NAG Library Function Document

nag_complex_cholesky (f01bnc)

1: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 1$ .
2: a[ $n \times tda$ ] – ComplexInput/Output: On entry: the lower triangle of the $n$ by $n$ positive definite Hermitian matrix $A$ . The elements of the array above the diagonal need not be set.

On exit: the off-diagonal elements of the upper triangular matrix $U$ . The lower triangle of $A$ is unchanged.
3: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
4: p[n] – doubleOutput: On exit: the reciprocals of the real diagonal elements of $U$ .
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $tda \geq n$ .
NE_DIAG_IMAG_NON_ZERO: Matrix diagonal element $a [(⟨value⟩) \times tda + ⟨value⟩]$ has nonzero imaginary part.
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
NE_NOT_POS_DEF: The matrix is not positive definite, possibly due to rounding errors.

The Cholesky factorization of a positive definite matrix is known for its remarkable numerical stability. The computed matrix

U

satisfies the relation

U^{H} U = A + E

where the 2-norms of

A

and

E

are related by

‖E‖ \leq c ε ‖A‖,

c

is a modest function of

n

, and

ε

is the machine precision.

Not applicable.

The time taken by nag_complex_cholesky (f01bnc) is approximately proportional to

n^{3}

To compute the Cholesky factorization of the well-conditioned positive definite Hermitian matrix

(\begin{matrix} 15 & 1 - 2 i & 2 & - 4 + 3 i \\ 1 + 2 i & 20 & - 2 + i & 3 - 3 i \\ 2 & - 2 - i & 18 & - 1 + 2 i \\ - 4 - 3 i & 3 + 3 i & - 1 - 2 i & 26 \end{matrix}) .