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A C++ class library for scientific computing

distance of <CODE>h</CODE>.  A 2nd-order accurate operator has error term 
<EM>O(h^2)</EM>; a 4th-order accurate operator has error term <EM>O(h^4)</EM>.
</P><P>

All of the stencils have factors associated with them.  For example, the
<CODE>central12</CODE> operator is a discrete first derivative which is 2nd-order
accurate.  Its factor is 2h; this means that to get the first derivative of
an array A, you need to use <CODE>central12(A,firstDim)</CODE><EM>/(2h)</EM>.
Typically when designing stencils, one factors out all of the <EM>h</EM> terms
for efficiency.
</P><P>

The factor terms always consist of an integer multiplier (often 1) and a
power of <EM>h</EM>.  For ease of use, all of the operators listed below are
provided in a second "normalized" version in which the integer multiplier
is 1.  The normalized versions have an <CODE>n</CODE> appended to the name, for
example <CODE>central12n</CODE> is the normalized version of <CODE>central12</CODE>, and
has factor <EM>h</EM> instead of <EM>2h</EM>.
</P><P>

These operators are defined in <CODE>blitz/array/stencilops.h</CODE> if you wish
to see the implementation.
</P><P>

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<H3> 4.4.1 Central differences </H3>
<!--docid::SEC109::-->
<P>

<DL COMPACT>
<DT><CODE>central12(A,dimension)</CODE>
<DD>1st derivative, 2nd order accurate.  Factor: <EM>2h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-1</td><td>0</td><td>1</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>central22(A,dimension)</CODE>
<DD>2nd derivative, 2nd order accurate.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-1</td><td>0</td><td>1</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000060"><font color="#ffffff">-2</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>central32(A,dimension)</CODE>
<DD>3rd derivative, 2nd order accurate.  Factor: <EM>2h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">2</font></td><td></td><td bgcolor="#000000"><font color="#ffffff">-2</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>central42(A,dimension)</CODE>
<DD>4th derivative, 2nd order accurate.  Factor: <EM>h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000060"><font color="#ffffff">6</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>central14(A,dimension)</CODE>
<DD>1st derivative, 4th order accurate.  Factor: <EM>12h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-8</font></td><td></td><td bgcolor="#000000"><font color="#ffffff">8</font></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td></tr>
</table>
<P>

<DT><CODE>central24(A,dimension)</CODE>
<DD>2nd derivative, 4th order accurate.  Factor: <EM>12h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td bgcolor="#000060"><font color="#ffffff">-30</font></td><td bgcolor="#000000"><font color="#ffffff">16</font></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td></tr>
</table>
<P>

<DT><CODE>central34(A,dimension)</CODE>
<DD>3rd derivative, 4th order accurate.  Factor: <EM>8h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-8</font></td><td bgcolor="#000000"><font color="#ffffff">13</font></td><td></td><td bgcolor="#000000"><font color="#ffffff">-13</font></td><td bgcolor="#000000"><font color="#ffffff">8</font></td></tr>
</table>
<P>

<DT><CODE>central44(A,dimension)</CODE>
<DD>4th derivative, 4th order accurate.  Factor: <EM>6h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">12</font></td><td bgcolor="#000000"><font color="#ffffff">-39</font></td><td bgcolor="#000060"><font color="#ffffff">56</font></td><td bgcolor="#000000"><font color="#ffffff">-39</font></td><td bgcolor="#000000"><font color="#ffffff">12</font></td></tr>
</table>
</DL>
<P>

Note that the above are available in normalized versions <CODE>central12n</CODE>,
<CODE>central22n</CODE>, ..., <CODE>central44n</CODE> which have factors of <EM>h</EM>,
<EM>h^2</EM>, <EM>h^3</EM>, or <EM>h^4</EM> as appropriate.  
</P><P>

These are available in multicomponent versions: for example,
<CODE>central12(A,component,dimension)</CODE> gives the central12 operator for the
specified component (Components are numbered 0, 1, ... N-1).  
</P><P>

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<H3> 4.4.2 Forward differences </H3>
<!--docid::SEC110::-->
<P>

