Annotation of rpl/lapack/lapack/zlahr2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
! 2: *
! 3: * -- LAPACK auxiliary routine (version 3.2.1) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * -- April 2009 --
! 7: *
! 8: * .. Scalar Arguments ..
! 9: INTEGER K, LDA, LDT, LDY, N, NB
! 10: * ..
! 11: * .. Array Arguments ..
! 12: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ),
! 13: $ Y( LDY, NB )
! 14: * ..
! 15: *
! 16: * Purpose
! 17: * =======
! 18: *
! 19: * ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
! 20: * matrix A so that elements below the k-th subdiagonal are zero. The
! 21: * reduction is performed by an unitary similarity transformation
! 22: * Q' * A * Q. The routine returns the matrices V and T which determine
! 23: * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.
! 24: *
! 25: * This is an auxiliary routine called by ZGEHRD.
! 26: *
! 27: * Arguments
! 28: * =========
! 29: *
! 30: * N (input) INTEGER
! 31: * The order of the matrix A.
! 32: *
! 33: * K (input) INTEGER
! 34: * The offset for the reduction. Elements below the k-th
! 35: * subdiagonal in the first NB columns are reduced to zero.
! 36: * K < N.
! 37: *
! 38: * NB (input) INTEGER
! 39: * The number of columns to be reduced.
! 40: *
! 41: * A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
! 42: * On entry, the n-by-(n-k+1) general matrix A.
! 43: * On exit, the elements on and above the k-th subdiagonal in
! 44: * the first NB columns are overwritten with the corresponding
! 45: * elements of the reduced matrix; the elements below the k-th
! 46: * subdiagonal, with the array TAU, represent the matrix Q as a
! 47: * product of elementary reflectors. The other columns of A are
! 48: * unchanged. See Further Details.
! 49: *
! 50: * LDA (input) INTEGER
! 51: * The leading dimension of the array A. LDA >= max(1,N).
! 52: *
! 53: * TAU (output) COMPLEX*16 array, dimension (NB)
! 54: * The scalar factors of the elementary reflectors. See Further
! 55: * Details.
! 56: *
! 57: * T (output) COMPLEX*16 array, dimension (LDT,NB)
! 58: * The upper triangular matrix T.
! 59: *
! 60: * LDT (input) INTEGER
! 61: * The leading dimension of the array T. LDT >= NB.
! 62: *
! 63: * Y (output) COMPLEX*16 array, dimension (LDY,NB)
! 64: * The n-by-nb matrix Y.
! 65: *
! 66: * LDY (input) INTEGER
! 67: * The leading dimension of the array Y. LDY >= N.
! 68: *
! 69: * Further Details
! 70: * ===============
! 71: *
! 72: * The matrix Q is represented as a product of nb elementary reflectors
! 73: *
! 74: * Q = H(1) H(2) . . . H(nb).
! 75: *
! 76: * Each H(i) has the form
! 77: *
! 78: * H(i) = I - tau * v * v'
! 79: *
! 80: * where tau is a complex scalar, and v is a complex vector with
! 81: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
! 82: * A(i+k+1:n,i), and tau in TAU(i).
! 83: *
! 84: * The elements of the vectors v together form the (n-k+1)-by-nb matrix
! 85: * V which is needed, with T and Y, to apply the transformation to the
! 86: * unreduced part of the matrix, using an update of the form:
! 87: * A := (I - V*T*V') * (A - Y*V').
! 88: *
! 89: * The contents of A on exit are illustrated by the following example
! 90: * with n = 7, k = 3 and nb = 2:
! 91: *
! 92: * ( a a a a a )
! 93: * ( a a a a a )
! 94: * ( a a a a a )
! 95: * ( h h a a a )
! 96: * ( v1 h a a a )
! 97: * ( v1 v2 a a a )
! 98: * ( v1 v2 a a a )
! 99: *
! 100: * where a denotes an element of the original matrix A, h denotes a
! 101: * modified element of the upper Hessenberg matrix H, and vi denotes an
! 102: * element of the vector defining H(i).
! 103: *
! 104: * This subroutine is a slight modification of LAPACK-3.0's DLAHRD
! 105: * incorporating improvements proposed by Quintana-Orti and Van de
! 106: * Gejin. Note that the entries of A(1:K,2:NB) differ from those
! 107: * returned by the original LAPACK-3.0's DLAHRD routine. (This
! 108: * subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
! 109: *
! 110: * References
! 111: * ==========
! 112: *
! 113: * Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
! 114: * performance of reduction to Hessenberg form," ACM Transactions on
! 115: * Mathematical Software, 32(2):180-194, June 2006.
