naginterfaces.library.linsys.real_symm_packed_solve¶
- naginterfaces.library.linsys.real_symm_packed_solve(uplo, n, ap, b)[source]¶
real_symm_packed_solve
computes the solution to a real system of linear equations , where is an symmetric matrix, stored in packed format and and are matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.For full information please refer to the NAG Library document for f04bj
https://www.nag.com/numeric/nl/nagdoc_29/flhtml/f04/f04bjf.html
- Parameters
- uplostr, length 1
If , the upper triangle of the matrix is stored.
If , the lower triangle of the matrix is stored.
- nint
The number of linear equations , i.e., the order of the matrix .
- apfloat, array-like, shape
The symmetric matrix , packed column-wise in a linear array. The th column of the matrix is stored in the array as follows:
- bfloat, array-like, shape
The matrix of right-hand sides .
- Returns
- apfloat, ndarray, shape
If no exception is raised, the block diagonal matrix and the multipliers used to obtain the factor or from the factorization or as computed by
lapacklin.dsptrf
, stored as a packed triangular matrix in the same storage format as .- ipivint, ndarray, shape
If no exception is raised, details of the interchanges and the block structure of , as determined by
lapacklin.dsptrf
.If , then rows and columns and were interchanged, and is a diagonal block;
if and , then rows and columns and were interchanged and is a diagonal block;
if and , then rows and columns and were interchanged and is a diagonal block.
- bfloat, ndarray, shape
If the function exits successfully or = + 1, the solution matrix .
- rcondfloat
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix , computed as .
- errbndfloat
If the function exits successfully or = + 1, an estimate of the forward error bound for a computed solution , such that , where is a column of the computed solution returned in the array and is the corresponding column of the exact solution . If is less than machine precision, is returned as unity.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, not one of ‘U’ or ‘u’ or ‘L’ or ‘l’: .
- (errno )
Diagonal block of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.
- Warns
- NagAlgorithmicWarning
- (errno )
A solution has been computed, but is less than machine precision so that the matrix is numerically singular.
- Notes
The diagonal pivoting method is used to factor as , if , or , if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, and is symmetric and block diagonal with and diagonal blocks. The factored form of is then used to solve the system of equations .
- References
Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug
Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia