Annotation of rpl/lapack/lapack/dlaorhr_col_getrfnp2.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DLAORHR_COL_GETRFNP2
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLAORHR_GETRF2NP + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaorhr_col_getrfnp2.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * INTEGER INFO, LDA, M, N
! 25: * ..
! 26: * .. Array Arguments ..
! 27: * DOUBLE PRECISION A( LDA, * ), D( * )
! 28: * ..
! 29: *
! 30: *
! 31: *> \par Purpose:
! 32: * =============
! 33: *>
! 34: *> \verbatim
! 35: *>
! 36: *> DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
! 37: *> pivoting of a real general M-by-N matrix A. The factorization has
! 38: *> the form:
! 39: *>
! 40: *> A - S = L * U,
! 41: *>
! 42: *> where:
! 43: *> S is a m-by-n diagonal sign matrix with the diagonal D, so that
! 44: *> D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
! 45: *> as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
! 46: *> i-1 steps of Gaussian elimination. This means that the diagonal
! 47: *> element at each step of "modified" Gaussian elimination is at
! 48: *> least one in absolute value (so that division-by-zero not
! 49: *> possible during the division by the diagonal element);
! 50: *>
! 51: *> L is a M-by-N lower triangular matrix with unit diagonal elements
! 52: *> (lower trapezoidal if M > N);
! 53: *>
! 54: *> and U is a M-by-N upper triangular matrix
! 55: *> (upper trapezoidal if M < N).
! 56: *>
! 57: *> This routine is an auxiliary routine used in the Householder
! 58: *> reconstruction routine DORHR_COL. In DORHR_COL, this routine is
! 59: *> applied to an M-by-N matrix A with orthonormal columns, where each
! 60: *> element is bounded by one in absolute value. With the choice of
! 61: *> the matrix S above, one can show that the diagonal element at each
! 62: *> step of Gaussian elimination is the largest (in absolute value) in
! 63: *> the column on or below the diagonal, so that no pivoting is required
! 64: *> for numerical stability [1].
! 65: *>
! 66: *> For more details on the Householder reconstruction algorithm,
! 67: *> including the modified LU factorization, see [1].
! 68: *>
! 69: *> This is the recursive version of the LU factorization algorithm.
! 70: *> Denote A - S by B. The algorithm divides the matrix B into four
! 71: *> submatrices:
! 72: *>
! 73: *> [ B11 | B12 ] where B11 is n1 by n1,
! 74: *> B = [ -----|----- ] B21 is (m-n1) by n1,
! 75: *> [ B21 | B22 ] B12 is n1 by n2,
! 76: *> B22 is (m-n1) by n2,
! 77: *> with n1 = min(m,n)/2, n2 = n-n1.
! 78: *>
! 79: *>
! 80: *> The subroutine calls itself to factor B11, solves for B21,
! 81: *> solves for B12, updates B22, then calls itself to factor B22.
! 82: *>
! 83: *> For more details on the recursive LU algorithm, see [2].
! 84: *>
! 85: *> DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
! 86: *> routine DLAORHR_COL_GETRFNP, which uses blocked code calling
! 87: *. Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
! 88: *> is self-sufficient and can be used without DLAORHR_COL_GETRFNP.
! 89: *>
! 90: *> [1] "Reconstructing Householder vectors from tall-skinny QR",
! 91: *> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
! 92: *> E. Solomonik, J. Parallel Distrib. Comput.,
! 93: *> vol. 85, pp. 3-31, 2015.
! 94: *>
! 95: *> [2] "Recursion leads to automatic variable blocking for dense linear
! 96: *> algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,
! 97: *> vol. 41, no. 6, pp. 737-755, 1997.
! 98: *> \endverbatim
! 99: *
! 100: * Arguments:
! 101: * ==========
! 102: *
! 103: *> \param[in] M
! 104: *> \verbatim
! 105: *> M is INTEGER
! 106: *> The number of rows of the matrix A. M >= 0.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in] N
! 110: *> \verbatim
! 111: *> N is INTEGER
! 112: *> The number of columns of the matrix A. N >= 0.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[in,out] A
! 116: *> \verbatim
! 117: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 118: *> On entry, the M-by-N matrix to be factored.
! 119: *> On exit, the factors L and U from the factorization
! 120: *> A-S=L*U; the unit diagonal elements of L are not stored.
! 121: *> \endverbatim
! 122: *>
! 123: *> \param[in] LDA
! 124: *> \verbatim
! 125: *> LDA is INTEGER
! 126: *> The leading dimension of the array A. LDA >= max(1,M).
! 127: *> \endverbatim
! 128: *>
! 129: *> \param[out] D
! 130: *> \verbatim
! 131: *> D is DOUBLE PRECISION array, dimension min(M,N)
! 132: *> The diagonal elements of the diagonal M-by-N sign matrix S,
! 133: *> D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
! 134: *> be only plus or minus one.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[out] INFO
! 138: *> \verbatim
! 139: *> INFO is INTEGER
! 140: *> = 0: successful exit
! 141: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 142: *> \endverbatim
! 143: *>
! 144: * Authors:
! 145: * ========
! 146: *
! 147: *> \author Univ. of Tennessee
! 148: *> \author Univ. of California Berkeley
! 149: *> \author Univ. of Colorado Denver
! 150: *> \author NAG Ltd.
