f04bjf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ symmetric matrix, stored in packed format 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 f04bjf or nagf_linsys_real_symm_packed_solve.
3Description
The diagonal pivoting method is used to factor $A$ as $A=UD{U}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'U'}$, or $A=LD{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is symmetric and block diagonal with $1\times 1$ and $2\times 2$ diagonal blocks. 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: $\mathbf{nrhs}$ – IntegerInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}}\times ({\mathbf{n}}+1)/2)$.
On entry: the $n\times n$ symmetric matrix $A$, packed column-wise in a linear array. The $j$th column of the matrix $A$ is stored in the array ap as follows:
More precisely,
if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j(j-1)/2\right)$ for $i\le j$;
if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+(2n-j)(j-1)/2\right)$ for $i\ge j$.
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by f07pdf, stored as a packed triangular matrix in the same storage format as $A$.
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, details of the interchanges and the block structure of $D$, as determined by f07pdf.
If ${\mathbf{ipiv}}\left(k\right)>0$, then rows and columns $k$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged, and ${d}_{kk}$ is a $1\times 1$ diagonal block;
if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k-1\right)<0$, then rows and columns $k-1$ and $-{\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k-1:k,k-1:k}$ is a $2\times 2$ diagonal block;
if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k+1\right)<0$, then rows and columns $k+1$ and $-{\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k:k+1,k:k+1}$ is a $2\times 2$ diagonal block.
6: $\mathbf{b}({\mathbf{ldb}},*)$ – 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 $X$.
7: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04bjf is called.
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
9: $\mathbf{errbnd}$ – 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.
10: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ 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 $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Diagonal block $\u27e8\mathit{\text{value}}\u27e9$ of the block diagonal matrix is zero. The factorization has been completed, but 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, ${\mathbf{nrhs}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-7$
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 $2\times {\mathbf{n}}$. 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. f04bjf 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
f04bjf 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 packed storage scheme is illustrated by the following example when $n=4$ and ${\mathbf{uplo}}=\text{'U'}$. Two-dimensional storage of the symmetric matrix $A$:
The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $(\frac{1}{3}{n}^{3}+2{n}^{2}r)$. 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.
The complex analogues of f04bjf are f04cjf for complex Hermitian matrices, and f04djf for complex symmetric matrices.