523 lines
16 KiB
FortranFixed
523 lines
16 KiB
FortranFixed
SUBROUTINE ZSYTF2( UPLO, N, A, LDA, IPIV, INFO )
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*
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* -- LAPACK routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER INFO, LDA, N
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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COMPLEX*16 A( LDA, * )
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* ..
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*
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* Purpose
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* =======
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*
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* ZSYTF2 computes the factorization of a complex symmetric matrix A
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* using the Bunch-Kaufman diagonal pivoting method:
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*
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* A = U*D*U' or A = L*D*L'
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*
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* where U (or L) is a product of permutation and unit upper (lower)
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* triangular matrices, U' is the transpose of U, and D is symmetric and
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* block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
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*
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* This is the unblocked version of the algorithm, calling Level 2 BLAS.
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*
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* Arguments
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* =========
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*
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* UPLO (input) CHARACTER*1
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* Specifies whether the upper or lower triangular part of the
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* symmetric matrix A is stored:
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* = 'U': Upper triangular
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* = 'L': Lower triangular
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*
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* N (input) INTEGER
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* The order of the matrix A. N >= 0.
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*
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* A (input/output) COMPLEX*16 array, dimension (LDA,N)
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* On entry, the symmetric matrix A. If UPLO = 'U', the leading
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* n-by-n upper triangular part of A contains the upper
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* triangular part of the matrix A, and the strictly lower
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* triangular part of A is not referenced. If UPLO = 'L', the
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* leading n-by-n lower triangular part of A contains the lower
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* triangular part of the matrix A, and the strictly upper
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* triangular part of A is not referenced.
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*
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* On exit, the block diagonal matrix D and the multipliers used
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* to obtain the factor U or L (see below for further details).
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* IPIV (output) INTEGER array, dimension (N)
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* Details of the interchanges and the block structure of D.
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* If IPIV(k) > 0, then rows and columns k and IPIV(k) were
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* interchanged and D(k,k) is a 1-by-1 diagonal block.
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* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
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* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
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* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
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* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
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* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -k, the k-th argument had an illegal value
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* > 0: if INFO = k, D(k,k) is exactly zero. The factorization
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* has been completed, but the block diagonal matrix D is
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* exactly singular, and division by zero will occur if it
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* is used to solve a system of equations.
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*
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* Further Details
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* ===============
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*
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* 09-29-06 - patch from
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* Bobby Cheng, MathWorks
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*
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* Replace l.209 and l.377
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* IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
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* by
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* IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
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*
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* 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
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* Company
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*
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* If UPLO = 'U', then A = U*D*U', where
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* U = P(n)*U(n)* ... *P(k)U(k)* ...,
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* i.e., U is a product of terms P(k)*U(k), where k decreases from n to
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* 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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* defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
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* that if the diagonal block D(k) is of order s (s = 1 or 2), then
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*
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* ( I v 0 ) k-s
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* U(k) = ( 0 I 0 ) s
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* ( 0 0 I ) n-k
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* k-s s n-k
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*
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* If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
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* If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
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* and A(k,k), and v overwrites A(1:k-2,k-1:k).
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*
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* If UPLO = 'L', then A = L*D*L', where
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* L = P(1)*L(1)* ... *P(k)*L(k)* ...,
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* i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
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* n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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* defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
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* that if the diagonal block D(k) is of order s (s = 1 or 2), then
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*
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* ( I 0 0 ) k-1
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* L(k) = ( 0 I 0 ) s
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* ( 0 v I ) n-k-s+1
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* k-1 s n-k-s+1
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*
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* If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
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* If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
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* and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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DOUBLE PRECISION EIGHT, SEVTEN
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PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
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COMPLEX*16 CONE
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PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL UPPER
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INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
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DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
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COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
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* ..
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* .. External Functions ..
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LOGICAL DISNAN, LSAME
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INTEGER IZAMAX
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EXTERNAL DISNAN, LSAME, IZAMAX
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZSCAL, ZSWAP, ZSYR
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
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* ..
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* .. Statement Functions ..
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DOUBLE PRECISION CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -4
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZSYTF2', -INFO )
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RETURN
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END IF
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*
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* Initialize ALPHA for use in choosing pivot block size.
