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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$A$. The matrix is assumued to have first been decomposed into triangular factors

using functions from Chapter F07.

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

If A$A$ is positive definite, then U = L^{T}$U={L}^{\mathrm{T}}$, and the determinant is the product of the squares of the diagonal elements of L$L$. Otherwise, the functions in this chapter use the Dolittle form of the LU$LU$ decomposition, where L$L$ has unit elements on its diagonal. The determinant is then the product of the diagonal elements of U$U$, 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 d2$d2$ is an integer and

detA = d1 × 2 ^{d2}
$$\mathrm{det}A=d1\times {2}^{d2}$$ |

(1/16) ≤ |d1| < 1
.
$$\frac{1}{16}\le \left|d1\right|<1\text{.}$$ |

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

Most of the original functions of the chapter were based on those published in the book edited by Wilkinson and Reinsch (1971). We are very grateful to the late Dr J H Wilkinson FRS for his help and interest during the implementation of this chapter of the Library.

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$A$ 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$A$ is positive definite, i.e., all eigenvalues of A > 0$A>0$. A real symmetric matrix A$A$ not known to be positive definite must be treated as a general real matrix.
In all other cases either the band function or the general functions must be used.

The functions in this chapter are general purpose functions. These give the value of the determinant in its scaled form, d1$d1$ and d2$d2$, given the triangular decomposition of the matrix from a suitable function from Chapter F07.

Is A$A$ a real matrix? | _ yes |
Is A$A$ a symmetric positive definite matrix? | _ yes |
Is A$A$ a band matrix? | _ yes |
nag_lapack_dpbtrf (f07hd) and nag_det_real_band_sym (f03bh) |

| | | | no | |
||||

| | | | nag_lapack_dpotrf (f07fd) and nag_det_real_sym (f03bf) | ||||

| | no | |
|||||

| | nag_lapack_dgetrf (f07ad) and nag_det_real_gen (f03ba) | |||||

no | |
||||||

nag_lapack_zgetrf (f07ar) and nag_det_complex_gen (f03bn) |

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013