Adds two multivectors component-wise.

Examples

iex> a = Galixir.Algebras.Complex1.new(scalar: 2)
iex> b = Galixir.Algebras.Complex1.new(scalar: 3)
iex> Galixir.Algebras.Complex1.add(a, b)
Galixir.Algebras.Complex1.new(scalar: 5)

Computes the complex argument.

Returns the angle in radians:

atan2(imaginary, real)

Checks whether a multivector is a blade.

A blade is a multivector containing components from at most one grade.

Scalars are considered blades.

Examples

iex> Galixir.Algebras.Complex1.blade?(Galixir.Algebras.Complex1.new(e1: 2))
true

iex> Galixir.Algebras.Complex1.blade?(Galixir.Algebras.Complex1.new(scalar: 2))
true

iex> Galixir.Algebras.Complex1.blade?(Galixir.Algebras.Complex1.new(scalar: 2, e1: 1))
false

Returns the mapping between blade names and storage indices.

Blade coefficients are stored in a fixed-size tuple. This map translates canonical blade names into their corresponding tuple index.

Example

iex> Galixir.Algebras.Complex1.blade_indices()[:e1]
1

Returns the canonical sign of a multivector.

The canonical sign is determined by the first non-zero coefficient in storage order.

Returns:

• 1 if the first non-zero coefficient is positive
• -1 if the first non-zero coefficient is negative
• 0 if all coefficients are zero

Examples

iex> Elixir.Galixir.Algebras.Complex1.canonical_sign(Elixir.Galixir.Algebras.Complex1.new(e1: 2))
1

iex> Elixir.Galixir.Algebras.Complex1.canonical_sign(Elixir.Galixir.Algebras.Complex1.new(e1: -2))
-1

iex> Elixir.Galixir.Algebras.Complex1.canonical_sign(Elixir.Galixir.Algebras.Complex1.new())
0

Returns the coefficient of a basis blade.

The requested blade can be given in canonical form or as any registered blade alias. Aliases are automatically converted to the canonical blade and the appropriate sign is applied.

## Examples

iex> Elixir.Galixir.Algebras.Complex1.coefficient( ...> Elixir.Galixir.Algebras.Complex1.new(e1: 3), ...> :e1 ...> ) 3.0

Constructs a complex number from real and imaginary parts.

Creates:

real + imag*j

Computes the complex conjugate.

For:

z = a + bj

returns:

z̄ = a - bj

Returns the dimension of the algebra.

This is the number of basis vectors defined by the signature.

Computes the scalar product of two multivectors.

Computes the dual of a multivector.

The dual maps each basis blade to its complementary blade with the appropriate orientation sign. The complement is determined by the full pseudoscalar of the algebra.

The operation is linear and applies independently to every coefficient.

Examples

iex> Galixir.Algebras.Complex1.dual(Galixir.Algebras.Complex1.new(e1: 1)) |> inspect Galixir.Algebras.Complex1.new(scalar: 1.0) |> inspect

Computes the complex exponential.

For:

z = a + bj

computes:

exp(z) = exp(a)(cos(b) + j sin(b))

Computes the geometric product of two multivectors.

The geometric product is the fundamental multiplication operation of geometric algebra. It combines the outer product and metric-dependent inner product into a single associative operation.

The result depends on the algebra's metric signature.

Examples

iex> Galixir.Algebras.Complex1.gp(
...>   Galixir.Algebras.Complex1.new(e1: 1),
...>   Galixir.Algebras.Complex1.new(e1: 1)
...> )
Galixir.Algebras.Complex1.new(scalar: -1)

Extracts the grade-g component of a multivector.

All coefficients whose basis blades are not of grade g are set to zero.

Raises ArgumentError if g is outside the range 0..dimension().

Examples

iex> Galixir.Algebras.Complex1.grade(
...>   Galixir.Algebras.Complex1.new(scalar: 1, e1: 2),
...>   1
...> )
Galixir.Algebras.Complex1.new(e1: 2)

iex> Galixir.Algebras.Complex1.grade(
...>   Galixir.Algebras.Complex1.new(scalar: 1, e1: 2),
...>   0
...> )
Galixir.Algebras.Complex1.new(scalar: 1)

Returns the grades present in a multivector.

The returned list contains every grade with at least one non-zero coefficient, ordered from lowest to highest.

