f01brf factorizes a real sparse matrix. The routine either forms the $LU$ factorization of a permutation of the entire matrix, or, optionally, first permutes the matrix to block lower triangular form and then only factorizes the diagonal blocks.
The routine may be called by the names f01brf or nagf_matop_real_gen_sparse_lu.
3Description
Given a real sparse matrix $A$, f01brf may be used to obtain the $LU$ factorization of a permutation of $A$,
$$PAQ=LU$$
where $P$ and $Q$ are permutation matrices, $L$ is unit lower triangular and $U$ is upper triangular. The routine uses a sparse variant of Gaussian elimination, and the pivotal strategy is designed to compromise between maintaining sparsity and controlling loss of accuracy through round-off.
Optionally the routine first permutes the matrix into block lower triangular form and then only factorizes the diagonal blocks. For some matrices this gives a considerable saving in storage and execution time.
Extensive data checks are made; duplicated nonzeros can be accumulated.
The factorization is intended to be used by f04axf to solve sparse systems of linear equations $Ax=b$ or ${A}^{\mathrm{T}}x=b$. If several matrices of the same sparsity pattern are to be factorized, f01bsf should be used for the second and subsequent matrices.
A more recent algorithm for the same calculation is provided by f11mef.
4References
Duff I S (1977) MA28 – a set of Fortran subroutines for sparse unsymmetric linear equations AERE Report R8730 HMSO
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}>0$.
2: $\mathbf{nz}$ – IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint:
${\mathbf{nz}}>0$.
3: $\mathbf{a}\left({\mathbf{licn}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{a}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$, must contain the nonzero elements of the sparse matrix $A$. They can be in any order since f01brf will reorder them.
On exit: the nonzero elements in the $LU$ factorization. The array must not be changed by you between a call of f01brf and a call of f04axf.
4: $\mathbf{licn}$ – IntegerInput
On entry: the dimension of the arrays a and icn as declared in the (sub)program from which f01brf is called. Since the factorization is returned in a and icn, licn should be large enough to accommodate this and should ordinarily be $2$ to $4$ times as large as nz.
On entry: ${\mathbf{irn}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$, must contain the row index of the nonzero element stored in ${\mathbf{a}}\left(i\right)$.
On exit: irn is overwritten and is not needed for subsequent calls of f01bsforf04axf.
6: $\mathbf{lirn}$ – IntegerInput
On entry: the dimension of the array irn as declared in the (sub)program from which f01brf is called. It need not be as large as licn; normally it will not need to be very much greater than nz.
On entry: ${\mathbf{icn}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$, must contain the column index of the nonzero element stored in ${\mathbf{a}}\left(i\right)$.
On exit: icn contains the column indices of the nonzero elements in the factorization. The array must not be changed by you between a call of f01brf and subsequent calls of f01bsforf04axf.
8: $\mathbf{pivot}$ – Real (Kind=nag_wp)Input
On entry: should have a value in the range $0.0\le {\mathbf{pivot}}\le 0.9999$ and is used to control the choice of pivots. If ${\mathbf{pivot}}<0.0$, the value $0.0$ is assumed, and if ${\mathbf{pivot}}>0.9999$, the value $0.9999$ is assumed. When searching a row for a pivot, any element is excluded which is less than pivot times the largest of those elements in the row available as pivots. Thus decreasing pivot biases the algorithm to maintaining sparsity at the expense of stability.
Suggested value:
${\mathbf{pivot}}=0.1$ has been found to work well on test examples.
11: $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{grow}}=\mathrm{.TRUE.}$, ${\mathbf{w}}\left(1\right)$ contains an estimate (an upper bound) of the increase in size of elements encountered during the factorization (see grow); the rest of the array is used as workspace.
If ${\mathbf{grow}}=\mathrm{.FALSE.}$, the array is not used.
12: $\mathbf{lblock}$ – LogicalInput
On entry: if ${\mathbf{lblock}}=\mathrm{.TRUE.}$, the matrix is preordered into block lower triangular form before the $LU$ factorization is performed; otherwise the entire matrix is factorized.
