Developer Reference for Intel® oneAPI Math Kernel Library for C
?sysv_rk
Computes the solution to system of linear equations A * X = B for SY matrices.
Syntax
lapack_intLAPACKE_ssysv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , float*A , lapack_intlda , float*e , lapack_int*ipiv , float*B , lapack_intldb );
lapack_intLAPACKE_dsysv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , double*A , lapack_intlda , double*e , lapack_int*ipiv , double*B , lapack_intldb );
lapack_intLAPACKE_csysv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , lapack_complex_float*A , lapack_intlda , lapack_complex_float*e , lapack_int*ipiv , lapack_complex_float*B , lapack_intldb );
lapack_intLAPACKE_zsysv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , lapack_complex_double*A , lapack_intlda , lapack_complex_double*e , lapack_int*ipiv , lapack_complex_double*B , lapack_intldb );
Description
?sysv_rk computes the solution to a real or complex system of linear equations A * X = B, where A is an n -by- n symmetric matrix and X and B are n -by- nrhs matrices.
The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A= P*U*D*(U T )*(P T ), if uplo = 'U' , or A= P*L*D*(L T )*(P T ), if uplo = 'L' , where U (or L) is unit upper (or lower) triangular matrix, U T (or L T ) is the transpose of U (or L), P is a permutation matrix, P T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
?sytrf_rk is called to compute the factorization of a real or complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ?sytrs_3 .
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
- uplo
-
CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- n
-
INTEGER The number of linear equations; that is, the order of the matrix A. n ≥ 0.
- nrhs
-
INTEGER The number of right-hand sides; that is, the number of columns of the matrix B. nrhs ≥ 0.
- A
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk
Array of size max(1, lda * n ). On entry, the symmetric matrix A. If uplo = 'U' , the leading n -by- n 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 = 'L' , the leading n -by- n 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.
- lda
-
INTEGER
The leading dimension of the array A .
- B
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk
Array of size max(1, ldb * nrhs ). On entry, the n -by- nrhs right-hand side matrix B.
- ldb
-
INTEGER
The leading dimension of the array B . ldb ≥ max(1, n ) for column-major layout and ldb ≥ nrhs for row-major layout.
- lwork
-
INTEGER The length of the array work . If lwork = -1 , a workspace query is assumed; the routine calculates only the optimal size of the work array for factorization stage and returns this value as the first entry of the work array, and no error message related to lwork is issued by XERBLA.
Output Parameters
- A
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk On exit, if info = 0, the diagonal of the block diagonal matrix \(D\) and factors U or L as computed by ?sytrf_rk :
- e
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk
Array of size n . On exit, contains the output computed by the factorization routine ?sytrf_rk ; that is, the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix \(D\) with 1-by-1 or 2-by-2 diagonal blocks. If uplo = 'U' , e( i ) = D( i- 1, i ), i =1:N - 1, and e(1) is set to 0. If uplo = 'L' , e( i ) = D( i +1, i ), i =1:N - 1, and e( n ) is set to 0.
- ipiv
-
INTEGER
Array of size n . Details of the interchanges and the block structure of D, as determined by ?sytrf_rk . For more information, see the description of the ?sytrf_rk routine.
- B
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk On exit, if info = 0, the n -by- nrhs solution matrix X.
- work
-
REAL for ssysv_rk DOUBLE PRECISION for dsysv_rk COMPLEX for csysv_rk COMPLEX*16 for zsysv_rk Array, dimension ( MAX(1, lwork ) ). Work array used in the factorization stage. On exit, if info = 0, work (1) returns the optimal lwork .
Return Values
This function returns a value info .
If info = 0, the execution is successful.
If info = -k , the kth argument had an illegal value.
If info = k , the matrix A is singular. If uplo = 'U' , column k in the upper triangular part of A contains all zeros. If uplo = 'L' , column k in the lower triangular part of A contains all zeros. Therefore \(D_{k,k}\) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. 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.