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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
	<LANGUAGE>eng</LANGUAGE>
	<TITLE>backslash</TITLE>
	<TYPE>Scilab Function</TYPE>
	<DATE>April 1993</DATE>
	<SHORT_DESCRIPTION name="backslash (\)">left matrix division.</SHORT_DESCRIPTION>
	<CALLING_SEQUENCE>
		<CALLING_SEQUENCE_ITEM>x=A\b</CALLING_SEQUENCE_ITEM>
	</CALLING_SEQUENCE>
	<DESCRIPTION>
		<P>
			Backslash denotes left matrix division. 
			<VERB>x=A\b</VERB> is a solution to <VERB>A*x=b</VERB>.</P>
		<P>
			If <VERB>A</VERB> is square and nonsingular <VERB>x=A\b</VERB> (uniquely defined) is equivalent to <VERB>x=inv(A)*b</VERB> (but the computations are much cheaper).
		</P>
		<P>
			If <VERB>A</VERB> is not square, <VERB>x</VERB> is a least square solution.
			i.e. <VERB>norm(A*x-b)</VERB> is minimal (euclidian norm). If <VERB>A</VERB> is full
			column rank, the least square solution, <VERB>x=A\b</VERB>, is uniquely 
			defined (there is a unique <VERB>x</VERB> which minimizes <VERB>norm(A*x-b)</VERB>).
			If <VERB>A</VERB> is not full column rank, then the least square
			solution is not unique, and <VERB>x=A\b</VERB>, in general, is not the solution
			with minimum norm (the minimum norm solution is <VERB>x=pinv(A)*b</VERB>).
		</P>
		<P>
			<VERB>A.\B</VERB>  is the matrix with <VERB>(i,j)</VERB> entry <VERB>A(i,j)\B(i,j)</VERB>.
			If <VERB>A</VERB> (or <VERB>B</VERB>) is a scalar <VERB>A.\B</VERB> is equivalent to 
			<VERB>A*ones(B).\B</VERB> (or <VERB>A.\(B*ones(A))</VERB>
		</P>
		<P>
			<VERB>A\.B</VERB>  is an operator with no predefined meaning. It may be used
			to define a new operator (see overloading) with  the same precedence as * or /.
		</P>
	</DESCRIPTION>
	
<EXAMPLE>
<![CDATA[
A=rand(3,2);b=[1;1;1]; x=A\b; y=pinv(A)*b;  x-y
A=rand(2,3);b=[1;1]; x=A\b; y=pinv(A)*b; x-y, A*x-b, A*y-b
A=rand(3,1)*rand(1,2); b=[1;1;1]; x=A\b; y=pinv(A)*b; A*x-b, A*y-b
A=rand(2,1)*rand(1,3); b=[1;1]; x=A\b; y=pinv(A)*b; A*x-b, A*y-b 
]]>
</EXAMPLE>
	<SEE_ALSO>
		<SEE_ALSO_ITEM>
			<LINK>slash</LINK>
		</SEE_ALSO_ITEM>
		<SEE_ALSO_ITEM>
			<LINK>inv</LINK>
		</SEE_ALSO_ITEM>
		<SEE_ALSO_ITEM>
			<LINK>pinv</LINK>
		</SEE_ALSO_ITEM>
		<SEE_ALSO_ITEM>
			<LINK>percent</LINK>
		</SEE_ALSO_ITEM>
		<SEE_ALSO_ITEM>
			<LINK>ieee</LINK>
		</SEE_ALSO_ITEM>
	</SEE_ALSO>
</MAN>