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*
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ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
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*
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IF( UPPER ) THEN
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*
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* Factorize A as U*D*U' using the upper triangle of A
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*
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* K is the main loop index, decreasing from N to 1 in steps of
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* 1 or 2
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*
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K = N
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10 CONTINUE
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*
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* If K < 1, exit from loop
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*
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IF( K.LT.1 )
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$ GO TO 70
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KSTEP = 1
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*
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* Determine rows and columns to be interchanged and whether
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* a 1-by-1 or 2-by-2 pivot block will be used
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*
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ABSAKK = CABS1( A( K, K ) )
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*
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* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value
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*
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IF( K.GT.1 ) THEN
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IMAX = IZAMAX( K-1, A( 1, K ), 1 )
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COLMAX = CABS1( A( IMAX, K ) )
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ELSE
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COLMAX = ZERO
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END IF
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*
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IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
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*
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* Column K is zero or contains a NaN: set INFO and continue
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*
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IF( INFO.EQ.0 )
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$ INFO = K
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KP = K
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ELSE
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IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE
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*
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* JMAX is the column-index of the largest off-diagonal
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* element in row IMAX, and ROWMAX is its absolute value
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*
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JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
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ROWMAX = CABS1( A( IMAX, JMAX ) )
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IF( IMAX.GT.1 ) THEN
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JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 )
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ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
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END IF
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*
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IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
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*
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* interchange rows and columns K and IMAX, use 1-by-1
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* pivot block
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*
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KP = IMAX
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ELSE
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*
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* interchange rows and columns K-1 and IMAX, use 2-by-2
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* pivot block
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*
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KP = IMAX
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KSTEP = 2
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END IF
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END IF
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*
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KK = K - KSTEP + 1
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IF( KP.NE.KK ) THEN
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*
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* Interchange rows and columns KK and KP in the leading
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* submatrix A(1:k,1:k)
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*
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CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
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CALL ZSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
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$ LDA )
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T = A( KK, KK )
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A( KK, KK ) = A( KP, KP )
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A( KP, KP ) = T
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IF( KSTEP.EQ.2 ) THEN
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T = A( K-1, K )
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A( K-1, K ) = A( KP, K )
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A( KP, K ) = T
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END IF
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END IF
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*
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* Update the leading submatrix
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*
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IF( KSTEP.EQ.1 ) THEN
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*
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* 1-by-1 pivot block D(k): column k now holds
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*
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* W(k) = U(k)*D(k)
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*
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* where U(k) is the k-th column of U
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*
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* Perform a rank-1 update of A(1:k-1,1:k-1) as
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*
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* A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
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*
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R1 = CONE / A( K, K )
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CALL ZSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
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*
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* Store U(k) in column k
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*
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CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
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ELSE
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*
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* 2-by-2 pivot block D(k): columns k and k-1 now hold
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*
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* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
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*
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* where U(k) and U(k-1) are the k-th and (k-1)-th columns
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* of U
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*
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* Perform a rank-2 update of A(1:k-2,1:k-2) as
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*
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* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
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* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
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*
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IF( K.GT.2 ) THEN
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*
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D12 = A( K-1, K )
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D22 = A( K-1, K-1 ) / D12
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D11 = A( K, K ) / D12
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T = CONE / ( D11*D22-CONE )
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D12 = T / D12
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*
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DO 30 J = K - 2, 1, -1
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WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
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WK = D12*( D22*A( J, K )-A( J, K-1 ) )
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DO 20 I = J, 1, -1
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A( I, J ) = A( I, J ) - A( I, K )*WK -
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$ A( I, K-1 )*WKM1
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20 CONTINUE
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A( J, K ) = WK
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A( J, K-1 ) = WKM1
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30 CONTINUE
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*
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END IF
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*
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END IF
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END IF
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*
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* Store details of the interchanges in IPIV
