# Math Programming

Templates can be created for mathematical programming. You can create basic math templates or add iterative program loops, format descriptors and conditional branches to create more complicated mathematical templates.

Templex can also be used to create templates that generate multiple output files and solve linear equations.
Tip: To use the examples shown in the sections below, copy and paste them into a new file.

## Basic Templates

This is an example of a simple plain text template:
The square root of 2 is approximately equal to 1.4.
Because there are no Templex statements, variables, or expressions in the template, the content of the template is copied to the output stream verbatim. This template generates the following output:
The square root of 2 is approximately equal to 1.4.
Expressions, variables, and Templex statements must be placed between braces, { }, in order for Templex to evaluate them. Otherwise, they are treated as literal text and are not processed. The previous template can be modified to solve for the square root of 2 by including the math function, sqrt.
The square root of 2 is approximately equal to {sqrt(2)}.
The template now calculates the square root of 2 and generates the following output:
The square root of 2 is approximately equal to 1.414213.

## Iterative Program Loops

Iterative program loops can be included in a template for repetitively evaluating math expressions and formatting output text.
{for (x=0; x<=5; x++)}
x={x}, Square root of x={sqrt(x)}
{endloop}
This template produces the following output:
x=0, Square root of x = 0
x=1, Square root of x = 1
x=2, Square root of x = 1.41421
x=3, Square root of x = 1.73205
x=4, Square root of x = 2
x=5, Square root of x = 2.23607

## Format Descriptors

Templex can also format the output text according to the format descriptors you place in the template.
{for (x=0; x<=5; x++)}
x={x}, Square root of x={sqrt(x), %5.3f}
{endloop}
The output from the template is now formatted to show three decimal places:
x=0, Square root of x = 0.000
x=1, Square root of x = 1.000
x=2, Square root of x = 1.414
x=3, Square root of x = 1.732
x=4, Square root of x = 2.000
x=5, Square root of x = 2.236
Any characters on a line that follow a single quote ,‘, are a comment. No action is taken on any function, operator, or expression that follows the quote on that line. Text or expressions before the single quote are processed normally, but the items following the single quote are considered comments and are not processed.
{'This template evaluates the square root of}
{'the numbers 0 through 5}
{for (x=0; x<=5; x++)}
x={x}, Square root of x={sqrt(x), %5.3f}
{endloop}
Note that the output from the template does not include the commented text:
x=0, Square root of x = 0.000
x=1, Square root of x = 1.000
x=2, Square root of x = 1.414
x=3, Square root of x = 1.732
x=4, Square root of x = 2.000
x=5, Square root of x = 2.236

## Conditional Branches

Conditional branches can be included in a template to change the output text based on the value of one or more variables.
{'This template evaluates the square root of the}
{'numbers 0 through 5 and prints the output when}
{'the value of the square root is less than or}
{'equal to 1.5}
{for (x=0; x<=5; x++)}
{if (sqrt(x)<=1.5)}
x={x}, Square root of x={sqrt(x), %5.3f}
{endif}
{endloop}
The output from the template includes only the values less than or equal to 2:
x=0, Square root of x = 0.000
x=1, Square root of x = 1.000
x=2, Square root of x = 1.414

## Multiple Output Files

An array of string variables is used to generate multiple output files.
{ofile = {"st_sqr.out", "st_sqrt.out", "st_sine.out"}}
{open ofile[0]}
{for (x=0; x<=5; x++)}
x={x}; Square of x= {x^2, %5.3f}
{endloop}
{close}
{open ofile[1]}
{for (x=0; x<=5; x++)}
x={x}; Square root of x= {sqrt(x), %5.3f}
{endloop}
{close}
{open ofile[2]}
{for (x=0; x<=5; x++)}
x={x}; Sine of x= {sin(x), %5.3f}
{endloop}
{close}
This template creates three files called st_sqr.out, st_sqrt.out, and st_sine.out with the corresponding results.

## Solving Linear Equations

The following template uses vector and matrix functions and operators to solve a linear system of equations.

To solve the following linear system of equations:
3x + 4y + 9z = 29
5x - 3y +  z = 16
x +  y +  z =  4
{x_coeffs = {  3,  5, 1 }  }
{y_coeffs = {  4, -3, 1 }  }
{z_coeffs = {  9,  1, 1 }  }
{rhs      = { 29, 16, 4 }' }
{lhs = {x_coeffs, y_coeffs, z_coeffs}' }
{format, %5.2f, matrix}
coefficient matrix:
{lhs}
right hand side:
{rhs}
determinant:
{determinant(lhs)}
inverse:
{inverse(lhs) , %7.4f}
inverse times original:
{inverse(lhs) * lhs}
{inverse(lhs) * rhs}

This template produces the following output.

