Developer Reference for Intel® oneAPI Math Kernel Library for C
?sytrf_rook
Computes the bounded Bunch-Kaufman factorization of a symmetric matrix.
Syntax
lapack_intLAPACKE_ssytrf_rook ( intmatrix_layout , charuplo , lapack_intn , float*a , lapack_intlda , lapack_int*ipiv );
lapack_intLAPACKE_dsytrf_rook ( intmatrix_layout , charuplo , lapack_intn , double*a , lapack_intlda , lapack_int*ipiv );
lapack_intLAPACKE_csytrf_rook ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_int*ipiv );
lapack_intLAPACKE_zsytrf_rook ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_int*ipiv );
Include Files
mkl.h
Description
ssytrf_rook dsytrf_rook csytrf_rook zsytrf_rook sytrf_rook
The routine computes the factorization of a real/complex symmetric matrix A using the bounded Bunch-Kaufman (“rook”) 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 \(D\) .
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array b is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
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 \(U*D*U^{T}\) .
If uplo = 'L' , the array a stores the lower triangular part of the matrix \(A\) , and \(A\) is factored as \(L*D*L^{T}\) .
n
The order of matrix A ; n ≥ 0.
a
Array, size (lda,n)lda * n . The array a contains either the upper or the lower triangular part of the matrix A (see uplo ).
lda
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
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 ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) 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
If ipiv(k) > 0 , then rows and columns k and ipiv(k) were interchanged and \(D_{k, k}\) is a 1-by-1 diagonal block.
If uplo = 'U' and ipiv(k) < 0 and ipiv(k - 1) < 0 , then rows and columns k and - ipiv(k) were interchanged, rows and columns k - 1 and - ipiv(k-1) were interchanged, and \(D_{k-1:k, k-1:k}\) is a 2-by-2 diagonal block.
If uplo = 'L' and ipiv(k) < 0 and ipiv(k + 1) < 0 , then rows and columns k and -ipiv ( k ) were interchanged, rows and columns k + 1 and -ipiv ( k + 1) were interchanged, and \(D_{k:k+1, k:k+1}\) is a 2-by-2 diagonal block.
Return Values
This function returns a value info .
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
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The total number of floating-point operations is approximately \((1/3)n^{3}\) for real flavors or \((4/3)n^{3}\) for complex flavors.
After calling this routine, you can call the following routines:
?sytrs_rook (Solves a system of linear equations with a UDU- or LDL-factored symmetric coefficient matrix.)
to solve \(A X = B\)
?sycon_rook (Estimates the reciprocal of the condition number of a symmetric matrix.) ?sycon_rook (Fortran only)
to estimate the condition number of \(A\)
?sytri_rook (Computes the inverse of a symmetric matrix using \(U*D*U^T\) or \(L*D*L^T\) bounded Bunch-Kaufman factorization.) ?sytri_rook (Fortran only)
to compute the inverse of \(A\) .
If uplo = 'U' , then \(A = U D U^{T}\) , where
U = P(n)*U(n)* ... *P(k)*U(k)*...,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 -1] .
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
\[U(k) = \begin{pmatrix} I & v & 0 \\ 0 & I & 0 \\ \underbrace{0}_{k-s} & \underbrace{0}_{s} & \underbrace{I}_{n-k} \end{pmatrix} \begin{matrix} & k-s \\ & s \\ & n-k \\ & \end{matrix}\]
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 \(A = L D L^{T}\) , where
L = P(1)*L(1)* ... *P(k)*L(k)*...,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
\[L(k) = \begin{pmatrix} I & 0 & 0 \\ 0 & I & 0 \\ \underbrace{0}_{k-1} & \underbrace{v}_{s} & \underbrace{I}_{n-k-s+1} \end{pmatrix} \begin{matrix} & k-1 \\ & s \\ & n-k+s-1 \\ & \end{matrix}\]
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 \(A(k+2:n, k:k+1)\).