Math


Let's have a look at Agena's functions for:



As you will see, you actually do not have to know much about the interpreter to do some undergraduate math.

Calculus

Define a function, for example f(x) = sin(x):


> f := << x -> sin(x) >>

Print a  table of values:


> for x from -1 to 1 by 0.5 do
>    print(x, f(x))
> od;
-1      -0.8414709848079
-0.5    -0.4794255386042
0       0
0.5     0.4794255386042
1       0.8414709848079

Same, with a little formatting:


> for x from -0.5 to 0.5 by 0.25 do
>    printf('%+05.2f %+10.6f\n', x, f(x))
> od;
-0.50  -0.479426
-0.25  -0.247404
+0.00  +0.000000
+0.25  +0.247404
+0.50  +0.479426

Determine all the zeros over [-5, 5]:


> calc.zeros(f, -5, 5):
seq(-3.1415926535898, 0, 3.1415926535898)

Differentiate f at point 0:


> calc.differ(f, 0):
1

Evaluate the third derivative of f at point 0:


> calc.differ(f, 0, deriv=3):
-0.99999999999983

Compute the minimum and maximum values on the interval [-10, 10]:


> calc.minimum(f, -10, 10):
seq(-7.8539815784491, -1.5707963196727, 4.7123889665853)

> calc.maximum(f, -10, 10):
seq(-4.7123889665853, 1.5707963196727, 7.8539815784491)

Integrate f over [0, Pi]:


> calc.integ(f, 0, Pi):
2

Compute the series `Sum(1/n!, n=0 .. 100)` to return an approximation of Euler's number:


> calc.fsum(<< n -> 1/fact(n) >>, 0, 100):
2.718281828459

Linear Algebra

Define two vectors in different fashions: In the simple form, just pass all components explicitly; or pass only the non-zero components:


> a := < 1, 2, 3 >:
< 1, 2, 3 >

> b := vector(3, [1 ~ 2]):
< 2, 0, 0 >

Check whether a and b are parallel and have the same direction:


> abs(a+b) = abs(a) + abs(b):
false

Set a vector component by indexing:


> b[3] := 1;

Now read the modified vector and its rightmost component  - a negative integral index n depicts the |n|-th element from the right::


> b:
< 2, 0, 1 >

> b[3], b[-1]:
1       1

Addition and subtraction:


> a + b:
< 3, 2, 4 >

> a - b:
< -1, 2, 2 >

Scalar, dot and cross product:


> 2 * a:
< 2, 4, 6 >

> a * a:
14

> linalg.crossprod(a, b):
< 2, 5, -4 >

Find the vector x which satisfies the matrix equation A x = b. The matrix constructor expects row vectors.


> A :=  < < 1, 2, -4 >, < 2, 1, 3 >, < -3, 1, 6 > >:
[  1, 2, -4 ]
[  2, 1,  3 ]
[ -3, 1,  6 ]

> b := < -6, 5, -2 >:
< -6, 5, -2 >

> linalg.linsolve(A, b):
< 2, -2, 1 >

The determinant:


> linalg.det(A):
-59

Statistics

First we define a distribution:


> s := seq(10, 8, 1, 6, 5, 2, 9, 7, 3, 4):
seq(10, 8, 1, 6, 5, 2, 9, 7, 3, 4)

Minimum and maximum values in one result:


> stats.minmax(s):
seq(1, 10)

Minimum and maximum observations along with their position in the distribution:


> stats.min(s):
1       3

> stats.max(s):
10      1

Arithmetic mean:


> stats.amean(s):
5.5

The median:


> stats.median(s):
5.5

For the first quartile, the median and the third quartile of a distribution, along with the minimum, the maximum observation, and the arithmetic mean, in this order, enter:


> stats.fivenum(s):
seq(2.75, 5.5, 8.25, 1, 10, 5.5)

Standard and absolute deviation:


> stats.sd(s):
2.872281323269

> stats.ad(s):
2.5

Outliers:


> s := seq(-100, 8, 10, 1, 6, 5, 2, 9, 7, 3, 4, 1000):
seq(-100, 8, 10, 1, 6, 5, 2, 9, 7, 3, 4, 1000)

> stats.chauvenet(s):
seq(1000, -100)

Combinatorics

The Cartesian product:


> combinat.cartprod([[1, 2, 3], [30], [50, 100]]):
[[1, 30, 50], [1, 30, 100], [2, 30, 50], [2, 30, 100], [3, 30, 50], [3, 30, 100]]

Combinations and number of combinations:


> combinat.chosse(3, 2):
[[1, 2], [1, 3], [2, 3]]

> combinat.numbcomb(3, 2):
3

Permutations and number of permutations:


> combinat.permute([1, 2, 3], 3):
[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

> combinat.numbperm(3, 2):
6


Functions and Operators


+

Addition

^

Exponentiation with rational power

-

Subtraction

**

Exponentiation with integral power

*

Multiplication

%

Modulus

/

Division

\

Integer division

sin(x)

Sine

sinh(x)

Hyperbolic sine

cos(x)

Cosine

cosh(x)

Hyperbolic cosine

tan(x)

Tangent

tanh(x)

Hyperbolic tangent

sec(x)

Secant

arcsinh(x)

Inverse hyperbolic sine

csc(x)

Cosecant

arccosh(x)

Inverse hyperbolic cosine

cot(x)

Cotangent

arctanh(x)

Inverse hyperbolic tangent

arcsin(x)

Inverse sine

sinc(x)

Cardinal sine

arccos(x)

Inverse cosine

cosc(x)

Cardinal cosine

arctan(x)

Inverse tangent

tanc(x)

Cardinal tangent

exp(x)

Exponentiation e^x

sqrt(x)

Square root

ln(x)

Natural logarithm

cbrt(x)

Cubic root

log(x, b)

Logarithm of x to base b

root(x, n)

Non-principal n-th root of x

hypot(x, y)

Hypotenuse

proot(x, n)

Principal n-th root of x

gamma(x)

Gamma function

fact(n)

Factorial

lngamma(x)

Logarithmic Gamma function

erf(x)

Error function

abs(x)

Absolute value/magnitude

int(x)

Rounds to the nearest integer towards zero

sign(x)

Sign

frac(x

Fractional part

entier(x)

Rounds downwards to the nearest integer

round(x, d)

Rounds x to d-th digit

floor(x)

Rounds downwards to the nearest integer (same as entier)

even(x)

Checks for even number

ceil(x)

Rounds upwards to the nearest integer

odd(x)

Checks for odd  number

Constants


Eps

Equals 1.4901161193847656e-08

infinity

Infinity

DoubleEps

Equals 1.084202172485504434e-19

undefined

An expression stating that it is undefined, e.g. a singularity

degrees

180/Pi to convert radians to degrees

Pi

3.14159265358979323846

radians

Factor Pi/180 to convert degrees to radians

Exp

e = exp(1) = 2.71828182845904523536

I

Imaginary unit

Phi

Golden ratio (1+ sqrt(5))/2