$$A={U}^{\mathrm{T}}U\text{\hspace{1em} or \hspace{1em}}A=L{L}^{\mathrm{T}}$$
to compute the solution to a real system of linear equations
$$AX=B\text{,}$$
where $A$ is an $n\times n$ symmetric positive definite matrix and $X$ and $B$ are $n\times r$ matrices. Error bounds on the solution and a condition estimate are also provided.
Whether or not the system will be equilibrated depends on the scaling of the matrix $A$, but if equilibration is used, $A$ is overwritten by ${D}_{S}A{D}_{S}$ and $B\times {D}_{S}B$.
2.If ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if ${\mathbf{fact}}=\text{'E'}$) 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 matrix and $L$ is a lower triangular matrix.
3.If the leading $i\times i$ principal minor of $A$ is not positive definite, then the routine returns with ${\mathbf{info}}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\mathbf{info}}={\mathbf{n}+{\mathbf{1}}}$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below.
4.The system of equations is solved for $X$ using the factored form of $A$.
5.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
6.If equilibration was used, the matrix $X$ is premultiplied by ${D}_{S}$ so that it solves the original system before equilibration.
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5Arguments
1: $\mathbf{fact}$ – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ is supplied on entry, and if not, whether the matrix $A$ should be equilibrated before it is factorized.
${\mathbf{fact}}=\text{'F'}$
af contains the factorized form of $A$. If ${\mathbf{equed}}=\text{'Y'}$, the matrix $A$ has been equilibrated with scaling factors given by s. a and af will not be modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to af and factorized.
${\mathbf{fact}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to af and factorized.
Constraint:
${\mathbf{fact}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2: $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ symmetric matrix $A$.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'Y'}$, a must have been equilibrated by the scaling factor in s as ${D}_{S}A{D}_{S}$.
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{fact}}=\text{'F'}$ or $\text{'N'}$, or if ${\mathbf{fact}}=\text{'E'}$ and ${\mathbf{equed}}=\text{'N'}$, a is not modified.
If ${\mathbf{fact}}=\text{'E'}$ and ${\mathbf{equed}}=\text{'Y'}$, a is overwritten by ${D}_{S}A{D}_{S}$.
6: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fbf is called.
7: $\mathbf{af}({\mathbf{ldaf}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array af
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, af contains the triangular 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. If ${\mathbf{equed}}\ne \text{'N'}$, af is the factorized form of the equilibrated matrix ${D}_{S}A{D}_{S}$.
On exit: if ${\mathbf{fact}}=\text{'N'}$, af returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$ of the original matrix $A$.
If ${\mathbf{fact}}=\text{'E'}$, af returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$ of the equilibrated matrix $A$ (see the description of a for the form of the equilibrated matrix).
8: $\mathbf{ldaf}$ – IntegerInput
On entry: the first dimension of the array af as declared in the (sub)program from which f07fbf is called.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, equed need not be set.
If ${\mathbf{fact}}=\text{'F'}$, equed must specify the form of the equilibration that was performed as follows:
if ${\mathbf{equed}}=\text{'N'}$, no equilibration;
if ${\mathbf{equed}}=\text{'Y'}$, equilibration was performed, i.e., $A$ has been replaced by ${D}_{S}A{D}_{S}$.
On exit: if ${\mathbf{fact}}=\text{'F'}$, equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of the equilibration that was performed as specified above.
Constraint:
if ${\mathbf{fact}}=\text{'F'}$, ${\mathbf{equed}}=\text{'N'}$ or $\text{'Y'}$.
10: $\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array s
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, s need not be set.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'Y'}$, s must contain the scale factors, ${D}_{S}$, for $A$; each element of s must be positive.
On exit: if ${\mathbf{fact}}=\text{'F'}$, s is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{equed}}=\text{'Y'}$, s contains the scale factors, ${D}_{S}$, for $A$; each element of s is positive.
11: $\mathbf{b}({\mathbf{ldb}},*)$ – 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 $n\times r$ right-hand side matrix $B$.
On exit: if ${\mathbf{equed}}=\text{'N'}$, b is not modified.
If ${\mathbf{equed}}=\text{'Y'}$, b is overwritten by ${D}_{S}B$.
12: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07fbf is called.
13: $\mathbf{x}({\mathbf{ldx}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n\times r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{equed}}=\text{'Y'}$, and the solution to the equilibrated system is ${D}_{S}^{-1}X$.
14: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07fbf is called.
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{rcond}}=1.0/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
16: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_{j}-{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ where ${\hat{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
17: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
18: $\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
The leading minor of order $\u27e8\mathit{\text{value}}\u27e9$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. ${\mathbf{rcond}}=0.0$ is returned.
${\mathbf{info}}={\mathbf{n}}+1$
$U$ (or $L$) is nonsingular, but rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of rcond would suggest.
7Accuracy
For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $(A+E)x=b$, where
if ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
if ${\mathbf{uplo}}=\text{'L'}$,$\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon $ is the machine precision. See Section 10.1 of Higham (2002) for further details.
If $\hat{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
where
$\mathrm{cond}(A,\hat{x},b)={\Vert \left|{A}^{-1}\right|(\left|A\right|\left|\hat{x}\right|+\left|b\right|)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left|{A}^{-1}\right|\left|A\right|\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If $\hat{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${\Vert x-\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in ${\mathbf{ferr}}\left(j\right)$. 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.
f07fbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fbf 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 factorization of $A$ requires approximately $\frac{1}{3}{n}^{3}$ floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of $4{n}^{2}$ floating-point operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix $A$ are also output.