! 151: *
! 152: *> \date November 2019
! 153: *
! 154: *> \ingroup doubleGEcomputational
! 155: *
! 156: *> \par Contributors:
! 157: * ==================
! 158: *>
! 159: *> \verbatim
! 160: *>
! 161: *> November 2019, Igor Kozachenko,
! 162: *> Computer Science Division,
! 163: *> University of California, Berkeley
! 164: *>
! 165: *> \endverbatim
! 166: *
! 167: * =====================================================================
! 168: RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
! 169: IMPLICIT NONE
! 170: *
! 171: * -- LAPACK computational routine (version 3.9.0) --
! 172: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 173: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 174: * November 2019
! 175: *
! 176: * .. Scalar Arguments ..
! 177: INTEGER INFO, LDA, M, N
! 178: * ..
! 179: * .. Array Arguments ..
! 180: DOUBLE PRECISION A( LDA, * ), D( * )
! 181: * ..
! 182: *
! 183: * =====================================================================
! 184: *
! 185: * .. Parameters ..
! 186: DOUBLE PRECISION ONE
! 187: PARAMETER ( ONE = 1.0D+0 )
! 188: * ..
! 189: * .. Local Scalars ..
! 190: DOUBLE PRECISION SFMIN
! 191: INTEGER I, IINFO, N1, N2
! 192: * ..
! 193: * .. External Functions ..
! 194: DOUBLE PRECISION DLAMCH
! 195: EXTERNAL DLAMCH
! 196: * ..
! 197: * .. External Subroutines ..
! 198: EXTERNAL DGEMM, DSCAL, DTRSM, XERBLA
! 199: * ..
! 200: * .. Intrinsic Functions ..
! 201: INTRINSIC ABS, DSIGN, MAX, MIN
! 202: * ..
! 203: * .. Executable Statements ..
! 204: *
! 205: * Test the input parameters
! 206: *
! 207: INFO = 0
! 208: IF( M.LT.0 ) THEN
! 209: INFO = -1
! 210: ELSE IF( N.LT.0 ) THEN
! 211: INFO = -2
! 212: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 213: INFO = -4
! 214: END IF
! 215: IF( INFO.NE.0 ) THEN
! 216: CALL XERBLA( 'DLAORHR_COL_GETRFNP2', -INFO )
! 217: RETURN
! 218: END IF
! 219: *
! 220: * Quick return if possible
! 221: *
! 222: IF( MIN( M, N ).EQ.0 )
! 223: $ RETURN
! 224:
! 225: IF ( M.EQ.1 ) THEN
! 226: *
! 227: * One row case, (also recursion termination case),
! 228: * use unblocked code
! 229: *
! 230: * Transfer the sign
! 231: *
! 232: D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
! 233: *
! 234: * Construct the row of U
! 235: *
! 236: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
! 237: *
! 238: ELSE IF( N.EQ.1 ) THEN
! 239: *
! 240: * One column case, (also recursion termination case),
! 241: * use unblocked code
! 242: *
! 243: * Transfer the sign
! 244: *
! 245: D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
! 246: *
! 247: * Construct the row of U
! 248: *
! 249: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
! 250: *
! 251: * Scale the elements 2:M of the column
! 252: *
! 253: * Determine machine safe minimum
! 254: *
! 255: SFMIN = DLAMCH('S')
! 256: *
! 257: * Construct the subdiagonal elements of L
! 258: *
! 259: IF( ABS( A( 1, 1 ) ) .GE. SFMIN ) THEN
! 260: CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
! 261: ELSE
! 262: DO I = 2, M
! 263: A( I, 1 ) = A( I, 1 ) / A( 1, 1 )
! 264: END DO
! 265: END IF
! 266: *
! 267: ELSE
! 268: *
! 269: * Divide the matrix B into four submatrices
! 270: *
! 271: N1 = MIN( M, N ) / 2
! 272: N2 = N-N1
! 273:
! 274: *
! 275: * Factor B11, recursive call
! 276: *
! 277: CALL DLAORHR_COL_GETRFNP2( N1, N1, A, LDA, D, IINFO )
! 278: *
! 279: * Solve for B21
! 280: *
! 281: CALL DTRSM( 'R', 'U', 'N', 'N', M-N1, N1, ONE, A, LDA,
! 282: $ A( N1+1, 1 ), LDA )
! 283: *
! 284: * Solve for B12
! 285: *
! 286: CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
! 287: $ A( 1, N1+1 ), LDA )
! 288: *
! 289: * Update B22, i.e. compute the Schur complement
! 290: * B22 := B22 - B21*B12
! 291: *
! 292: CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
! 293: $ A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
! 294: *
! 295: * Factor B22, recursive call
! 296: *
! 297: CALL DLAORHR_COL_GETRFNP2( M-N1, N2, A( N1+1, N1+1 ), LDA,
! 298: $ D( N1+1 ), IINFO )
! 299: *
! 300: END IF
! 301: RETURN
! 302: *
! 303: * End of DLAORHR_COL_GETRFNP2
! 304: *
! 305: END
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