NAG Library Function Document

1Purpose

nag_herm_packed_lin_solve (f04cjc) computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ complex Hermitian matrix, stored in packed format and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

2Specification

 #include #include
 void nag_herm_packed_lin_solve (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, Complex ap[], Integer ipiv[], Complex b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)

3Description

The diagonal pivoting method is used to factor $A$ as $A=UD{U}^{\mathrm{H}}$, if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, or $A=LD{L}^{\mathrm{H}}$, if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is Hermitian and block diagonal with $1$ by $1$ and $2$ by $2$ 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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
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:    $\mathbf{ap}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ Hermitian 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:
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{H}}$ or $A=LD{L}^{\mathrm{H}}$ as computed by nag_zhptrf (f07prc), stored as a packed triangular matrix in the same storage format as $A$.
6:    $\mathbf{ipiv}\left[{\mathbf{n}}\right]$IntegerOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, details of the interchanges and the block structure of $D$, as determined by nag_zhptrf (f07prc).
• If ${\mathbf{ipiv}}\left[k-1\right]>0$, then rows and columns $k$ and ${\mathbf{ipiv}}\left[k-1\right]$ were interchanged, and ${d}_{kk}$ is a $1$ by $1$ diagonal block;
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ and ${\mathbf{ipiv}}\left[k-1\right]={\mathbf{ipiv}}\left[k-2\right]<0$, then rows and columns $k-1$ and $-{\mathbf{ipiv}}\left[k-1\right]$ were interchanged and ${d}_{k-1:k,k-1:k}$ is a $2$ by $2$ diagonal block;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$ and ${\mathbf{ipiv}}\left[k-1\right]={\mathbf{ipiv}}\left[k\right]<0$, then rows and columns $k+1$ and $-{\mathbf{ipiv}}\left[k-1\right]$ were interchanged and ${d}_{k:k+1,k:k+1}$ is a $2$ by $2$ diagonal block.
7:    $\mathbf{b}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the $n$ by $r$ solution matrix $X$.
8:    $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
9:    $\mathbf{rcond}$double *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
10:  $\mathbf{errbnd}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ 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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
The double allocatable memory required is n, and the Complex allocatable memory required is $2×{\mathbf{n}}$. Allocation failed before the solution could be computed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR
Diagonal block $〈\mathit{\text{value}}〉$ of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.

7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. nag_herm_packed_lin_solve (f04cjc) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8Parallelism and Performance

nag_herm_packed_lin_solve (f04cjc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The packed storage scheme is illustrated by the following example when $n=4$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$. Two-dimensional storage of the Hermitian matrix $A$:
 $a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 aij = a-ji .$
Packed storage of the upper triangle of $A$:
 $ap= a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 .$
The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $\left(\frac{1}{3}{n}^{3}+2{n}^{2}r\right)$. 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.
Function nag_complex_sym_packed_lin_solve (f04djc) is for complex symmetric matrices, and the real analogue of nag_herm_packed_lin_solve (f04cjc) is nag_real_sym_packed_lin_solve (f04bjc).

10Example

This example solves the equations
 $AX=B,$
where $A$ is the Hermitian indefinite matrix
 $A= -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00$
and
 $B= 2.98-10.18i 28.68-39.89i -9.58+03.88i -24.79-08.40i -0.77-16.05i 4.23-70.02i 7.79+05.48i -35.39+18.01i .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

10.1Program Text

Program Text (f04cjce.c)

10.2Program Data

Program Data (f04cjce.d)

10.3Program Results

Program Results (f04cjce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017