The function may be called by the names: f01mcc, nag_matop_real_vband_posdef_fac or nag_real_cholesky_skyline.
3Description
f01mcc determines the unit lower triangular matrix $L$ and the diagonal matrix $D$ in the Cholesky factorization $A=LD{L}^{\mathrm{T}}$ of a symmetric positive definite variable-bandwidth matrix $A$ of order $n$. (Such a matrix is sometimes called a ‘sky-line’ matrix.)
The matrix $A$ is represented by the elements lying within the envelope of its lower triangular part, that is, between the first nonzero of each row and the diagonal (see Section 10 for an example). The width ${\mathbf{row}}\left[i\right]$ of the $i$th row is the number of elements between the first nonzero element and the element on the diagonal, inclusive. Although, of course, any matrix possesses an envelope as defined, this function is primarily intended for the factorization of symmetric positive definite matrices with an average bandwidth which is small compared with $n$ (also see Section 9).
The method is based on the property that during Cholesky factorization there is no fill-in outside the envelope.
The determination of $L$ and $D$ is normally the first of two steps in the solution of the system of equations $Ax=b$. The remaining step, viz. the solution of ${LDL}^{\mathrm{T}}x=b$ may be carried out using f04mcc.
4References
Jennings A (1966) A compact storage scheme for the solution of symmetric linear simultaneous equations Comput. J.9 281–285
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
On entry: the elements within the envelope of the lower triangle of the positive definite symmetric matrix $A$, taken in row by row order. The following code assigns the matrix elements within the envelope to the correct elements of the array
On entry: ${\mathbf{row}}\left[i\right]$ must contain the width of row $i$ of the matrix $A$, i.e., the number of elements between the first (left-most) nonzero element and the element on the diagonal, inclusive.
Constraint:
$1\le {\mathbf{row}}\left[\mathit{i}\right]\le \mathit{i}+1$, for $\mathit{i}=0,1,\dots ,n-1$.
On exit: the elements within the envelope of the lower triangular matrix $L$, taken in row by row order. The envelope of $L$ is identical to that of the lower triangle of $A$. The unit diagonal elements of $L$ are stored explicitly. See also Section 9
On exit: the diagonal elements of the diagonal matrix $D$. Note that the determinant of $A$ is equal to the product of these diagonal elements. If the value of the determinant is required it should not be determined by forming the product explicitly, because of the possibility of overflow or underflow. The logarithm of the determinant may safely be formed from the sum of the logarithms of the diagonal elements.
7: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_2_INT_ARG_GT
On entry, ${\mathbf{row}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$ while $i=\u27e8\mathit{\text{value}}\u27e9$. These arguments must satisfy ${\mathbf{row}}\left[i\right]\le i+1$.
NE_2_INT_ARG_LT
On entry, ${\mathbf{lal}}=\u27e8\mathit{\text{value}}\u27e9$ while ${\mathbf{row}}\left[0\right]+\cdots +{\mathbf{row}}\left[n-1\right]=\u27e8\mathit{\text{value}}\u27e9$. These arguments must satisfy ${\mathbf{lal}}\ge {\mathbf{row}}\left[0\right]+\cdots +{\mathbf{row}}\left[n-1\right]$.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{row}}\left[\u27e8\mathit{\text{value}}\u27e9\right]$ must not be less than 1: ${\mathbf{row}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
NE_NOT_POS_DEF
The matrix is not positive definite, possibly due to rounding errors.
NE_NOT_POS_DEF_FACT
The matrix is not positive definite, possibly due to rounding errors. The factorization has been completed but may be very inaccurate.
7Accuracy
On successful exit then the computed$L$ and $D$ satisfy the relation ${LDL}^{\mathrm{T}}=A+F$, where
where $k$ is a constant of order unity, $m$ is the largest value of ${\mathbf{row}}\left[i\right]$, and $\epsilon $ is the machine precision. See pages 25–27 and 54–55 or Wilkinson and Reinsch (1971). If the error NE_NOT_POS_DEF_FACT is reported then the factorization has been completed although the matrix was not positive definite. However the factorization may be very inaccurate and should be used only with great caution. For instance, if it is used to solve a set of equations $Ax=b$ using f04mcc, the residual vector $b-Ax$ should be checked.
8Parallelism and Performance
f01mcc is not threaded in any implementation.
9Further Comments
The time taken by f01mcc is approximately proportional to the sum of squares of the values of ${\mathbf{row}}\left[i\right]$.
The distribution of row widths may be very non-uniform without undue loss of efficiency. Moreover, the function has been designed to be as competitive as possible in speed with functions designed for full or uniformly banded matrices, when applied to such matrices.
The function may be called with the same actual array supplied for arguments a and al, in which case $L$ overwrites the lower triangle of $A$.
10Example
To obtain the Cholesky factorization of the symmetric matrix, whose lower triangle is
$$\text{.}$$
For this matrix, the elements of row must be set to $1$, $2$, $2$, $1$, $5$, $3$, and the elements within the envelope must be supplied in row order as