Suggested value:
${\mathbf{lblock}}=\mathrm{.TRUE.}$ unless the matrix is known to be irreducible, or is singular and an upper bound on the rank is required.
13: $\mathbf{grow}$ – LogicalInput
On entry: if ${\mathbf{grow}}=\mathrm{.TRUE.}$, then on exit ${\mathbf{w}}\left(1\right)$ contains an estimate (an upper bound) of the increase in size of elements encountered during the factorization. If the matrix is well-scaled (see Section 9.2), then a high value for ${\mathbf{w}}\left(1\right)$ indicates that the $LU$ factorization may be inaccurate and you should be wary of the results and perhaps increase the argument pivot for subsequent runs (see Section 7).
On entry: if ${\mathbf{abort}}\left(1\right)=\mathrm{.TRUE.}$, f01brf will exit immediately on detecting a structural singularity (one that depends on the pattern of nonzeros) and return ${\mathbf{ifail}}={\mathbf{1}}$; otherwise it will complete the factorization (see Section 9.3).
If ${\mathbf{abort}}\left(2\right)=\mathrm{.TRUE.}$, f01brf will exit immediately on detecting a numerical singularity (one that depends on the numerical values) and return ${\mathbf{ifail}}={\mathbf{2}}$; otherwise it will complete the factorization (see Section 9.3).
If ${\mathbf{abort}}\left(3\right)=\mathrm{.TRUE.}$, f01brf will exit immediately (with ${\mathbf{ifail}}={\mathbf{5}}$) when the arrays a and icn are filled up by the previously factorized, active and unfactorized parts of the matrix; otherwise it continues so that better guidance on necessary array sizes can be given in ${\mathbf{idisp}}\left(6\right)$ and ${\mathbf{idisp}}\left(7\right)$, and will exit with ifail in the range $4$ to $6$. Note that there is always an immediate error exit if the array irn is too small.
If ${\mathbf{abort}}\left(4\right)=\mathrm{.TRUE.}$, f01brf exits immediately (with ${\mathbf{ifail}}={\mathbf{13}}$) if it finds duplicate elements in the input matrix.
If ${\mathbf{abort}}\left(4\right)=\mathrm{.FALSE.}$, f01brf proceeds using a value equal to the sum of the duplicate elements. In either case details of each duplicate element are output on the current advisory message unit (see x04abf), unless suppressed by the value of ifail on entry.
On exit: contains information about the factorization.
${\mathbf{idisp}}\left(1\right)$ and ${\mathbf{idisp}}\left(2\right)$ indicate the position in arrays a and icn of the first and last elements in the $LU$ factorization of the diagonal blocks. (${\mathbf{idisp}}\left(2\right)$ gives the number of nonzeros in the factorization.) ${\mathbf{idisp}}\left(1\right)$ and ${\mathbf{idisp}}\left(2\right)$ must not be changed by you between a call of f01brf and subsequent calls to f01bsforf04axf.
${\mathbf{idisp}}\left(3\right)$ and ${\mathbf{idisp}}\left(4\right)$ monitor the adequacy of ‘elbow room’ in the arrays irn and a (and icn) respectively, by giving the number of times that the data in these arrays has been compressed during the factorization to release more storage. If either ${\mathbf{idisp}}\left(3\right)$ or ${\mathbf{idisp}}\left(4\right)$ is quite large (say greater than $10$), it will probably pay you to increase the size of the corresponding array(s) for subsequent runs. If either is very low or zero, then you can perhaps save storage by reducing the size of the corresponding array(s).
${\mathbf{idisp}}\left(5\right)$, when ${\mathbf{lblock}}=\mathrm{.FALSE.}$, gives an upper bound on the rank of the matrix; when ${\mathbf{lblock}}=\mathrm{.TRUE.}$, gives an upper bound on the sum of the ranks of the lower triangular blocks.
${\mathbf{idisp}}\left(6\right)$ and ${\mathbf{idisp}}\left(7\right)$ give the minimum size of arrays irn and a (and icn) respectively which would enable a successful run on an identical matrix (but some ‘elbow-room’ should be allowed – see Section 9).
