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Diffstat (limited to 'src/fortran/lapack/zlatrd.f')
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diff --git a/src/fortran/lapack/zlatrd.f b/src/fortran/lapack/zlatrd.f new file mode 100644 index 0000000..5fef7b5 --- /dev/null +++ b/src/fortran/lapack/zlatrd.f @@ -0,0 +1,279 @@ + SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) +* +* -- LAPACK auxiliary routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER UPLO + INTEGER LDA, LDW, N, NB +* .. +* .. Array Arguments .. + DOUBLE PRECISION E( * ) + COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * ) +* .. +* +* Purpose +* ======= +* +* ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to +* Hermitian tridiagonal form by a unitary similarity +* transformation Q' * A * Q, and returns the matrices V and W which are +* needed to apply the transformation to the unreduced part of A. +* +* If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a +* matrix, of which the upper triangle is supplied; +* if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a +* matrix, of which the lower triangle is supplied. +* +* This is an auxiliary routine called by ZHETRD. +* +* Arguments +* ========= +* +* UPLO (input) CHARACTER*1 +* Specifies whether the upper or lower triangular part of the +* Hermitian matrix A is stored: +* = 'U': Upper triangular +* = 'L': Lower triangular +* +* N (input) INTEGER +* The order of the matrix A. +* +* NB (input) INTEGER +* The number of rows and columns to be reduced. +* +* A (input/output) COMPLEX*16 array, dimension (LDA,N) +* On entry, the Hermitian matrix A. If UPLO = 'U', the leading +* n-by-n upper triangular part of A contains the upper +* triangular part of the matrix A, and the strictly lower +* triangular part of A is not referenced. If UPLO = 'L', the +* leading n-by-n lower triangular part of A contains the lower +* triangular part of the matrix A, and the strictly upper +* triangular part of A is not referenced. +* On exit: +* if UPLO = 'U', the last NB columns have been reduced to +* tridiagonal form, with the diagonal elements overwriting +* the diagonal elements of A; the elements above the diagonal +* with the array TAU, represent the unitary matrix Q as a +* product of elementary reflectors; +* if UPLO = 'L', the first NB columns have been reduced to +* tridiagonal form, with the diagonal elements overwriting +* the diagonal elements of A; the elements below the diagonal +* with the array TAU, represent the unitary matrix Q as a +* product of elementary reflectors. +* See Further Details. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* E (output) DOUBLE PRECISION array, dimension (N-1) +* If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal +* elements of the last NB columns of the reduced matrix; +* if UPLO = 'L', E(1:nb) contains the subdiagonal elements of +* the first NB columns of the reduced matrix. +* +* TAU (output) COMPLEX*16 array, dimension (N-1) +* The scalar factors of the elementary reflectors, stored in +* TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. +* See Further Details. +* +* W (output) COMPLEX*16 array, dimension (LDW,NB) +* The n-by-nb matrix W required to update the unreduced part +* of A. +* +* LDW (input) INTEGER +* The leading dimension of the array W. LDW >= max(1,N). +* +* Further Details +* =============== +* +* If UPLO = 'U', the matrix Q is represented as a product of elementary +* reflectors +* +* Q = H(n) H(n-1) . . . H(n-nb+1). +* +* 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), +* and tau in TAU(i-1). +* +* If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), +* and tau in TAU(i). +* +* The elements of the vectors v together form the n-by-nb matrix V +* which is needed, with W, to apply the transformation to the unreduced +* part of the matrix, using a Hermitian rank-2k update of the form: +* A := A - V*W' - W*V'. +* +* The contents of A on exit are illustrated by the following examples +* with n = 5 and nb = 2: +* +* if UPLO = 'U': if UPLO = 'L': +* +* ( a a a v4 v5 ) ( d ) +* ( a a v4 v5 ) ( 1 d ) +* ( a 1 v5 ) ( v1 1 a ) +* ( d 1 ) ( v1 v2 a a ) +* ( d ) ( v1 v2 a a a ) +* +* where d denotes a diagonal element of the reduced