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SUBROUTINE <a name="DSYTRF.1"></a><a href="dsytrf.f.html#DSYTRF.1">DSYTRF</a>( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), WORK( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DSYTRF.19"></a><a href="dsytrf.f.html#DSYTRF.1">DSYTRF</a> computes the factorization of a real symmetric matrix A using
</span><span class="comment">*</span><span class="comment"> the Bunch-Kaufman diagonal pivoting method. The form of the
</span><span class="comment">*</span><span class="comment"> factorization is
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A = U*D*U**T or A = L*D*L**T
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where U (or L) is a product of permutation and unit upper (lower)
</span><span class="comment">*</span><span class="comment"> triangular matrices, and D is symmetric and block diagonal with
</span><span class="comment">*</span><span class="comment"> 1-by-1 and 2-by-2 diagonal blocks.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This is the blocked version of the algorithm, calling Level 3 BLAS.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> UPLO (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'U': Upper triangle of A is stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangle of A is stored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> On entry, the symmetric matrix A. If UPLO = 'U', the leading
</span><span class="comment">*</span><span class="comment"> N-by-N upper triangular part of A contains the upper
</span><span class="comment">*</span><span class="comment"> triangular part of the matrix A, and the strictly lower
</span><span class="comment">*</span><span class="comment"> triangular part of A is not referenced. If UPLO = 'L', the
</span><span class="comment">*</span><span class="comment"> leading N-by-N lower triangular part of A contains the lower
</span><span class="comment">*</span><span class="comment"> triangular part of the matrix A, and the strictly upper
</span><span class="comment">*</span><span class="comment"> triangular part of A is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, the block diagonal matrix D and the multipliers used
</span><span class="comment">*</span><span class="comment"> to obtain the factor U or L (see below for further details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IPIV (output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> Details of the interchanges and the block structure of D.
</span><span class="comment">*</span><span class="comment"> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
</span><span class="comment">*</span><span class="comment"> interchanged and D(k,k) is a 1-by-1 diagonal block.
</span><span class="comment">*</span><span class="comment"> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
</span><span class="comment">*</span><span class="comment"> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
</span><span class="comment">*</span><span class="comment"> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
</span><span class="comment">*</span><span class="comment"> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
</span><span class="comment">*</span><span class="comment"> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The length of WORK. LWORK >=1. For best performance
</span><span class="comment">*</span><span class="comment"> LWORK >= N*NB, where NB is the block size returned by <a name="ILAENV.71"></a><a href="hfy-index.html#ILAENV">ILAENV</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment"> only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment"> this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment"> message related to LWORK is issued by <a name="XERBLA.76"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment"> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
</span><span class="comment">*</span><span class="comment"> has been completed, but the block diagonal matrix D is
</span><span class="comment">*</span><span class="comment"> exactly singular, and division by zero will occur if it
</span><span class="comment">*</span><span class="comment"> is used to solve a system of equations.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If UPLO = 'U', then A = U*D*U', where
</span><span class="comment">*</span><span class="comment"> U = P(n)*U(n)* ... *P(k)U(k)* ...,
</span><span class="comment">*</span><span class="comment"> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
</span><span class="comment">*</span><span class="comment"> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
</span><span class="comment">*</span><span class="comment"> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
</span><span class="comment">*</span><span class="comment"> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
</span><span class="comment">*</span><span class="comment"> that if the diagonal block D(k) is of order s (s = 1 or 2), then
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( I v 0 ) k-s
</span><span class="comment">*</span><span class="comment"> U(k) = ( 0 I 0 ) s
</span><span class="comment">*</span><span class="comment"> ( 0 0 I ) n-k
</span><span class="comment">*</span><span class="comment"> k-s s n-k
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
</span><span class="comment">*</span><span class="comment"> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
</span><span class="comment">*</span><span class="comment"> and A(k,k), and v overwrites A(1:k-2,k-1:k).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If UPLO = 'L', then A = L*D*L', where
</span><span class="comment">*</span><span class="comment"> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
</span><span class="comment">*</span><span class="comment"> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
</span><span class="comment">*</span><span class="comment"> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
</span><span class="comment">*</span><span class="comment"> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
</span><span class="comment">*</span><span class="comment"> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
</span><span class="comment">*</span><span class="comment"> that if the diagonal block D(k) is of order s (s = 1 or 2), then
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( I 0 0 ) k-1
</span><span class="comment">*</span><span class="comment"> L(k) = ( 0 I 0 ) s
</span><span class="comment">*</span><span class="comment"> ( 0 v I ) n-k-s+1
</span><span class="comment">*</span><span class="comment"> k-1 s n-k-s+1
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
</span><span class="comment">*</span><span class="comment"> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
</span><span class="comment">*</span><span class="comment"> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> LOGICAL LQUERY, UPPER
INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> LOGICAL <a name="LSAME.130"></a><a href="lsame.f.html#LSAME.1">LSAME</a>
INTEGER <a name="ILAENV.131"></a><a href="hfy-index.html#ILAENV">ILAENV</a>
EXTERNAL <a name="LSAME.132"></a><a href="lsame.f.html#LSAME.1">LSAME</a>, <a name="ILAENV.132"></a><a href="hfy-index.html#ILAENV">ILAENV</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL <a name="DLASYF.135"></a><a href="dlasyf.f.html#DLASYF.1">DLASYF</a>, <a name="DSYTF2.135"></a><a href="dsytf2.f.html#DSYTF2.1">DSYTF2</a>, <a name="XERBLA.135"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
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