f04bff computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ symmetric positive definite band matrix of band width $2k+1$, and $X$ and $B$ are $n\times r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.
The routine may be called by the names f04bff or nagf_linsys_real_posdef_band_solve.
3Description
The Cholesky factorization is used to factor $A$ as $A={U}^{\mathrm{T}}U$, if ${\mathbf{uplo}}=\text{'U'}$, or $A=L{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular band matrix with $k$ superdiagonals, and $L$ is a lower triangular band matrix with $k$ subdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5Arguments
1: $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$ – IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{kd}$ – IntegerInput
On entry: the number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.
Constraint:
${\mathbf{kd}}\ge 0$.
4: $\mathbf{nrhs}$ – IntegerInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{ab}({\mathbf{ldab}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ symmetric band matrix $A$. The upper or lower triangular part of the symmetric matrix is stored in the first ${\mathbf{kd}}+1$ rows of the array. The $j$th column of $A$ is stored in the $j$th column of the array ab as follows:
The matrix is stored in rows $1$ to $k+1$, more precisely,
if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}(k+1+i-j,j)\text{ for}\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,j-k)\le i\le j$;
if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}(1+i-j,j)\text{ for}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(n,j+k)\text{.}$
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.
6: $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f04bff is called.
Constraint:
${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7: $\mathbf{b}({\mathbf{ldb}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On entry: the $n\times r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n\times r$ solution matrix $X$.
8: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04bff is called.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
10: $\mathbf{errbnd}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${\Vert \hat{x}-x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{errbnd}}$, where $\hat{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.
11: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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).
The principal minor of order $\u27e8\mathit{\text{value}}\u27e9$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
On entry, uplo not one of 'U' or 'u' or 'L' or 'l': ${\mathbf{uplo}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{kd}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{kd}}\ge 0$.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{nrhs}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{ldab}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{kd}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
${\mathbf{ifail}}=-8$
On entry, ${\mathbf{ldb}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
${\mathbf{ifail}}=-99$
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.
The integer allocatable memory required is n, and the real allocatable memory required is $3\times {\mathbf{n}}$. Allocation failed before the solution could be computed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The computed solution for a single right-hand side, $\hat{x}$, satisfies an equation of the form
where $\kappa \left(A\right)={\Vert {A}^{-1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. f04bff uses the approximation ${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999)
for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f04bff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bff makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The band storage scheme for the array ab is illustrated by the following example, when $n=6$, $k=2$, and ${\mathbf{uplo}}=\text{'U'}$:
Array elements marked $*$ need not be set and are not referenced by the routine.
Assuming that $n\gg k$, the total number of floating-point operations required to solve the equations $AX=B$ is approximately ${n(k+1)}^{2}$ for the factorization and $4nkr$ for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.