${\mathbf{idisp}}\left(8\right)$ to $\left(10\right)$ are only used if ${\mathbf{lblock}}=\mathrm{.TRUE.}$.
${\mathbf{idisp}}\left(8\right)$ gives the structural rank of the matrix.
${\mathbf{idisp}}\left(9\right)$ gives the number of diagonal blocks.
${\mathbf{idisp}}\left(10\right)$ gives the size of the largest diagonal block.
You must not change idisp between a call of f01brf and subsequent calls to f01bsforf04axf.
16: $\mathbf{ifail}$ – IntegerInput/Output
This routine uses an ifail input value codification that differs from the normal case to distinguish between errors and warnings (see Section 4 in the Introduction to the NAG Library FL Interface).
On entry: ifail must be set to one of the values below to set behaviour on detection of an error; these values have no effect when no error is detected. The behaviour relate to whether or not program execution is halted and whether or not messages are printed when an error or warning is detected.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $1$, $11$, $101$ or $111$ is recommended. If the printing of messages is undesirable, then the value $1$ is recommended. Otherwise, the recommended value is $110$. When the value $\mathbf{1}$, $\mathbf{11}$, $\mathbf{101}$ or $\mathbf{111}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
Matrix is structurally singular – decomposition aborted.
Matrix is structurally singular – decomposition aborted $\mathrm{RANK}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=2$
Matrix is numerically singular – decomposition aborted.
${\mathbf{ifail}}=3$
lirn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$.
lirn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$. To continue set lirn to at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=4$
licn much too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$.
licn much too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$. To continue set licn to at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=5$
licn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$.
licn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$. To continue set licn to at least $\u27e8\mathit{\text{value}}\u27e9$.
licn too small. For successful decomposition set licn to at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=6$
licn and lirn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$.
licn and lirn too small. Decomposition aborted at stage $\u27e8\mathit{\text{value}}\u27e9$ in block $\u27e8\mathit{\text{value}}\u27e9$ with first row $\u27e8\mathit{\text{value}}\u27e9$ and last row $\u27e8\mathit{\text{value}}\u27e9$. To continue set lirn to at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
licn not big enough for permutation – increase by $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=8$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{nz}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nz}}>0$.
${\mathbf{ifail}}=10$
On entry, ${\mathbf{licn}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nz}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{licn}}\ge {\mathbf{nz}}$.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{lirn}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nz}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lirn}}\ge {\mathbf{nz}}$.
${\mathbf{ifail}}=12$
On entry, ${\mathbf{irn}}\left(I\right)$ or ${\mathbf{icn}}\left(I\right)$ is out of range: $I=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{a}}\left(I\right)=\u27e8\mathit{\text{value}}\u27e9$${\mathbf{irn}}\left(I\right)=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{icn}}\left(I\right)=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=13$
On entry, duplicate elements found – see advisory messages.
${\mathbf{ifail}}=-1$
Matrix is structurally singular – decomposition completed.
${\mathbf{ifail}}=-2$
Matrix is numerically singular – decomposition completed.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The factorization obtained is exact for a perturbed matrix whose $(i,j)$th element differs from ${a}_{ij}$ by less than $3\epsilon \rho {m}_{ij}$ where $\epsilon $ is the machine precision, $\rho $ is the growth value returned in ${\mathbf{w}}\left(1\right)$ if ${\mathbf{grow}}=\mathrm{.TRUE.}$, and ${m}_{ij}$ the number of Gaussian elimination operations applied to element $(i,j)$. The value of ${m}_{ij}$ is not greater than $n$ and is usually much less. Small $\rho $ values, therefore, guarantee accurate results, but unfortunately large $\rho $ values may give a very pessimistic indication of accuracy.
8Parallelism and Performance
f01brf is not threaded in any implementation.
9Further Comments
9.1Timing
The time required may be estimated very roughly from the number $\tau $ of nonzeros in the factorized form (output as ${\mathbf{idisp}}\left(2\right)$) and for f01brf and its associates is
where our unit is the time for the inner loop of a full matrix code (e.g., solving a full set of equations takes about $\frac{1}{3}{n}^{3}$ units). Note that the faster f01bsf time makes it well worthwhile to use this for a sequence of problems with the same pattern.