! 116: *
! 117: * =====================================================================
! 118: *
! 119: * .. Parameters ..
! 120: COMPLEX*16 ZERO, ONE
! 121: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
! 122: $ ONE = ( 1.0D+0, 0.0D+0 ) )
! 123: * ..
! 124: * .. Local Scalars ..
! 125: INTEGER I
! 126: COMPLEX*16 EI
! 127: * ..
! 128: * .. External Subroutines ..
! 129: EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZGEMV, ZLACPY,
! 130: $ ZLARFG, ZSCAL, ZTRMM, ZTRMV, ZLACGV
! 131: * ..
! 132: * .. Intrinsic Functions ..
! 133: INTRINSIC MIN
! 134: * ..
! 135: * .. Executable Statements ..
! 136: *
! 137: * Quick return if possible
! 138: *
! 139: IF( N.LE.1 )
! 140: $ RETURN
! 141: *
! 142: DO 10 I = 1, NB
! 143: IF( I.GT.1 ) THEN
! 144: *
! 145: * Update A(K+1:N,I)
! 146: *
! 147: * Update I-th column of A - Y * V'
! 148: *
! 149: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
! 150: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
! 151: $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
! 152: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
! 153: *
! 154: * Apply I - V * T' * V' to this column (call it b) from the
! 155: * left, using the last column of T as workspace
! 156: *
! 157: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
! 158: * ( V2 ) ( b2 )
! 159: *
! 160: * where V1 is unit lower triangular
! 161: *
! 162: * w := V1' * b1
! 163: *
! 164: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
! 165: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
! 166: $ I-1, A( K+1, 1 ),
! 167: $ LDA, T( 1, NB ), 1 )
! 168: *
! 169: * w := w + V2'*b2
! 170: *
! 171: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
! 172: $ ONE, A( K+I, 1 ),
! 173: $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
! 174: *
! 175: * w := T'*w
! 176: *
! 177: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
! 178: $ I-1, T, LDT,
! 179: $ T( 1, NB ), 1 )
! 180: *
! 181: * b2 := b2 - V2*w
! 182: *
! 183: CALL ZGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
! 184: $ A( K+I, 1 ),
! 185: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
! 186: *
! 187: * b1 := b1 - V1*w
! 188: *
! 189: CALL ZTRMV( 'Lower', 'NO TRANSPOSE',
! 190: $ 'UNIT', I-1,
! 191: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
! 192: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
! 193: *
! 194: A( K+I-1, I-1 ) = EI
! 195: END IF
! 196: *
! 197: * Generate the elementary reflector H(I) to annihilate
! 198: * A(K+I+1:N,I)
! 199: *
! 200: CALL ZLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
! 201: $ TAU( I ) )
! 202: EI = A( K+I, I )
! 203: A( K+I, I ) = ONE
! 204: *
! 205: * Compute Y(K+1:N,I)
! 206: *
! 207: CALL ZGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
! 208: $ ONE, A( K+1, I+1 ),
! 209: $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
! 210: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1,
! 211: $ ONE, A( K+I, 1 ), LDA,
! 212: $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
! 213: CALL ZGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
! 214: $ Y( K+1, 1 ), LDY,
! 215: $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
! 216: CALL ZSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
! 217: *
! 218: * Compute T(1:I,I)
! 219: *
! 220: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
! 221: CALL ZTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
! 222: $ I-1, T, LDT,
! 223: $ T( 1, I ), 1 )
! 224: T( I, I ) = TAU( I )
! 225: *
! 226: 10 CONTINUE
! 227: A( K+NB, NB ) = EI
! 228: *
! 229: * Compute Y(1:K,1:NB)
! 230: *
! 231: CALL ZLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
! 232: CALL ZTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
! 233: $ 'UNIT', K, NB,
! 234: $ ONE, A( K+1, 1 ), LDA, Y, LDY )
! 235: IF( N.GT.K+NB )
! 236: $ CALL ZGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
! 237: $ NB, N-K-NB, ONE,
! 238: $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
! 239: $ LDY )
! 240: CALL ZTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
! 241: $ 'NON-UNIT', K, NB,
! 242: $ ONE, T, LDT, Y, LDY )
! 243: *
! 244: RETURN
! 245: *
! 246: * End of ZLAHR2
! 247: *
! 248: END
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