Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
1: – IntegerInput
On entry: , the number of rows of the matrix .
When then an immediate return is effected.
2: – IntegerInput
On entry: , the number of columns of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the leading upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the upper triangular part of a will contain the upper triangular matrix , and the upper trapezoidal part of a will contain details of the factorization as described in Section 3.
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01qgf is called.
5: – Real (Kind=nag_wp) arrayOutput
On exit: contains the scalar for the th transformation. If then , otherwise contains as described in Section 3 and is always in the range .
6: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, and .
On entry, .
On entry, and .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The computed factors and satisfy the relation
is the machine precision (see x02ajf), is a modest function of and and denotes the spectral (two) norm.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01qgf 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 approximate number of floating-point operations is given by .