Advanced Math in JTcl

I ( BAJ ) use a lot of numerical methods in my work and use the Apache Commons Math library (written in Java) in my code. This term I'm teaching a course in Computer Applications in Chemistry at http://www.umbc.edu and we're using MATLAB for part of it. That inspired me to think about adding a scriptable interface in the JTcl Interpreter to the Apache Commons Math Library.

See http://amath4jtcl.kenai.com/gettingstarted.html for info on getting started with the library. You can either just use the library (as jar file) or use the a version that embeds JTcl and Commons Math.

As With Swank, site above is closed. Maybe we could add this and the utility procs either here or to github where Swank and Jtcl currently are?

Load the utility procs and java code

source resource:/tcl/pkg/jtcllib/library/amath/amath.tcl

Vectors and matrices can be created either from Tcl lists:

    #A 2 row by 3 column matrix:

         set m {{1 2 3} {4 5 6}} 

   #A 3 element vector

        set v {1 2 3}

or by using the matrix command.

    # matrix create -rows numberOfRows -col numberOfCols ?-var varName? ?-identity? ?-randn sdevValue? ?-complex?

       set m [matrix create -rows 5 -cols 4 -randn 0.3]
    # or
       matrix create -rows 5 -cols 4 -randn 0.3 -var m

using the 2nd form ensures that, in interactive use, the contents of the matrix will not be converted to a string and displayed in the console, a bad situation for a really large matrix.

The = command

This command works much like Tcl's built-in expression command (and is in fact derived from it). The "expr" and "=" commands essentially provide a domain-specific language inside of Tcl for performing math. Some aspects of "=" were inspired by the Tcl Expr Patch

There are four main differences that the "=" command has compared to "expr".

1. There is no need to use braces around the arguments.
2. Variables do not need to be prefixed with a "$".
3. Variables can be math or vector objects
4. There are additional functions available.

As an example, the equation of a line would be expressed using the "expr" command as:

     set y [expr {$a*$x + $b}]

whereas with the "=" command it would be:

     set y [= a*x + b]

Since the arguments to the "=" command can be vectors or matrices one can often eliminate one or more levels of loops in calculations. So using the above equations to calculate the y coordinates for a series of x coordinates in Tcl would be something like:

   set xValues {1 2 3 4 5 6}
   set yValues [list]
   foreach x $xValues {
        set y [expr {$a*$x+$b}]
        lappend yValues $y
   }

with the "=" command it can be expressed as:

   set x {1 2 3 4 5 6}
   set y [= a*x+b]

More examples

source resource:/tcl/pkg/jtcllib/library/amath/amath.tcl
 
 # create vector from Tcl list
 set a {1 2 3}
 
 # Do math on vector (gets converted internally to an Apache Commons RealVector]
 set b [= a*4]
 set c [= a+b]

 # Create matrix with ::amath::matrix command
 
 set myMat [matrix create -rows 3 -cols 3]

 # Create a matrix and do some linear algebra (singular value decomposition)
 set m {{1 2} {3 4}}
 set m [= tr(m)]
 lassign [svd $m] u s v
 set vt [= tr(v)]
 set usvt [= u*s*vt]
 set diffM [= m-usvt]
 set diffNorm [norm $diffM]
 puts $diffNorm

 # Use math function on matrix
 set mlog [= log(m)]
 
 # Element-wise multiplication ( .* operator)
 
 set mElemSq [= m.*m]
 puts $mElemSq

 # use mset command and colon notation to set all elements of the first column to 5
 mset m(:,0) 5

# create a complex value and do some math

set xc {3+4i}

set yc [= 4*xc]
puts $yc