The routine may be called by the names f07jnf, nagf_lapacklin_zptsv or its LAPACK name zptsv.
f07jnf factors as . The factored form of is then used to solve the system of equations.
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – IntegerInput
On entry: , the order of the matrix .
2: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d
must be at least
On entry: the diagonal elements of the tridiagonal matrix .
On exit: the diagonal elements of the diagonal matrix from the factorization .
4: – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e
must be at least
On entry: the subdiagonal elements of the tridiagonal matrix .
On exit: the subdiagonal elements of the unit bidiagonal factor from the factorization of . (e can also be regarded as the superdiagonal of the unit bidiagonal factor from the factorization of .)
5: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
On entry: the right-hand side matrix .
On exit: if , the solution matrix .
6: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07jnf is called.
7: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order is not positive definite,
and the solution has not been computed.
The factorization has not been completed unless .
The computed solution for a single right-hand side, , satisfies an equation of the form
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
f07jpf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cgf solves and returns a forward error bound and condition estimate. f04cgf calls f07jnf to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07jnf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The number of floating-point operations required for the factorization of is proportional to , and the number of floating-point operations required for the solution of the equations is proportional to , where is the number of right-hand sides.