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*
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IF( KSTEP.EQ.1 ) THEN
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IPIV( K ) = KP
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ELSE
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IPIV( K ) = -KP
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IPIV( K-1 ) = -KP
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END IF
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*
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* Decrease K and return to the start of the main loop
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*
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K = K - KSTEP
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GO TO 10
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*
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ELSE
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*
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* Factorize A as L*D*L' using the lower triangle of A
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*
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* K is the main loop index, increasing from 1 to N in steps of
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* 1 or 2
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*
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K = 1
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40 CONTINUE
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*
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* If K > N, exit from loop
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*
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IF( K.GT.N )
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$ GO TO 70
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KSTEP = 1
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*
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* Determine rows and columns to be interchanged and whether
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* a 1-by-1 or 2-by-2 pivot block will be used
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*
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ABSAKK = CABS1( A( K, K ) )
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*
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* IMAX is the row-index of the largest off-diagonal element in
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* column K, and COLMAX is its absolute value
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*
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IF( K.LT.N ) THEN
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IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 )
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COLMAX = CABS1( A( IMAX, K ) )
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ELSE
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COLMAX = ZERO
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END IF
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*
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IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
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*
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* Column K is zero or contains a NaN: set INFO and continue
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*
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IF( INFO.EQ.0 )
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$ INFO = K
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KP = K
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ELSE
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IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE
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*
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* JMAX is the column-index of the largest off-diagonal
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* element in row IMAX, and ROWMAX is its absolute value
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*
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JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA )
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ROWMAX = CABS1( A( IMAX, JMAX ) )
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IF( IMAX.LT.N ) THEN
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JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
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ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
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END IF
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*
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IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
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*
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* no interchange, use 1-by-1 pivot block
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*
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KP = K
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ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
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*
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* interchange rows and columns K and IMAX, use 1-by-1
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* pivot block
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*
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KP = IMAX
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ELSE
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*
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* interchange rows and columns K+1 and IMAX, use 2-by-2
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* pivot block
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*
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KP = IMAX
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KSTEP = 2
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END IF
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END IF
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*
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KK = K + KSTEP - 1
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IF( KP.NE.KK ) THEN
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*
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* Interchange rows and columns KK and KP in the trailing
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* submatrix A(k:n,k:n)
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*
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IF( KP.LT.N )
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$ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
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CALL ZSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
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$ LDA )
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T = A( KK, KK )
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A( KK, KK ) = A( KP, KP )
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A( KP, KP ) = T
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IF( KSTEP.EQ.2 ) THEN
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T = A( K+1, K )
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A( K+1, K ) = A( KP, K )
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A( KP, K ) = T
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END IF
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END IF
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*
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* Update the trailing submatrix
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*
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IF( KSTEP.EQ.1 ) THEN
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*
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* 1-by-1 pivot block D(k): column k now holds
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*
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* W(k) = L(k)*D(k)
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*
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* where L(k) is the k-th column of L
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*
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IF( K.LT.N ) THEN
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*
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* Perform a rank-1 update of A(k+1:n,k+1:n) as
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*
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* A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
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*
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R1 = CONE / A( K, K )
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CALL ZSYR( UPLO, N-K, -R1, A( K+1, K ), 1,
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$ A( K+1, K+1 ), LDA )
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*
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* Store L(k) in column K
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*
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CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
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END IF
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ELSE
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*
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* 2-by-2 pivot block D(k)
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*
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IF( K.LT.N-1 ) THEN
|
|
*
|
|
* Perform a rank-2 update of A(k+2:n,k+2:n) as
|
|
*
|
|
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
|
|
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
|
|
*
|
|
* where L(k) and L(k+1) are the k-th and (k+1)-th
|
|
* columns of L
|
|
*
|
|
D21 = A( K+1, K )
|
|
D11 = A( K+1, K+1 ) / D21
|
|
D22 = A( K, K ) / D21
|
|
T = CONE / ( D11*D22-CONE )
|
|
D21 = T / D21
|
|
*
|
|
DO 60 J = K + 2, N
|
|
WK = D21*( D11*A( J, K )-A( J, K+1 ) )
|
|
WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
|
|
DO 50 I = J, N
|
|
A( I, J ) = A( I, J ) - A( I, K )*WK -
|
|
$ A( I, K+1 )*WKP1
|
|
50 CONTINUE
|
|
A( J, K ) = WK
|
|
A( J, K+1 ) = WKP1
|
|
60 CONTINUE
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
* Store details of the interchanges in IPIV
|
|
*
|
|
IF( KSTEP.EQ.1 ) THEN
|
|
IPIV( K ) = KP
|
|
ELSE
|
|
IPIV( K ) = -KP
|
|
IPIV( K+1 ) = -KP
|
|
END IF
|
|
*
|
|
* Increase K and return to the start of the main loop
|
|
*
|
|
K = K + KSTEP
|
|
GO TO 40
|
|
*
|
|
END IF
|
|
*
|
|
70 CONTINUE
|
|
RETURN
|
|
*
|
|
* End of ZSYTF2
|
|
*
|
|
END
|