Annotation of rpl/lapack/lapack/zlahrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
! 2: *
! 3: * -- LAPACK auxiliary routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 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: * ZLAHRD 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 a 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 OBSOLETE auxiliary routine.
! 26: * This routine will be 'deprecated' in a future release.
! 27: * Please use the new routine ZLAHR2 instead.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * N (input) INTEGER
! 33: * The order of the matrix A.
! 34: *
! 35: * K (input) INTEGER
! 36: * The offset for the reduction. Elements below the k-th
! 37: * subdiagonal in the first NB columns are reduced to zero.
! 38: *
! 39: * NB (input) INTEGER
! 40: * The number of columns to be reduced.
! 41: *
! 42: * A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
! 43: * On entry, the n-by-(n-k+1) general matrix A.
! 44: * On exit, the elements on and above the k-th subdiagonal in
! 45: * the first NB columns are overwritten with the corresponding
! 46: * elements of the reduced matrix; the elements below the k-th
! 47: * subdiagonal, with the array TAU, represent the matrix Q as a
! 48: * product of elementary reflectors. The other columns of A are
! 49: * unchanged. See Further Details.
! 50: *
! 51: * LDA (input) INTEGER
! 52: * The leading dimension of the array A. LDA >= max(1,N).
! 53: *
! 54: * TAU (output) COMPLEX*16 array, dimension (NB)
! 55: * The scalar factors of the elementary reflectors. See Further
! 56: * Details.
! 57: *
! 58: * T (output) COMPLEX*16 array, dimension (LDT,NB)
! 59: * The upper triangular matrix T.
! 60: *
! 61: * LDT (input) INTEGER
! 62: * The leading dimension of the array T. LDT >= NB.
! 63: *
! 64: * Y (output) COMPLEX*16 array, dimension (LDY,NB)
! 65: * The n-by-nb matrix Y.
! 66: *
! 67: * LDY (input) INTEGER
! 68: * The leading dimension of the array Y. LDY >= max(1,N).
! 69: *
! 70: * Further Details
! 71: * ===============
! 72: *
! 73: * The matrix Q is represented as a product of nb elementary reflectors
! 74: *
! 75: * Q = H(1) H(2) . . . H(nb).
! 76: *
! 77: * Each H(i) has the form
! 78: *
! 79: * H(i) = I - tau * v * v'
! 80: *
! 81: * where tau is a complex scalar, and v is a complex vector with
! 82: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
! 83: * A(i+k+1:n,i), and tau in TAU(i).
! 84: *
! 85: * The elements of the vectors v together form the (n-k+1)-by-nb matrix
! 86: * V which is needed, with T and Y, to apply the transformation to the
! 87: * unreduced part of the matrix, using an update of the form:
! 88: * A := (I - V*T*V') * (A - Y*V').
! 89: *
! 90: * The contents of A on exit are illustrated by the following example
! 91: * with n = 7, k = 3 and nb = 2:
! 92: *
! 93: * ( a h a a a )
! 94: * ( a h a a a )
! 95: * ( a h a a a )
! 96: * ( h h a a a )
! 97: * ( v1 h a a a )
! 98: * ( v1 v2 a a a )
! 99: * ( v1 v2 a a a )
! 100: *
! 101: * where a denotes an element of the original matrix A, h denotes a
! 102: * modified element of the upper Hessenberg matrix H, and vi denotes an
! 103: * element of the vector defining H(i).
! 104: *
! 105: * =====================================================================
! 106: *
! 107: * .. Parameters ..
! 108: COMPLEX*16 ZERO, ONE
! 109: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
! 110: $ ONE = ( 1.0D+0, 0.0D+0 ) )
! 111: * ..
! 112: * .. Local Scalars ..
! 113: INTEGER I
! 114: COMPLEX*16 EI
! 115: * ..
! 116: * .. External Subroutines ..
! 117: EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL,
! 118: $ ZTRMV
! 119: * ..
! 120: * .. Intrinsic Functions ..
! 121: INTRINSIC MIN
! 122: * ..
! 123: * .. Executable Statements ..
! 124: *
! 125: * Quick return if possible
! 126: *
! 127: IF( N.LE.1 )
! 128: $ RETURN
! 129: *
! 130: DO 10 I = 1, NB
! 131: IF( I.GT.1 ) THEN
! 132: *
! 133: * Update A(1:n,i)
! 134: *
! 135: * Compute i-th column of A - Y * V'
! 136: *
! 137: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
! 138: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY,
! 139: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 )
! 140: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA )
! 141: *
! 142: * Apply I - V * T' * V' to this column (call it b) from the
! 143: * left, using the last column of T as workspace
! 144: *
! 145: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
! 146: * ( V2 ) ( b2 )
! 147: *
! 148: * where V1 is unit lower triangular
! 149: *
! 150: * w := V1' * b1
! 151: *
! 152: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
! 153: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1,
! 154: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
! 155: *
! 156: * w := w + V2'*b2
! 157: *
! 158: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
! 159: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE,
! 160: $ T( 1, NB ), 1 )
! 161: *
! 162: * w := T'*w
! 163: *
! 164: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1,
! 165: $ T, LDT, T( 1, NB ), 1 )
! 166: *
! 167: * b2 := b2 - V2*w
! 168: *
! 169: CALL ZGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ),
! 170: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
! 171: *
! 172: * b1 := b1 - V1*w
! 173: *
! 174: CALL ZTRMV( 'Lower', 'No transpose', 'Unit', I-1,
! 175: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 )
! 176: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
! 177: *
! 178: A( K+I-1, I-1 ) = EI
! 179: END IF
! 180: *
! 181: * Generate the elementary reflector H(i) to annihilate
! 182: * A(k+i+1:n,i)
! 183: *
! 184: EI = A( K+I, I )
! 185: CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1,
! 186: $ TAU( I ) )
! 187: A( K+I, I ) = ONE
! 188: *
! 189: * Compute Y(1:n,i)
! 190: *
! 191: CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA,
! 192: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 )
! 193: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE,
! 194: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ),
! 195: $ 1 )
! 196: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1,
! 197: $ ONE, Y( 1, I ), 1 )
! 198: CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 )
! 199: *
! 200: * Compute T(1:i,i)
! 201: *
! 202: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
! 203: CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT,
! 204: $ T( 1, I ), 1 )
! 205: T( I, I ) = TAU( I )
! 206: *
! 207: 10 CONTINUE
! 208: A( K+NB, NB ) = EI
! 209: *
! 210: RETURN
! 211: *
! 212: * End of ZLAHRD
! 213: *
! 214: END
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