<DL COMPACT>
<DT><CODE>forward11(A,dimension)</CODE>
<DD>1st derivative, 1st order accurate.  Factor: <EM>h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>forward21(A,dimension)</CODE>
<DD>2nd derivative, 1st order accurate.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-2</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>forward31(A,dimension)</CODE>
<DD>3rd derivative, 1st order accurate.  Factor: <EM>h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">3</font></td><td bgcolor="#000000"><font color="#ffffff">-3</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>forward41(A,dimension)</CODE>
<DD>4th derivative, 1st order accurate.  Factor: <EM>h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000000"><font color="#ffffff">6</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000000"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>forward12(A,dimension)</CODE>
<DD>1st derivative, 2nd order accurate.  Factor: <EM>2h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">-3</font></td><td bgcolor="#000000"><font color="#ffffff">4</font></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td></tr>
</table>
<P>

<DT><CODE>forward22(A,dimension)</CODE>
<DD>2nd derivative, 2nd order accurate.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">2</font></td><td bgcolor="#000000"><font color="#ffffff">-5</font></td><td bgcolor="#000000"><font color="#ffffff">4</font></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td></tr>
</table>
<P>

<DT><CODE>forward32(A,dimension)</CODE>
<DD>3rd derivative, 2nd order accurate.  Factor: <EM>2h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">-5</font></td><td bgcolor="#000000"><font color="#ffffff">18</font></td><td bgcolor="#000000"><font color="#ffffff">-24</font></td><td bgcolor="#000000"><font color="#ffffff">14</font></td><td bgcolor="#000000"><font color="#ffffff">-3</font></td></tr>
</table>
<P>

<DT><CODE>forward42(A,dimension)</CODE>
<DD>4th derivative, 2nd order accurate.  Factor: <EM>h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr align=right><td></td><td bgcolor="#000060"><font color="#ffffff">3</font></td><td bgcolor="#000000"><font color="#ffffff">-14</font></td><td bgcolor="#000000"><font color="#ffffff">26</font></td><td bgcolor="#000000"><font color="#ffffff">-24</font></td><td bgcolor="#000000"><font color="#ffffff">11</font></td><td bgcolor="#000000"><font color="#ffffff">-2</font></td></tr>
</table>
</DL>
<P>

Note that the above are available in normalized versions <CODE>forward11n</CODE>,
<CODE>forward21n</CODE>, ..., <CODE>forward42n</CODE> which have factors of <EM>h</EM>,
<EM>h^2</EM>, <EM>h^3</EM>, or <EM>h^4</EM> as appropriate.  
</P><P>

These are available in multicomponent versions: for example,
<CODE>forward11(A,component,dimension)</CODE> gives the forward11 operator for the
specified component (Components are numbered 0, 1, ... N-1).
</P><P>

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<H3> 4.4.3 Backward differences </H3>
<!--docid::SEC111::-->
<P>

<DL COMPACT>
<DT><CODE>backward11(A,dimension)</CODE>
<DD>1st derivative, 1st order accurate.  Factor: <EM>h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000060"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>backward21(A,dimension)</CODE>
<DD>2nd derivative, 1st order accurate.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-2</font></td><td bgcolor="#000060"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>backward31(A,dimension)</CODE>
<DD>3rd derivative, 1st order accurate.  Factor: <EM>h^3</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">3</font></td><td bgcolor="#000000"><font color="#ffffff">-3</font></td><td bgcolor="#000060"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>backward41(A,dimension)</CODE>
<DD>4th derivative, 1st order accurate.  Factor: <EM>h^4</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-4</td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000000"><font color="#ffffff">6</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000060"><font color="#ffffff">1</font></td></tr>
</table>
<P>

<DT><CODE>backward12(A,dimension)</CODE>
<DD>1st derivative, 2nd order accurate.  Factor: <EM>2h</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">1</font></td><td bgcolor="#000000"><font color="#ffffff">-4</font></td><td bgcolor="#000060"><font color="#ffffff">3</font></td></tr>
</table>
<P>

<DT><CODE>backward22(A,dimension)</CODE>
<DD>2nd derivative, 2nd order accurate.  Factor: <EM>h^2</EM>
<table cellpadding=2 rules=all>
<tr align=right><td></td><td>-3</td><td>-2</td><td>-1</td><td>0</td></tr><tr align=right><td></td><td bgcolor="#000000"><font color="#ffffff">-1</font></td><td bgcolor="#000000"><font color="#ffffff">4</font></td><td bgcolor="#000000"><font color="#ffffff">-5</font></td><td bgcolor="#000060"><font color="#ffffff">2</font></td></tr>
</table>