The routine may be called by the names f08nhf, nagf_lapackeig_dgebal or its LAPACK name dgebal.
f08nhf balances a real general matrix . The term ‘balancing’ covers two steps, each of which involves a similarity transformation of . The routine can perform either or both of these steps.
1.The routine first attempts to permute to block upper triangular form by a similarity transformation:
where is a permutation matrix, and and are upper triangular. Then the diagonal elements of and are eigenvalues of . The rest of the eigenvalues of are the eigenvalues of the central diagonal block , in rows and columns to . Subsequent operations to compute the eigenvalues of (or its Schur factorization) need only be applied to these rows and columns; this can save a significant amount of work if and . If no suitable permutation exists (as is often the case), the routine sets and , and is the whole of .
2.The routine applies a diagonal similarity transformation to , to make the rows and columns of as close in norm as possible:
This scaling can reduce the norm of the matrix (i.e., ) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)Input
On entry: indicates whether is to be permuted and/or scaled (or neither).
is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
is permuted but not scaled.
is scaled but not permuted.
is both permuted and scaled.
, , or .
2: – IntegerInput
On entry: , the order 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 matrix .
On exit: a is overwritten by the balanced matrix. If , a is not referenced.
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08nhf is called.
5: – IntegerOutput
6: – IntegerOutput
On exit: the values and such that on exit is zero if and or .
If or , and .
7: – Real (Kind=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to . More precisely, if is the index of the row and column interchanged with row and column and is the scaling factor used to balance row and column then
The order in which the interchanges are made is to then to .
8: – 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 errors are negligible.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08nhf 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.
If the matrix is balanced by f08nhf, then any eigenvectors computed subsequently are eigenvectors of the matrix (see Section 3) and hence f08njfmust then be called to transform them back to eigenvectors of .
If the Schur vectors of are required, then this routine must not be called with or , because then the balancing transformation is not orthogonal. If this routine is called with , then any Schur vectors computed subsequently are Schur vectors of the matrix , and f08njfmust be called (with
to transform them back to Schur vectors of .
The total number of floating-point operations is approximately proportional to .
This example computes all the eigenvalues and right eigenvectors of the matrix , where
The program first calls f08nhf to balance the matrix; it then computes the Schur factorization of the balanced matrix, by reduction to Hessenberg form and the algorithm. Then it calls f08qkf to compute the right eigenvectors of the balanced matrix, and finally calls f08njf to transform the eigenvectors back to eigenvectors of the original matrix .