1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
|
.TH backslash 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
backslash (\\) - left matrix division.
.SH CALLING SEQUENCE
.nf
x=A\\b
.fi
.SH DESCRIPTION
Backslash denotes left matrix division.
\fVx=A\\b\fR is a solution to \fVA*x=b\fR.
.LP
If \fVA\fR is nonsingular \fVx=A\\b\fR (uniquely defined)
is equivalent to \fVx=inv(A)*b\fR.
.LP
If \fVA\fR is singular, \fVx\fR is a least square solution.
i.e. \fVnorm(A*x-b)\fR is minimal. If \fVA\fR is full
column rank, the least square solution, \fVx=A\\b\fR, is uniquely
defined (there is a unique \fVx\fR which minimizes \fVnorm(A*x-b)\fR).
If \fVA\fR is not full column rank, then the least square
solution is not unique, and \fVx=A\\b\fR, in general, is not the solution
with minimum norm (the minimum norm solution is \fVx=pinv(A)*b\fR).
.LP
\fVA.\\B\fR is the matrix with \fV(i,j)\fR entry \fVA(i,j)\\B(i,j)\fR.
If \fVA\fR (or \fVB\fR) is a scalar \fVA.\\B\fR is equivalent to
\fVA*ones(B).\\B\fR (or \fVA.\\(B*ones(A))\fR
.SH EXAMPLE
.nf
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
.fi
.SH SEE ALSO
slash, inv, pinv, percent, ieee
|
