dsytrf (l)  Linux Manuals
dsytrf: computes the factorization of a real symmetric matrix A using the BunchKaufman diagonal pivoting method
NAME
DSYTRF  computes the factorization of a real symmetric matrix A using the BunchKaufman diagonal pivoting methodSYNOPSIS
 SUBROUTINE DSYTRF(
 UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
 CHARACTER UPLO
 INTEGER INFO, LDA, LWORK, N
 INTEGER IPIV( * )
 DOUBLE PRECISION A( LDA, * ), WORK( * )
PURPOSE
DSYTRF computes the factorization of a real symmetric matrix A using the BunchKaufman diagonal pivoting method. The form of the factorization isA
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1by1 and 2by2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
ARGUMENTS
 UPLO (input) CHARACTER*1

= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.  N (input) INTEGER
 The order of the matrix A. N >= 0.
 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
 On entry, the symmetric matrix A. If UPLO = aqUaq, the leading NbyN 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 = aqLaq, the leading NbyN 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 IPIV (output) INTEGER array, dimension (N)
 Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1by1 diagonal block. If UPLO = aqUaq and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 diagonal block. If UPLO = aqLaq and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2by2 diagonal block.
 WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
FURTHER DETAILS
If UPLO = aqUaq, then A = U*D*Uaq, whereU
i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1by1 and 2by2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
U(k)
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1:k).
If UPLO = aqLaq, then A = L*D*Laq, where
L
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1by1 and 2by2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then
L(k)
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).