matrix, a denotes +* an element of the original matrix that is unchanged, and vi denotes +* an element of the vector defining H(i). +* +* ===================================================================== +* +* .. Parameters .. + COMPLEX*16 ZERO, ONE, HALF + PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), + $ ONE = ( 1.0D+0, 0.0D+0 ), + $ HALF = ( 0.5D+0, 0.0D+0 ) ) +* .. +* .. Local Scalars .. + INTEGER I, IW + COMPLEX*16 ALPHA +* .. +* .. External Subroutines .. + EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL +* .. +* .. External Functions .. + LOGICAL LSAME + COMPLEX*16 ZDOTC + EXTERNAL LSAME, ZDOTC +* .. +* .. Intrinsic Functions .. + INTRINSIC DBLE, MIN +* .. +* .. Executable Statements .. +* +* Quick return if possible +* + IF( N.LE.0 ) + $ RETURN +* + IF( LSAME( UPLO, 'U' ) ) THEN +* +* Reduce last NB columns of upper triangle +* + DO 10 I = N, N - NB + 1, -1 + IW = I - N + NB + IF( I.LT.N ) THEN +* +* Update A(1:i,i) +* + A( I, I ) = DBLE( A( I, I ) ) + CALL ZLACGV( N-I, W( I, IW+1 ), LDW ) + CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), + $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) + CALL ZLACGV( N-I, W( I, IW+1 ), LDW ) + CALL ZLACGV( N-I, A( I, I+1 ), LDA ) + CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), + $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) + CALL ZLACGV( N-I, A( I, I+1 ), LDA ) + A( I, I ) = DBLE( A( I, I ) ) + END IF + IF( I.GT.1 ) THEN +* +* Generate elementary reflector H(i) to annihilate +* A(1:i-2,i) +* + ALPHA = A( I-1, I ) + CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) ) + E( I-1 ) = ALPHA + A( I-1, I ) = ONE +* +* Compute W(1:i-1,i) +* + CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, + $ ZERO, W( 1, IW ), 1 ) + IF( I.LT.N ) THEN + CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE, + $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO, + $ W( I+1, IW ), 1 ) + CALL ZGEMV( 'No transpose', I-1, N-I, -ONE, + $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, + $ W( 1, IW ), 1 ) + CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE, + $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO, + $ W( I+1, IW ), 1 ) + CALL ZGEMV( 'No transpose', I-1, N-I, -ONE, + $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, + $ W( 1, IW ), 1 ) + END IF + CALL ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) + ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1, + $ A( 1, I ), 1 ) + CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) + END IF +* + 10 CONTINUE + ELSE +* +* Reduce first NB columns of lower triangle +* + DO 20 I = 1, NB +* +* Update A(i:n,i) +* + A( I, I ) = DBLE( A( I, I ) ) + CALL ZLACGV( I-1, W( I, 1 ), LDW ) + CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), + $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) + CALL ZLACGV( I-1, W( I, 1 ), LDW ) + CALL ZLACGV( I-1, A( I, 1 ), LDA ) + CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), + $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) + CALL ZLACGV( I-1, A( I, 1 ), LDA ) + A( I, I ) = DBLE( A( I, I ) ) + IF( I.LT.N ) THEN +* +* Generate elementary reflector H(i) to annihilate +* A(i+2:n,i) +* + ALPHA = A( I+1, I ) + CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, + $ TAU( I ) ) + E( I ) = ALPHA + A( I+1, I ) = ONE +* +* Compute W(i+1:n,i) +* + CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, + $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) + CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE, + $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO, + $ W( 1, I ), 1 ) + CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), + $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) + CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE, + $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, + $ W( 1, I ), 1 ) + CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), + $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) + CALL ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) + ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1, + $ A( I+1, I ), 1 ) + CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) + END IF +* + 20 CONTINUE + END IF +* + RETURN +* +* End of ZLATRD +* + END |