Singular Systems

Jep Extensions

Console application

This console application illustrates features of the Jep Extensions package.

Field:

This example has been cross compiled to javascript using the GWTJep package.

Field selection

This console can perform calculations using various different fields including integers, fractions, decimals with a specific number of decimal places. Use

  • setfield double for standard floating point numbers
  • setfield bigdec 30 for big decimal numbers with 30 decimal places
  • setfield bigdec unlimited for big decimal numbers with unlimited size
  • setfield integer all calculations in integer
  • setfield exactint all calculations in integers throwing exceptions on overflow
  • setfield bigint all calculations using unlimited precision BigIntegers
  • setfield rational for fractions (rational numbers unlimited precision)
  • setfield complex all calculations using complex numbers
  • setfield simple closest match to standard jep operation
  • setfield mixed 30 combines rational numbers and bigdecimal number with 30 dp
Notes:
  • Changing the field resets all variables.
  • Different fields have different sets of functions and constants defiend
  • The JavaScript versions may not always give the same results as the Java version as JavaScript has different ways of representing numbers.

Structured programming

The console allows simple structured programming constructs like loops and if statments. It supports 

    for(i=1;i<10;++i) { ... }
    while(i<10) { ... }  while loops
    break;   (inside a loop)
    continue;   (inside a loop)
    if(i<10) { ... } else { ... }
    statement; statement
    { statement; statement }
    print(a,b,c)
    println(a,b,c)

Examples

A simple loop can add the numbers from 1 to 10 

sum=0; for(i=1; i<=10; ++i) { sum += i; }

Symbolic operations

  • diff(x^2,x) differentiate x^2 with respect to x
  • rules prints the set of differentiation rules used
  • clean(0+1*x) cleans a expression
  • simplify(2x+3x) simplifies a expression using the polynomial simplification algorithm
  • expand((x+1)*(x-1)) expands a polynomial'
  • compare((x+1)^2, x^2+2x+1) compares without expansion
  • coeffs(x^2+2x+1,x) extract coefficients as an array
  • subst(x*y*z,y=x+1,z=x-1) substitution rhs equations into the lhs expression
Symbolic assignment
  • f := x^2 set f to have and equation from the rhs
  • eqn(f) extract equation from a symbolic variable
  • verbose on/verbose off switch verbose mode on or off

Matrix operations

  • m = [[1,2],[3,4]] - creation of matrix
  • u = [5,6] - creation of vector
  • m[1][2] - find the 2nd element in the 1st column
  • det(m) - the determinate of a matrix
  • trace(m) - the trace of a matrix
  • trans(m) - transpose of a matrix
  • id(3) - creates a 3x3 identity matrix
  • zeroMat(3,2) - creates a 3 by matrix of zeros
  • zeroVec(3) - creates a vector of zeros length 3
  • size(m) - finds the length of a vector or size of a matrix
  • inv(m) - find the inverse of m
  • v=solve(m,u) - find solution of m * v = u

Statistical functions

  • count(v) - count the number of elements
  • min(v) - finds the min value in a vector or matrix
  • max(v) - finds the max value in a vector or matrix
  • sum(v) - finds the sum of the elements
  • product(v) - finds the product of the elements
  • mean(v) - finds the mean value in a vector or matrix
  • var(v) - finds the variance of the values
  • sd(v) - finds the standard deviation of the values
  • median(v) - finds the mean value in a vector or matrix
  • ranks(d,data) - finds the rank of element d in data array
  • ranks(v) - finds the ranks all the elements
  • mode(v) - finds the mode of the values

Advanced Examples

Using the factorial(x) function to find number of digits of precision

Input
setfield integer
factorial(10)
factorial(20)
factorial(21)
factorial(22)
Result
Setting field INTEGER
3628800
2432902008176640000
51090942171709440000
1.1240007277776077e+21

Doubles work the same 

setfield double
factorial(21)
factorial(22)

Setting field DOUBLE
51090942171709440000
1.1240007277776077e+21
BigIntegers allow much larger values 
setfield bigint
factorial(20)
factorial(30)
factorial(40)
factorial(50)

Setting field BIGINT
2432902008176640000
265252859812191058636308480000000
815915283247897734345611269596115894272000000000
30414093201713378043612608166064768844377641568960512000000000000

Calculations with fractions

setfield rational
1/6*2/5
1/6+1/2

Setting field RATIONAL
2/3
1/15

Calculation of pi using Ramanujan's formula Ramanujan's formula 

s=1103; a =1; c=1; d=1; \\
for(k=1;k<10;++k) {\\
  a*=(4*k-3)*(4*k-2)*(4*k-1)*(4*k);  \\
  b =1103 +  26390*k; \\
  c *= k*k*k*k; d *= 396^4; s+= a*b/(c*d); \\
  v = 9801/(2*sqrt(2)*s); println(v); }

3.141592653589793877998905826306015
3.141592653589793238462649065702759
3.141592653589793238462643383279558
3.141592653589793238462643383279506
3.141592653589793238462643383279506
3.141592653589793238462643383279506
3.141592653589793238462643383279506
3.141592653589793238462643383279506
3.141592653589793238462643383279506

Calculation of e 

s=1; f=1; for(k=1;k<30;++k) { f*=k; s+=1/f; println(s) }

2
2.5
2.666666666666666666666666666666667
2.708333333333333333333333333333334
2.716666666666666666666666666666667
2.718055555555555555555555555555556
2.718253968253968253968253968253969
...
2.718281828459045235360287471352545
2.718281828459045235360287471352658

Continued fraction for pi 

setfield double
a=zeroVec(20);
n=pi; for(i=1;i<=20;++i) { b=floor(n); n = 1/(n-b); a[i]=b  }
a

Setting field DOUBLE
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
4
[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1, 1, 7, 4]
Reconstructing values from array representation of a continued fraction 
a=[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2]
for(i=1;i<=20;++i) { \\
  s=a[i]; \\
  for(j=i-1;j>0;--j) { \\
    s = a[j]+1/s } \\
  println(s) }

[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2]
3
3.142857142857142857142857142857143
3.141509433962264150943396226415094
...
3.141592653589793239014009759199591
3.141592653589793238386377506390380
3.141592653589793238493875058011561
Solving an expression using Newton method. Uses symbolic differentiation and symbolic assignment f:= ... 
f := x^2 - x - 1
g := diff(f,x)
x=1
for(i=0;i<10;++i) { x -= f/g; println(x,f); }

f:=x^2-x-1
g:=2*x-1
x=1
2, 1
1.666666666666666666666666666666667, 0.111111111111111111111111111111112
1.619047619047619047619047619047619, 0.002267573696145124716553287981859
1.618034447821681864235055724417427, 0.000001026515933067055100295739241
1.618033988749989097047296779290725, 2.10746819100131229750E-13
1.618033988749894848204586838338167, 8.882845E-27
1.618033988749894848204586834365638, 0E-33
1.618033988749894848204586834365638, 0E-33
1.618033988749894848204586834365638, 0E-33
1.618033988749894848204586834365638, 0E-33