# Matrix Residual Norm Type#

Matrix residual norm type = {char_string}

## Description / Usage#

This optional card selects the type of norm that is used to measure the size of the residuals occurring during the solution of the linear matrix system $$r \left( z \right) = b - Az$$, where $$z$$ is an approximation to the solution $$x$$ of the linear matrix problem $$Ax = b$$. The types of norms used by the linear solver are controlled by values of {char_string}:

{char_string}

Norm type

r0

$$\frac{\lVert r \rVert_2}{\lVert r^0 \rVert_2}$$

rhs

$$\frac{\lVert r \rVert_2}{\lVert b \rVert_2}$$

Anorm

$$\frac{\lVert r \rVert_2}{\lVert A \rVert_{\infty}}$$

sol

$$\frac{\lVert r \rVert_{\infty}}{\lVert A \rVert_{\infty} \lVert x \rVert_1 + \lVert b \rVert_{\infty} }$$

noscaled

$$\lVert r \rVert_2$$

The (0) superscript for the r0 specification indicates the initial value of the residual.

If the Matrix residual norm type card is omitted, the default is r0.

## Examples#

Following is a sample card:

Matrix residual norm type = r0

## Technical Discussion#

For direct factorization linear solution algorithms, the norm should become very small in the single iteration that is performed. This card is more pertinent when an iterative solution algorithm has been specified.

Note the distinction between the residual for the overall global Newton iteration and use of the term residual to describe an aspect of the linear solver iteration. For the linear matrix systems, a residual $$r$$ may be computed for any guess of the solution to $$Ax = b$$ as $$r(z) = b - Az$$. If $$z = x$$, the actual solution, then the residual is zero; otherwise, it is some vector with a nonzero norm.