To solve the following linear system of equations:
3x + 4y + 9z = 29
5x - 3y +  z = 16
x +  y +  z =  4
coefficient matrix:
|    3.00    4.00    9.00   |
|    5.00   -3.00    1.00   |
|    1.00    1.00    1.00   |
right hand side:
|   29.00   |
|   16.00   |
|    4.00   |
determinant:
44.00
inverse:
|   -0.0909    0.1136    0.7045   |
|   -0.0909   -0.1364    0.9545   |
|    0.1818    0.0227   -0.6591   |
inverse times original:
|    1.00    0.00    0.00   |
|    0.00    1.00    0.00   |
|    0.00    0.00    1.00   |
|    2.00   |
|   -1.00   |
|    3.00   |

Multiple statements can be placed between a single set of braces, { }. Additionally, multiple statements can be placed on the same line by separating the statements with semi-colons, ;. A semi-colon is not required at the end of the line.

Placing multiple statements on one line or several statements between one set of braces gives you flexibility in the way you write your templates.

For example:
{
pt1 = {10.1, 20.2, 30.3}; pt2 = {11.1, 21.2, 33.3};
diff = pt1 - pt2
dist = sqrt(diff[0]^2+diff[1]^2+diff[2]^2)
}
The distance between pt1 and pt2 is {dist}.
This templates yields the following output:
The distance between pt1 and pt2 is 3.31662.
Templates can be extended to solve advanced mathematical problems. The following template uses different integration techniques to solve a first order differential equation:
{
'Solution to the First Order Differential Equation,
'Initial Value Problem: df/dx= -f + x + 1 where f(0)= 1
'using Euler, Modified Euler and Runge Kutta explicit
'integration techniques.
}
{
n=10
x = array(n+1)
Fexact = array(n+1)
Feu = array(n+1)
Fme = array(n+1)
Frk = array(n+1)
Fexact[0] = 1; Feu[0] = 1; Fme[0] = 1; Frk[0] = 1;
h=1/n
}
-X-    -Fexact-   -Feuler-  -Fmodeuler- -Frungkutta-
{for(i=0; i<10; i++)}
{
x[i+1] = (i+1) * h
'Exact Solution
Fexact[i+1] = x[i+1] + exp(-x[i+1])
'Euler Explicit Integration
g = -Feu[i] + x[i] + 1
Feu[i+1] = Feu[i] + h*g
'Modified Euler Integration
g_me1 = -Fme[i] + x[i] + 1
F_me = Fme[i] + 0.5*h*g_me1
g_me2 = -F_me + (x[i]+0.5*h) + 1
Fme[i+1] = Fme[i] + h*g_me2
'Runge Kutta
k1 = -Frk[i] + x[i] + 1
k2 = -(Frk[i] + 0.5*h*k1) + x[i] + 0.5*h + 1
k3 = -(Frk[i] + 0.5*h*k2) + x[i] + 0.5*h + 1
k4 = -(Frk[i] + h*k3) + x[i] + h + 1
Frk[i+1] = Frk[i] + (h/6)*(k1+2*k2+2*k3+k4)
}
{endloop}
{
table(x, Fexact, Feu, Fme, Frk, "%4.2f  %9.6f  %9.6f  %9.6f    %9.6f", 0, 10)
}
The template yields the following output:
-X-    -Fexact-   -Feuler-  -Fmodeuler- -Frungkutta-
0.00   1.000000   1.000000   1.000000     1.000000
0.10   1.004837   1.000000   1.005000     1.004838
0.20   1.018731   1.010000   1.019025     1.018731
0.30   1.040818   1.029000   1.041218     1.040818
0.40   1.070320   1.056100   1.070802     1.070320
0.50   1.106531   1.090490   1.107076     1.106531
0.60   1.148812   1.131441   1.149404     1.148812
0.70   1.196585   1.178297   1.197210     1.196586
0.80   1.249329   1.230467   1.249975     1.249329
0.90   1.306570   1.287420   1.307228     1.306570
1.00   1.367879   1.348678   1.368541     1.367880