NAG Library Function Document

nag_complex_sym_lin_solve (f04dhc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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:    $\mathbf{a}\left[\mathit{dim}\right]$ – ComplexInput/Output

Note: the dimension, dim, of the array a must be at least $\max \left(1,\mathbf{pda}\times \mathbf{n}\right)$.

The $\left(i,j\right)$th element of the matrix $A$ is stored in

• $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{a}\left[\left(i-1\right)\times \mathbf{pda}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $n$ by $n$ complex symmetric matrix $A$.

If $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the leading n by n upper triangular part of the array a contains the upper triangular part of the matrix $A$, and the strictly lower triangular part of a is not referenced.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the leading n by n lower triangular part of the array a contains the lower triangular part of the matrix $A$, and the strictly upper triangular part of a is not referenced.

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 $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by nag_zsytrf (f07nrc).

6:    $\mathbf{pda}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

7:    $\mathbf{ipiv}\left[\mathbf{n}\right]$ – 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_zsytrf (f07nrc).

• 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.

8:    $\mathbf{b}\left[\mathit{dim}\right]$ – ComplexInput/Output

Note: the dimension, dim, of the array b must be at least

• $\max \left(1,\mathbf{pdb}\times \mathbf{nrhs}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{n}\times \mathbf{pdb}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $B$ is stored in

• $\mathbf{b}\left[\left(j-1\right)\times \mathbf{pdb}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{b}\left[\left(i-1\right)\times \mathbf{pdb}+j-1\right]$ 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 $X$.

9:    $\mathbf{pdb}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdb}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdb}\ge \max \left(1,\mathbf{nrhs}\right)$.

10:  $\mathbf{rcond}$ – double *Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $\mathbf{rcond}=1/\left({‖A‖}_1{‖A^{-1}‖}_1\right)$.

11:  $\mathbf{errbnd}$ – 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 $\hat{x}$, such that ${‖\hat{x}-x‖}_1/{‖x‖}_1\le \mathbf{errbnd}$, 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.

12:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
The double allocatable memory required is n, and the Complex allocatable memory required is $\max \left(2\times \mathbf{n},\mathbf{lwork}\right)$, where lwork is the optimum workspace required by nag_zsysv (f07nnc). If this failure occurs it may be possible to solve the equations by calling the packed storage version of nag_complex_sym_lin_solve (f04dhc), nag_complex_sym_packed_lin_solve (f04djc), or by calling nag_zsysv (f07nnc) directly with less than the optimum workspace (see Chapter f07).
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

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, $\mathbf{pda}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}>0$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}>0$.

NE_INT_2

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}\ge \max \left(1,\mathbf{n}\right)$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$ and $\mathbf{nrhs}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}\ge \max \left(1,\mathbf{nrhs}\right)$.

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f04dhce.c)

10.2

Program Data

Program Data (f04dhce.d)

10.3

Program Results

Program Results (f04dhce.r)