Annotation of rpl/lapack/lapack/zgeqrt3.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZGEQRT3
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGEQRT3 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeqrt3.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeqrt3.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeqrt3.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * RECURSIVE SUBROUTINE ZGEQRT3( M, N, A, LDA, T, LDT, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * INTEGER INFO, LDA, M, N, LDT
! 25: * ..
! 26: * .. Array Arguments ..
! 27: * COMPLEX*16 A( LDA, * ), T( LDT, * )
! 28: * ..
! 29: *
! 30: *
! 31: *> \par Purpose:
! 32: * =============
! 33: *>
! 34: *> \verbatim
! 35: *>
! 36: *> ZGEQRT3 recursively computes a QR factorization of a complex M-by-N
! 37: *> matrix A, using the compact WY representation of Q.
! 38: *>
! 39: *> Based on the algorithm of Elmroth and Gustavson,
! 40: *> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
! 41: *> \endverbatim
! 42: *
! 43: * Arguments:
! 44: * ==========
! 45: *
! 46: *> \param[in] M
! 47: *> \verbatim
! 48: *> M is INTEGER
! 49: *> The number of rows of the matrix A. M >= N.
! 50: *> \endverbatim
! 51: *>
! 52: *> \param[in] N
! 53: *> \verbatim
! 54: *> N is INTEGER
! 55: *> The number of columns of the matrix A. N >= 0.
! 56: *> \endverbatim
! 57: *>
! 58: *> \param[in,out] A
! 59: *> \verbatim
! 60: *> A is COMPLEX*16 array, dimension (LDA,N)
! 61: *> On entry, the complex M-by-N matrix A. On exit, the elements on
! 62: *> and above the diagonal contain the N-by-N upper triangular matrix R;
! 63: *> the elements below the diagonal are the columns of V. See below for
! 64: *> further details.
! 65: *> \endverbatim
! 66: *>
! 67: *> \param[in] LDA
! 68: *> \verbatim
! 69: *> LDA is INTEGER
! 70: *> The leading dimension of the array A. LDA >= max(1,M).
! 71: *> \endverbatim
! 72: *>
! 73: *> \param[out] T
! 74: *> \verbatim
! 75: *> T is COMPLEX*16 array, dimension (LDT,N)
! 76: *> The N-by-N upper triangular factor of the block reflector.
! 77: *> The elements on and above the diagonal contain the block
! 78: *> reflector T; the elements below the diagonal are not used.
! 79: *> See below for further details.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[in] LDT
! 83: *> \verbatim
! 84: *> LDT is INTEGER
! 85: *> The leading dimension of the array T. LDT >= max(1,N).
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[out] INFO
! 89: *> \verbatim
! 90: *> INFO is INTEGER
! 91: *> = 0: successful exit
! 92: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 93: *> \endverbatim
! 94: *
! 95: * Authors:
! 96: * ========
! 97: *
! 98: *> \author Univ. of Tennessee
! 99: *> \author Univ. of California Berkeley
! 100: *> \author Univ. of Colorado Denver
! 101: *> \author NAG Ltd.
! 102: *
! 103: *> \date November 2011
! 104: *
! 105: *> \ingroup complex16GEcomputational
! 106: *
! 107: *> \par Further Details:
! 108: * =====================
! 109: *>
! 110: *> \verbatim
! 111: *>
! 112: *> The matrix V stores the elementary reflectors H(i) in the i-th column
! 113: *> below the diagonal. For example, if M=5 and N=3, the matrix V is
! 114: *>
! 115: *> V = ( 1 )
! 116: *> ( v1 1 )
! 117: *> ( v1 v2 1 )
! 118: *> ( v1 v2 v3 )
! 119: *> ( v1 v2 v3 )
! 120: *>
! 121: *> where the vi's represent the vectors which define H(i), which are returned
! 122: *> in the matrix A. The 1's along the diagonal of V are not stored in A. The
! 123: *> block reflector H is then given by
! 124: *>
! 125: *> H = I - V * T * V**H
! 126: *>
! 127: *> where V**H is the conjugate transpose of V.
! 128: *>
! 129: *> For details of the algorithm, see Elmroth and Gustavson (cited above).
