f03 – Determinants

This chapter is concerned with the calculation of determinants of square matrices.

The functions in this chapter compute the determinant of a square matrix $A$. The matrix is assumed to have first been decomposed into triangular factors

using functions from Chapter f07.

$$A=LU\text{,}$$ |

If $A$ is positive definite, then $U={L}^{\mathrm{T}}$, and the determinant is the product of the squares of the diagonal elements of $L$. Otherwise, the functions in this chapter use the Dolittle form of the $LU$ decomposition, where $L$ has unit elements on its diagonal. The determinant is then the product of the diagonal elements of $U$, taking account of possible sign changes due to row interchanges.

To avoid overflow or underflow in the computation of the determinant, some scaling is associated with each multiplication in the product of the relevant diagonal elements. The final value is represented by

where $id$ is an integer and

$$\mathrm{det}A=d\times {2}^{id}$$ |

$$\frac{1}{16}\le \left|d\right|<1\text{.}$$ |

For complex valued determinants the real and imaginary parts are scaled separately.

It is extremely wasteful of computer time and storage to use an inappropriate function, for example to use a function requiring a complex matrix when $A$ is real. Most programmers will know whether their matrix is real or complex, but may be less certain whether or not a real symmetric matrix $A$ is positive definite, i.e., all eigenvalues of $A>0$. A real symmetric matrix $A$ not known to be positive definite must be treated as a general real matrix.

Is $A$ a real matrix? | Is $A$ a symmetric positive definite matrix? | Is $A$ a banded matrix? | f07hdc and f03bhc | |||||||

yes | yes | yes | ||||||||

no | no | no | ||||||||

f07fdc and f03bfc | ||||||||||

f07adc and f03bac | ||||||||||

f07arc and f03bnc | ||||||||||

None.

The following lists all those functions that have been withdrawn since Mark 23 of the Library or are scheduled for withdrawal at one of the next two marks.

WithdrawnFunction | Mark ofWithdrawal | Replacement Function(s) |

nag_real_cholesky (f03aec) | 25 | nag_dpotrf (f07fdc) and nag_det_real_sym (f03bfc) |

nag_real_lu (f03afc) | 25 | nag_dgetrf (f07adc) and nag_det_real_gen (f03bac) |

nag_complex_lu (f03ahc) | 25 | nag_zgetrf (f07arc) and nag_det_complex_gen (f03bnc) |

Fox L (1964) *An Introduction to Numerical Linear Algebra* Oxford University Press

Wilkinson J H and Reinsch C (1971) *Handbook for Automatic Computation II, Linear Algebra* Springer–Verlag