The routine may be called by the names f06tpf or nagf_blas_zutr1.
f06tpf performs a factorization of an upper triangular matrix which has been modified by a rank-1 update:
where and are complex upper triangular matrices with real diagonal elements, and are -element complex vectors, is a complex scalar, and is an complex unitary matrix.
is formed as the product of two sequences of plane rotations and a unitary diagonal matrix :
is a rotation in the plane, chosen to annihilate : thus , where and is the last column of the unit matrix;
is a rotation in the plane, chosen to annihilate the element of , and thus restore it to upper triangular form;
, with chosen to make real; .
The plane rotation part of or has the form
with real. The tangents of the rotations are returned in the array x; the cosines and sines of these rotations can be recovered by calling f06bcf. The cosines and sines of the rotations are returned directly in the arrays c and s.
1: – IntegerInput
On entry: , the order of the matrices and .
2: – Complex (Kind=nag_wp)Input
On entry: the scalar .
3: – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array x
must be at least
On entry: the -element vector .
must be stored in , for .
On entry: the increment in the subscripts of y between successive elements of .
7: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the upper triangular matrix . The imaginary parts of the diagonal elements must be zero.
On exit: the upper triangular matrix . The imaginary parts of the diagonal elements must be zero.
8: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06tpf is called.
9: – Real (Kind=nag_wp) arrayOutput
On exit: the cosines of the rotations
, for .
10: – Complex (Kind=nag_wp) arrayOutput
On exit: the sines of the rotations
, for ; holds , the th diagonal element of .
6Error Indicators and Warnings
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f06tpf 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.