! 130: *> \endverbatim
! 131: *>
! 132: * =====================================================================
! 133: RECURSIVE SUBROUTINE ZGEQRT3( M, N, A, LDA, T, LDT, INFO )
! 134: *
! 135: * -- LAPACK computational routine (version 3.4.0) --
! 136: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 137: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 138: * November 2011
! 139: *
! 140: * .. Scalar Arguments ..
! 141: INTEGER INFO, LDA, M, N, LDT
! 142: * ..
! 143: * .. Array Arguments ..
! 144: COMPLEX*16 A( LDA, * ), T( LDT, * )
! 145: * ..
! 146: *
! 147: * =====================================================================
! 148: *
! 149: * .. Parameters ..
! 150: COMPLEX*16 ONE
! 151: PARAMETER ( ONE = (1.0D+00,0.0D+00) )
! 152: * ..
! 153: * .. Local Scalars ..
! 154: INTEGER I, I1, J, J1, N1, N2, IINFO
! 155: * ..
! 156: * .. External Subroutines ..
! 157: EXTERNAL ZLARFG, ZTRMM, ZGEMM, XERBLA
! 158: * ..
! 159: * .. Executable Statements ..
! 160: *
! 161: INFO = 0
! 162: IF( N .LT. 0 ) THEN
! 163: INFO = -2
! 164: ELSE IF( M .LT. N ) THEN
! 165: INFO = -1
! 166: ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
! 167: INFO = -4
! 168: ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
! 169: INFO = -6
! 170: END IF
! 171: IF( INFO.NE.0 ) THEN
! 172: CALL XERBLA( 'ZGEQRT3', -INFO )
! 173: RETURN
! 174: END IF
! 175: *
! 176: IF( N.EQ.1 ) THEN
! 177: *
! 178: * Compute Householder transform when N=1
! 179: *
! 180: CALL ZLARFG( M, A, A( MIN( 2, M ), 1 ), 1, T )
! 181: *
! 182: ELSE
! 183: *
! 184: * Otherwise, split A into blocks...
! 185: *
! 186: N1 = N/2
! 187: N2 = N-N1
! 188: J1 = MIN( N1+1, N )
! 189: I1 = MIN( N+1, M )
! 190: *
! 191: * Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
! 192: *
! 193: CALL ZGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
! 194: *
! 195: * Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
! 196: *
! 197: DO J=1,N2
! 198: DO I=1,N1
! 199: T( I, J+N1 ) = A( I, J+N1 )
! 200: END DO
! 201: END DO
! 202: CALL ZTRMM( 'L', 'L', 'C', 'U', N1, N2, ONE,
! 203: & A, LDA, T( 1, J1 ), LDT )
! 204: *
! 205: CALL ZGEMM( 'C', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
! 206: & A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
! 207: *
! 208: CALL ZTRMM( 'L', 'U', 'C', 'N', N1, N2, ONE,
! 209: & T, LDT, T( 1, J1 ), LDT )
! 210: *
! 211: CALL ZGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
! 212: & T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
! 213: *
! 214: CALL ZTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
! 215: & A, LDA, T( 1, J1 ), LDT )
! 216: *
! 217: DO J=1,N2
! 218: DO I=1,N1
! 219: A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
! 220: END DO
! 221: END DO
! 222: *
! 223: * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
! 224: *
! 225: CALL ZGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
! 226: & T( J1, J1 ), LDT, IINFO )
! 227: *
! 228: * Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
! 229: *
! 230: DO I=1,N1
! 231: DO J=1,N2
! 232: T( I, J+N1 ) = CONJG(A( J+N1, I ))
! 233: END DO
! 234: END DO
! 235: *
! 236: CALL ZTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
! 237: & A( J1, J1 ), LDA, T( 1, J1 ), LDT )
! 238: *
! 239: CALL ZGEMM( 'C', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
! 240: & A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
! 241: *
! 242: CALL ZTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
! 243: & T( 1, J1 ), LDT )
! 244: *
! 245: CALL ZTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
! 246: & T( J1, J1 ), LDT, T( 1, J1 ), LDT )
! 247: *
! 248: * Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
! 249: * [ 0 R2 ] [ 0 T2]
! 250: *
! 251: END IF
! 252: *
! 253: RETURN
! 254: *
! 255: * End of ZGEQRT3
! 256: *
! 257: END
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