SUBROUTINE CLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * -- LAPACK auxiliary routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ), \$ Y( LDY, NB ) * .. * * Purpose * ======= * * CLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) * matrix A so that elements below the k-th subdiagonal are zero. The * reduction is performed by a unitary similarity transformation * Q' * A * Q. The routine returns the matrices V and T which determine * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. * * This is an OBSOLETE auxiliary routine. * This routine will be 'deprecated' in a future release. * Please use the new routine CLAHR2 instead. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. * * K (input) INTEGER * The offset for the reduction. Elements below the k-th * subdiagonal in the first NB columns are reduced to zero. * * NB (input) INTEGER * The number of columns to be reduced. * * A (input/output) COMPLEX array, dimension (LDA,N-K+1) * On entry, the n-by-(n-k+1) general matrix A. * On exit, the elements on and above the k-th subdiagonal in * the first NB columns are overwritten with the corresponding * elements of the reduced matrix; the elements below the k-th * subdiagonal, with the array TAU, represent the matrix Q as a * product of elementary reflectors. The other columns of A are * unchanged. See Further Details. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * TAU (output) COMPLEX array, dimension (NB) * The scalar factors of the elementary reflectors. See Further * Details. * * T (output) COMPLEX array, dimension (LDT,NB) * The upper triangular matrix T. * * LDT (input) INTEGER * The leading dimension of the array T. LDT >= NB. * * Y (output) COMPLEX array, dimension (LDY,NB) * The n-by-nb matrix Y. * * LDY (input) INTEGER * The leading dimension of the array Y. LDY >= max(1,N). * * Further Details * =============== * * The matrix Q is represented as a product of nb elementary reflectors * * Q = H(1) H(2) . . . H(nb). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a complex scalar, and v is a complex vector with * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in * A(i+k+1:n,i), and tau in TAU(i). * * The elements of the vectors v together form the (n-k+1)-by-nb matrix * V which is needed, with T and Y, to apply the transformation to the * unreduced part of the matrix, using an update of the form: * A := (I - V*T*V') * (A - Y*V'). * * The contents of A on exit are illustrated by the following example * with n = 7, k = 3 and nb = 2: * * ( a h a a a ) * ( a h a a a ) * ( a h a a a ) * ( h h a a a ) * ( v1 h a a a ) * ( v1 v2 a a a ) * ( v1 v2 a a a ) * * where a denotes an element of the original matrix A, h denotes a * modified element of the upper Hessenberg matrix H, and vi denotes an * element of the vector defining H(i). * * ===================================================================== * * .. Parameters .. COMPLEX ZERO, ONE PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), \$ ONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. INTEGER I COMPLEX EI * .. * .. External Subroutines .. EXTERNAL CAXPY, CCOPY, CGEMV, CLACGV, CLARFG, CSCAL, \$ CTRMV * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Quick return if possible * IF( N.LE.1 ) \$ RETURN * DO 10 I = 1, NB IF( I.GT.1 ) THEN * * Update A(1:n,i) * * Compute i-th column of A - Y * V' * CALL CLACGV( I-1, A( K+I-1, 1 ), LDA ) CALL CGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, \$ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) CALL CLACGV( I-1, A( K+I-1, 1 ), LDA ) * * Apply I - V * T' * V' to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) * ( V2 ) ( b2 ) * * where V1 is unit lower triangular * * w := V1' * b1 * CALL CCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL CTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) * * w := w + V2'*b2 * CALL CGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, \$ A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE, \$ T( 1, NB ), 1 ) * * w := T'*w * CALL CTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1, \$ T, LDT, T( 1, NB ), 1 ) * * b2 := b2 - V2*w * CALL CGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), \$ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) * * b1 := b1 - V1*w * CALL CTRMV( 'Lower', 'No transpose', 'Unit', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL CAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) * A( K+I-1, I-1 ) = EI END IF * * Generate the elementary reflector H(i) to annihilate * A(k+i+1:n,i) * EI = A( K+I, I ) CALL CLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1, \$ TAU( I ) ) A( K+I, I ) = ONE * * Compute Y(1:n,i) * CALL CGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, \$ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL CGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, \$ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ), \$ 1 ) CALL CGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, \$ ONE, Y( 1, I ), 1 ) CALL CSCAL( N, TAU( I ), Y( 1, I ), 1 ) * * Compute T(1:i,i) * CALL CSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL CTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, \$ T( 1, I ), 1 ) T( I, I ) = TAU( I ) * 10 CONTINUE A( K+NB, NB ) = EI * RETURN * * End of CLAHRD * END