NAG Library Routine Document

f04bjf (real_symm_packed_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:     $\mathbf{n}$ – 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 $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$.

On entry: the $n$ by $n$ 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 $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ 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 (dsptrf), stored as a packed triangular matrix in the same storage format as $A$.

5:     $\mathbf{ipiv}\left(\mathbf{n}\right)$ – Integer arrayOutput

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, details of the interchanges and the block structure of $D$, as determined by f07pdf (dsptrf).

• 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 $1$ by $1$ 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 $2$ by $2$ 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 $2$ by $2$ diagonal block.

6:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

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

On entry: the $n$ by $r$ matrix of right-hand sides $B$.

On exit: if $\mathbf{ifail}=\mathbf{0}$ or $\mathbf{n}+\mathbf{1}$, the $n$ by $r$ 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.

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

8:     $\mathbf{rcond}$ – 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 $\mathbf{rcond}=1/\left({‖A‖}_1{‖A^{-1}‖}_1\right)$.

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 ${‖\hat{x}-x‖}_1/{‖x‖}_1\le \mathbf{errbnd}$, 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.

10:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ 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: $\mathbf{nrhs}\ge 0$.

$\mathbf{ifail}=-7$

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

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation 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 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f04bjfe.f90)

10.2

Program Data

Program Data (f04bjfe.d)

10.3

Program Results

Program Results (f04bjfe.r)