NAG FL Interface
f04dhf (complex_​symm_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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: $\mathbf{n}$ – Integer Input

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

3: $\mathbf{nrhs}$ – Integer Input

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: $\mathbf{a}(\mathbf{lda},*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array a must be at least $\max (1,\mathbf{n})$.

On entry: the $n\times n$ complex symmetric matrix $A$.

If $\mathbf{uplo}=\text{'U'}$, 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 $\mathbf{uplo}=\text{'L'}$, 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 $\mathbf{ifail}\ge \mathbf{0}$, 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 f07nrf.

5: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f04dhf is called.

Constraint: $\mathbf{lda}\ge \max (1,\mathbf{n})$.

6: $\mathbf{ipiv}\left(\mathbf{n}\right)$ – Integer array Output

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, details of the interchanges and the block structure of $D$, as determined by f07nrf.

• 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.

7: $\mathbf{b}(\mathbf{ldb},*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least $\max (1,\mathbf{nrhs})$.

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: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f04dhf is called.

Constraint: $\mathbf{ldb}\ge \max (1,\mathbf{n})$.

9: $\mathbf{rcond}$ – Real (Kind=nag_wp) Output

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

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: $\mathbf{ifail}$ – Integer Input/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).

6 Error Indicators and Warnings

$\mathbf{ifail}>0\hspace{0.17em}\text{and}\hspace{0.17em}\mathbf{ifail}\le \mathbf{n}$

Diagonal block $⟨\mathit{\text{value}}⟩$ 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': $\mathbf{uplo}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=-2$

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 0$.

$\mathbf{ifail}=-3$

On entry, $\mathbf{nrhs}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nrhs}\ge 0$.

$\mathbf{ifail}=-5$

On entry, $\mathbf{lda}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lda}\ge \max (1,\mathbf{n})$.

$\mathbf{ifail}=-8$

On entry, $\mathbf{ldb}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ldb}\ge \max (1,\mathbf{n})$.

$\mathbf{ifail}=-99$

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 real allocatable memory required is n, and the complex allocatable memory required is $\max (2\times \mathbf{n},\mathbf{lwork})$, where lwork is the optimum workspace required by f07nnf. If this failure occurs it may be possible to solve the equations by calling the packed storage version of f04dhf, f04djf, or by calling f07nnf directly with less than the optimum workspace (see Chapter F07).

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results