?sytrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?sytrf

Computes the Bunch-Kaufman factorization of a symmetric matrix.

Syntax

call ssytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call dsytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call csytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call zsytrf( uplo, n, a, lda, ipiv, work, lwork, info )

call sytrf( a [, uplo] [,ipiv] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine computes the factorization of a real/complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is:

if uplo='U', A = U*D*U^T
if uplo='L', A = L*D*L^T

where A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal (upper triangular for U and lower triangular for L), and D is a symmetric block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of D.

NOTE:

This routine supports the Progress Routine feature. See Progress Routine for details.

Input Parameters

uplo	CHARACTER1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored and how `A` is factored: If `uplo` = 'U', the array `a` stores the upper triangular part of the matrix `A`, and `A` is factored as `UDU^T`. If `uplo` = 'L', the array `a` stores the lower triangular part of the matrix `A`, and `A` is factored as `LD*L^T`.
n	INTEGER. The order of matrix `A`; `n`≥ 0.
a	REAL for ssytrf DOUBLE PRECISION for dsytrf COMPLEX for csytrf DOUBLE COMPLEX for zsytrf. Array, size `(lda,*)` . The array `a` contains either the upper or the lower triangular part of the matrix `A` (see `uplo`). The second dimension of `a` must be at least `max(1, n)`.
lda	INTEGER. The leading dimension of `a`; at least `max(1, n)`.
work	Same type as `a`. A workspace array, dimension at least `max(1,lwork)`.
lwork	INTEGER. The size of the `work` array `(lwork≥n)`. 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. See Application Notes for the suggested value of `lwork`.

Output Parameters

a	The upper or lower triangular part of a is overwritten by details of the block-diagonal matrix `D` and the multipliers used to obtain the factor `U` (or `L`).
`work(1)`	If `info=0`, on exit `work(1)` contains the minimum value of `lwork` required for optimum performance. Use this `lwork` for subsequent runs.
ipiv	INTEGER. Array, size at least `max(1, n)`. Contains details of the interchanges and the block structure of `D`. If `ipiv(i) = k >0`, then `d_ii` is a 1-by-1 block, and the `i`-th row and column of `A` was interchanged with the `k`-th row and column. If `uplo` = 'U' and `ipiv`(`i`) =`ipiv`(`i`-1) = -`m` < 0, then `D` has a 2-by-2 block in rows/columns `i` and `i-1`, and (`i-1`)-th row and column of `A` was interchanged with the `m`-th row and column. If `uplo` = 'L' and `ipiv`(`i`) =`ipiv`(`i`+1) = -`m` < 0, then `D` has a 2-by-2 block in rows/columns `i` and `i+1`, and (`i+1`)-th row and column of `A` was interchanged with the `m`-th row and column.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i-`th parameter had an illegal value. If `info = i`, `D_ii` is 0. The factorization has been completed, but `D` is exactly singular. Division by 0 will occur if you use `D` for solving a system of linear equations.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine sytrf interface are as follows:

a	holds the matrix `A` of size (`n`, `n`)
ipiv	holds the vector of length `n`
uplo	must be 'U' or 'L'. The default value is 'U'.

Application Notes

For better performance, try using lwork = n*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining elements of U and L are stored in the corresponding columns of the array a, but additional row interchanges are required to recover U or L explicitly (which is seldom necessary).

If ipiv(i) = i for all i =1...n , then all off-diagonal elements of U (L) are stored explicitly in the corresponding elements of the array a.

If uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where

|E| ≤
					c(n)ε
					P|U||D||U
					^T|P
					^T

c(n) is a modest linear function of n, and ε is the machine precision. A similar estimate holds for the computed L and D when uplo = 'L'.

The total number of floating-point operations is approximately (1/3)n³ for real flavors or (4/3)n³ for complex flavors.

After calling this routine, you can call the following routines:

?sytrs	to solve `A*X = B`
?sycon	to estimate the condition number of `A`
?sytri	to compute the inverse of `A`.

If uplo = 'U', then A = U*D*U', where

					U = P(n)*U(n)* ... *P(k)*U(k)*...,

that is, U is a product of terms P(k)*U(k), where

k decreases from n to 1 in steps of 1 and 2.
D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).
P(k) is a permutation matrix as defined by ipiv(k) .
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

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k) and A(k,k), and v overwrites A(1:k-2,k -1:k).

If uplo = 'L', then A = L*D*L', where

					L = P(1)*L(1)* ... *P(k)*L(k)*...,

that is, L is a product of terms P(k)*L(k), where

k increases from 1 to n in steps of 1 and 2.
D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).
P(k) is a permutation matrix as defined by ipiv(k).
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

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).

Parent topic: Matrix Factorization: LAPACK Computational Routines

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