naginterfaces.library.lapacklin.dgbequ

naginterfaces.library.lapacklin.dgbequ(m, kl, ku, ab)

dgbequ computes diagonal scaling matrices $D_R$ and $D_C$ intended to equilibrate a real $m\times n$ band matrix $A$ of band width $(k_l+k_u+1)$, and reduce its condition number.

For full information please refer to the NAG Library document for f07bf

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f07/f07bff.html

Parameters

mint

$m$, the number of rows of the matrix $A$.

klint

$k_l$, the number of subdiagonals of the matrix $A$.

kuint

$k_u$, the number of superdiagonals of the matrix $A$.

abfloat, array-like, shape $(kl+ku+1,n)$

The $m\times n$ band matrix $A$ whose scaling factors are to be computed.

See Further Comments for further details.

Returns

rfloat, ndarray, shape $\left(m\right)$

If no exception or warning is raised or $errno$ > $m$, $r$ contains the row scale factors, the diagonal elements of $D_R$. The elements of $r$ will be positive.

cfloat, ndarray, shape $\left(n\right)$

If no exception or warning is raised, $c$ contains the column scale factors, the diagonal elements of $D_C$. The elements of $c$ will be positive.

rowcndfloat

If no exception or warning is raised or $errno$ > $m$, $rowcnd$ contains the ratio of the smallest value of $r[i-1]$ to the largest value of $r[i-1]$. If $rowcnd\ge 0.1$ and $amax$ is neither too large nor too small, it is not worth scaling by $D_R$.

colcndfloat

If no exception or warning is raised, $colcnd$ contains the ratio of the smallest value of $c[i-1]$ to the largest value of $c[i-1]$.

If $colcnd\ge 0.1$, it is not worth scaling by $D_C$.

amaxfloat

$max\left|a_{ij}\right|$. If $amax$ is very close to overflow or underflow, the matrix $A$ should be scaled.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $m$.

Constraint: $m\ge 0$.

(errno $-2$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-4$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

Warns

NagAlgorithmicWarning

(errno $(i>0)\text{and}(i\le m)$)

Row $⟨value⟩$ of $A$ is exactly zero.

(errno $i>m$)

Column $⟨value⟩$ of $A$ is exactly zero.

Notes

dgbequ computes the diagonal scaling matrices. The diagonal scaling matrices are chosen to try to make the elements of largest absolute value in each row and column of the matrix $B$ given by

$$B=D_{R}A{D}_C$$

have absolute value $1$. The diagonal elements of $D_R$ and $D_C$ are restricted to lie in the safe range $(\delta ,1/\delta )$, where $\delta$ is the value returned by function machine.real_safe. Use of these scaling factors is not guaranteed to reduce the condition number of $A$ but works well in practice.