symcomp.md

symcomp.lua

the symbolic computation module

The common return value

All functions presented herein return a table with a so-called basic string at index 1 and a LaTeX representation of that string at index 2.

Example

res = symcomp.expr("1/2*x")

print(res[1]) -- "1/2*x"
print(res[2]) -- "\frac{1}{2} x"

---

Contents

• symcomp.evalAt
• symcomp.sub
• symcomp.solve
• symcomp.diff
• symcomp.integrate
• symcomp.identityMatrix
• symcomp.matrix
• symcomp.matrixSub
• symcomp.scalarMul
• symcomp.det
• symcomp.eigenvalues

---

symcomp.evalAt(f, x, x0) - evaluates f(x) at x=x0

symcomp.evalAt can be used to evaluate the function f(x) at x=x0.

Signature

res = symcomp.evalAt(f, x, x0)

Example

f = symcomp.expr("-1/2*x**3")

res = symcomp.evalAt(f, "x", 1)

Output (res[1]):

-1/2

---

symcomp.sub(a, b) - returns a - b

subtracts b from a where a, b can be arbitrary expressions (numbers, polynomials etc.).

Signature

res = symcomp.sub(a, b)

Example

f = symcomp.expr("2*x**4+7*x-3")
g = symcomp.expr("3*x**3-2*x+7")

res = symcomp.sub(f, g)

Output (res[1])

2*x**4-3*x**3+9*x-10

---

symcomp.solve(f, x) - solves f(x) == 0

returns a collection of values of x where f(x) == 0.

Signature

res = symcomp.solve(f, x)

The return value is a collection of tables as described above, e.g. if a function has zeros -1/2 and 5/2 then the return value will be the following:

{
    { "-1/2", "-\frac{1}{2}" },
    {  "5/2", "-\frac{5}{2}" },
}

Example

f = symcomp.expr("x**2-4")

res = symcomp.solve(f, "x")

Output (res[1][1] and res[2][1])

-2
2

---

symcomp.diff(f, x) - evaluates df/dx

differentiates f with respect to x

Signature

res = symcomp.diff(f, x)

Example

f = symcomp.expr("-1/2*x**3")

res = symcomp.diff(f, "x")

Output (res[1]):

(-3/2)*x**2

---

symcomp.integrate(f, x [, l , u]) - integrates f from l to u with respect to x

evaluates the integral of f from l to u with respect to x.

⚠️ Warning:
As of now, f must be a polynomial. Other types of functions can not be evaluated yet.

Signature

res = symcomp.integrate(f, x)
res = symcomp.integrate(f, x, l, u)

Example

f = symcomp.expr("1/3*x^2")
l = 0
u = 3

symcomp.integrate(f, "x")
symcomp.integrate(f, "x", l, u)

Output

x^3/9
3

---

symcomp.identityMatrix(s) - the identity matrix of size s

Signature

i = symcomp.identityMatrix(s)

Example

i3 = symcomp.identityMatrix(3)

Output (i3[1]):

[1, 0, 0] [0, 1, 0] [0, 0, 1]

---

symcomp.matrix(str) - a matrix

creates a matrix object from a string representation.

⚠️ Warning:
The format of a m x n matrix is [a_11, a_12, ..., a_1n] ... [a_m1, a_m2, ..., a_mn]. Note that every entry has to be specified in each row.

Signature

m = symcomp.matrix(str)

Example

m = symcomp.matrix("[1, 2] [3, 4]")

Output (m[1]):

[1, 2] [3, 4]

---

symcomp.matrixSub(a, b) - returns a - b

subtracts b from a where a, b are two matrices.

Signature

res = symcomp.matrixSub(a, b)

Example

a = "[1, 2] [3, 4]"
b = "[0, 1] [0, 2]"
 
res = symcomp.matrixSub(a, b)

Output (res[1])

[1, 1] [3, 2]

---

symcomp.scalarMul(k, a) - returns k*a

multiplies the matrix a by a scalar value k.

Signature

res = symcomp.scalarMul(k, a)

Example

a = "[1, 2] [3, 4]"
k = 4
 
res = symcomp.scalarMul(k, a)

Output (res[1])

[4, 8] [12, 16]

---

symcomp.det(a) - the determinant of a matrix

Signature

d = symcomp.det(a)

Example

a = "[1, 2] [3, 4]"

d = symcomp.det(a)

Output (d[1]):

-2

---

symcomp.eigenvalues(a) - the eigenvalues of a matrix a

Signature

evs = symcomp.eigenvalues(a)

Example

a = "[-5, -3] [-3, 3]"

evs = symcomp.eigenvalues(a)

Output:

-6
-4