It should be appreciated that $\tau $ varies widely from problem to problem. For network problems it may be little greater than nz, the number of nonzeros in $A$; for discretization of two-dimensional and three-dimensional partial differential equations it may be about $3n{\mathrm{log}}_{2}n$ and $\frac{1}{2}{n}^{5/3}$, respectively.
The time taken by f01brf to find the block lower triangular form (${\mathbf{lblock}}=\mathrm{.TRUE.}$) is typically $5\u201315\%$ of the time taken by the routine when it is not found (${\mathbf{lblock}}=\mathrm{.FALSE.}$). If the matrix is irreducible (${\mathbf{idisp}}\left(9\right)=1$ after a call with ${\mathbf{lblock}}=\mathrm{.TRUE.}$) then this time is wasted. Otherwise, particularly if the largest block is small (${\mathbf{idisp}}\left(10\right)\ll n$), the consequent savings are likely to be greater.
The time taken to estimate growth (${\mathbf{grow}}=\mathrm{.TRUE.}$) is typically under $20\%$ of the overall time.
The overall time may be substantially increased if there is inadequate ‘elbow-room’ in the arrays a, irn and icn. When the sizes of the arrays are minimal (${\mathbf{idisp}}\left(6\right)$ and ${\mathbf{idisp}}\left(7\right)$) it can execute as much as three times slower. Values of ${\mathbf{idisp}}\left(3\right)$ and ${\mathbf{idisp}}\left(4\right)$ greater than about $10$ indicate that it may be worthwhile to increase array sizes.
9.2Scaling
The use of a relative pivot tolerance pivot essentially presupposes that the matrix is well-scaled, i.e., that the matrix elements are broadly comparable in size. Practical problems are often naturally well-scaled but particular care is needed for problems containing mixed types of variables (for example millimetres and neutron fluxes).
9.3Singular and Rectangular Systems
It is envisaged that f01brf will almost always be called for square nonsingular matrices and that singularity indicates an error condition. However, even if the matrix is singular it is possible to complete the factorization. It is even possible for f04axf to solve a set of equations whose matrix is singular provided the set is consistent.
Two forms of singularity are possible. If the matrix would be singular for any values of the nonzeros (e.g., if it has a whole row of zeros), then we say it is structurally singular, and continue only if ${\mathbf{abort}}\left(1\right)=\mathrm{.FALSE.}$. If the matrix is nonsingular by virtue of the particular values of the nonzeros, then we say that it is numerically singular and continue only if ${\mathbf{abort}}\left(2\right)=\mathrm{.FALSE.}$, in which case an upper bound on the rank of the matrix is returned in ${\mathbf{idisp}}\left(5\right)$ when ${\mathbf{lblock}}=\mathrm{.FALSE.}$.
Rectangular matrices may be treated by setting n to the larger of the number of rows and numbers of columns and setting ${\mathbf{abort}}\left(1\right)=\mathrm{.FALSE.}$.
Note: the softfailure option should be used (last digit of ${\mathbf{ifail}}={\mathbf{1}}$) if you wish to factorize singular matrices with ${\mathbf{abort}}\left(1\right)$ or ${\mathbf{abort}}\left(2\right)$ set to .FALSE..
9.4Duplicated Nonzeros
The matrix $A$ may consist of a sum of contributions from different sub-systems (for example finite elements). In such cases you may rely on f01brf to perform assembly, since duplicated elements are summed.
9.5Determinant
To compute the determinant of $A$ after a call of f01brf, the following
code may be used to
det = 1.0
id = idisp(1)
Do 10 i = 1, n
idg = id + ikeep(3*n+i)
det = det*a(idg)
If (ikeep(n+i).ne.i)det = -det
If (ikeep(2*n+i).ne.i)det = -det
id = id + ikeep(i)
10 Continue
the real variable det will then hold the determinant.
This example program simply prints out some information about the factorization as returned by f01brf in ${\mathbf{w}}\left(1\right)$ and idisp. Normally the call of f01brf would be followed by a call of f04axf (see f04axf).