NAG FL Interface
f04bcf (real_​tridiag_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

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

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

2: $\mathbf{nrhs}$ – Integer Input

On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.

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

3: $\mathbf{dl}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: must contain the $(n-1)$ subdiagonal elements of the matrix $A$.

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, dl is overwritten by the $(n-1)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.

4: $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: must contain the $n$ diagonal elements of the matrix $A$.

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, d is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

5: $\mathbf{du}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: must contain the $(n-1)$ superdiagonal elements of the matrix $A$

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, du is overwritten by the $(n-1)$ elements of the first superdiagonal of $U$.

6: $\mathbf{du2}\left(\mathbf{n}-2\right)$ – Real (Kind=nag_wp) array Output

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, du2 returns the $(n-2)$ elements of the second superdiagonal of $U$.

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

On exit: if $\mathbf{ifail}\ge \mathbf{0}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row $\mathbf{ipiv}\left(i\right)$. $\mathbf{ipiv}\left(i\right)$ will always be either $i$ or $(i+1)$; $\mathbf{ipiv}\left(i\right)=i$ indicates a row interchange was not required.

8: $\mathbf{b}(\mathbf{ldb},*)$ – Real (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 $X$.

9: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f04bcf is called.

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

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

11: $\mathbf{errbnd}$ – 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.

12: $\mathbf{ifail}$ – 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

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

Diagonal element $⟨\mathit{\text{value}}⟩$ of the upper triangular factor 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, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 0$.

$\mathbf{ifail}=-2$

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

$\mathbf{ifail}=-9$

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 integer allocatable memory required is n, and the real allocatable memory required is $2\times \mathbf{n}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.

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