RK+1, KB+1 ), LDA )
336: END IF
337: *
338: * Recomputation of difficult columns.
339: *
340: 40 CONTINUE
341: IF( LSTICC.GT.0 ) THEN
342: ITEMP = NINT( VN2( LSTICC ) )
343: VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 )
344: *
345: * NOTE: The computation of VN1( LSTICC ) relies on the fact that
346: * SNRM2 does not fail on vectors with norm below the value of
347: * SQRT(DLAMCH('S'))
348: *
349: VN2( LSTICC ) = VN1( LSTICC )
350: LSTICC = ITEMP
351: GO TO 40
352: END IF
353: *
354: RETURN
355: *
356: * End of DLAQPS
357: *
358: END
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