NAG FL Interface
f07mbf (dsysvx)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{fact}$ – Character(1) Input

On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.

$\mathbf{fact}=\text{'F'}$

af and ipiv contain the factorized form of the matrix $A$. af and ipiv will not be modified.

$\mathbf{fact}=\text{'N'}$

The matrix $A$ will be copied to af and factorized.

Constraint: $\mathbf{fact}=\text{'F'}$ or $\text{'N'}$.

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: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{nrhs}$ – Integer Input

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) array Input

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

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

• 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.

6: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f07mbf is called.

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

7: $\mathbf{af}(\mathbf{ldaf},*)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: if $\mathbf{fact}=\text{'F'}$, af contains the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $\mathbf{a}=UD{U}^{\mathrm{T}}$ or $\mathbf{a}=LD{L}^{\mathrm{T}}$ as computed by f07mdf.

On exit: if $\mathbf{fact}=\text{'N'}$, af returns the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $\mathbf{a}=UD{U}^{\mathrm{T}}$ or $\mathbf{a}=LD{L}^{\mathrm{T}}$.

8: $\mathbf{ldaf}$ – Integer Input

On entry: the first dimension of the array af as declared in the (sub)program from which f07mbf is called.

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

9: $\mathbf{ipiv}\left(*\right)$ – Integer array Input/Output

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

On entry: if $\mathbf{fact}=\text{'F'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07mdf.

• if $\mathbf{ipiv}\left(i\right)=k>0$, $d_{ii}$ is a $1\times 1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if $\mathbf{uplo}=\text{'U'}$ and $\mathbf{ipiv}\left(i-1\right)=\mathbf{ipiv}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i-1)$th row and column of $A$ were interchanged with the $l$th row and column;
• if $\mathbf{uplo}=\text{'L'}$ and $\mathbf{ipiv}\left(i\right)=\mathbf{ipiv}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i+1)$th row and column of $A$ were interchanged with the $m$th row and column.

On exit: if $\mathbf{fact}=\text{'N'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07mdf, as described above.

10: $\mathbf{b}(\mathbf{ldb},*)$ – Real (Kind=nag_wp) array Input

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

On entry: the $n\times r$ right-hand side matrix $B$.

11: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07mbf is called.

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

12: $\mathbf{x}(\mathbf{ldx},*)$ – Real (Kind=nag_wp) array Output

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

On exit: if $\mathbf{info}=\mathbf{0}$ or $\mathbf{n}+\mathbf{1}$, the $n\times r$ solution matrix $X$.

13: $\mathbf{ldx}$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f07mbf is called.

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

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

On exit: the estimate of the reciprocal condition number of the matrix $A$. If $\mathbf{rcond}=0.0$, the matrix may be exactly singular. This condition is indicated by $\mathbf{info}>\mathbf{0}\hspace{0.17em}\text{and}\hspace{0.17em}\mathbf{info}\le \mathbf{n}$. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by $\mathbf{info}=\mathbf{n}+\mathbf{1}$.

15: $\mathbf{ferr}\left(*\right)$ – Real (Kind=nag_wp) array Output

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

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.

16: $\mathbf{berr}\left(*\right)$ – Real (Kind=nag_wp) array Output

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

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).

17: $\mathbf{work}\left(\max (1,\mathbf{lwork})\right)$ – Real (Kind=nag_wp) array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ returns the optimal lwork.

18: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f07mbf is called.

$\mathbf{lwork}\ge \max (1,3\times \mathbf{n})$, and for best performance, when $\mathbf{fact}=\text{'N'}$, $\mathbf{lwork}\ge \max (1,3\times \mathbf{n},\mathbf{n}\times \mathit{nb})$, where $\mathit{nb}$ is the optimal block size for f07mdf.

If $\mathbf{lwork}=\mathrm{-1}$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

19: $\mathbf{iwork}\left(*\right)$ – Integer array Workspace

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

20: $\mathbf{info}$ – Integer Output

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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

Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be computed. $\mathbf{rcond}=0.0$ is returned.

$\mathbf{info}=\mathbf{n}+1$

$D$ 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.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results