Examples

iex> Galixir.Algebras.Complex1.grades(
...>   Galixir.Algebras.Complex1.new(scalar: 1)
...> )
[0]

iex> Galixir.Algebras.Complex1.grades(
...>   Galixir.Algebras.Complex1.new(e1: 2)
...> )
[1]

iex> Galixir.Algebras.Complex1.grades(
...>   Galixir.Algebras.Complex1.new(scalar: 1, e1: 2)
...> )
[0, 1]

iex> Galixir.Algebras.Complex1.grades(
...>   Galixir.Algebras.Complex1.new()
...> )
[]

Extracts the imaginary part of a complex number.

Computes the inner product of two multivectors.

The inner product retains only those terms of the geometric product whose grade is the absolute difference of the operand grades.

For homogeneous blades A and B:

grade(inner(A, B)) = |grade(A) - grade(B)|

The exact result depends on the metric signature of the algebra.

Examples

iex> Galixir.Algebras.Complex1.inner(
...>   Galixir.Algebras.Complex1.new(e1: 2),
...>   Galixir.Algebras.Complex1.new(e1: 3)
...> )
Galixir.Algebras.Complex1.new(scalar: -6)

Computes the multiplicative inverse.

For a non-zero complex number:

z⁻¹ = conjugate(z) / |z|²

Computes the inverse of a multivector.

The inverse is computed using the reverse:

inverse(a) = reverse(a) / scalar_part(a * reverse(a))

This formula is valid when a * reverse(a) is a non-zero scalar.

Raises ArgumentError if the multivector is not invertible by this formula.

## Examples

    iex> Galixir.Algebras.Complex1.inverse(
    ...>   Galixir.Algebras.Complex1.new(e1: 2)
    ...> )|> inspect
    Galixir.Algebras.Complex1.new(e1: -0.5) |> inspect

Returns the imaginary unit.

In this algebra:

j² = -1

Computes the principal complex logarithm.

Returns:

log(z) = log(|z|) + arg(z)j

Computes the magnitude:

|z| = sqrt(|z|²)

Computes the squared magnitude:

|z|² = z * conjugate(z)

Returns the scalar value.

Returns the maximum absolute coefficient of a multivector.

Accepts either a multivector struct or the internal coefficient tuple.

Example

iex> Elixir.Galixir.Algebras.Complex1.max_abs_component(Elixir.Galixir.Algebras.Complex1.new(e1: 2, scalar: 5))
5.0

iex> Elixir.Galixir.Algebras.Complex1.max_abs_component(Elixir.Galixir.Algebras.Complex1.new(e1: 5, scalar: 2))
5.0

Multiplies two complex numbers.

This is equivalent to the geometric product:

a * b = gp(a,b)

Returns the norm of a multivector.

The norm is the square root of the absolute squared norm.

Example

iex> a = Galixir.Algebras.Complex1.new(scalar: 3)
iex> Galixir.Algebras.Complex1.norm(a)
3.0

Normalizes a multivector.

The result has unit norm while preserving the direction of the multivector.

Raises ArgumentError when attempting to normalize a null multivector.

Example

iex> a = Galixir.Algebras.Complex1.new(scalar: 2)
iex> Galixir.Algebras.Complex1.norm(Galixir.Algebras.Complex1.normalize(a))
1.0

Returns the multiplicative identity:

1 + 0i

Computes complex exponentiation.

Calculates:

z^t = exp(t * log(z))

Extracts the real part of a complex number.

Applies the reverse operation to a multivector.

Reverse (also called reversion) changes the sign of basis blades according to their grade:

grade 0:  +
grade 1:  +
grade 2:  -
grade 3:  -
grade 4:  +
...

For a blade with grade r, the sign is:

(-1)^(r(r-1)/2)

Examples

iex> Galixir.Algebras.Complex1.reverse(Galixir.Algebras.Complex1.new(e1: 2)) |> inspect
Galixir.Algebras.Complex1.new(e1: 2)|> inspect

iex> Galixir.Algebras.Complex1.reverse(Galixir.Algebras.Complex1.new(scalar: 3))|> inspect
Galixir.Algebras.Complex1.new(scalar: 3)|> inspect

Checks whether a multivector contains only a scalar component.

Components with an absolute value smaller than eps are considered zero.

Examples

iex> Galixir.Algebras.Complex1.scalar?(Galixir.Algebras.Complex1.new(scalar: 3))
true

iex> Galixir.Algebras.Complex1.scalar?(Galixir.Algebras.Complex1.new(e1: 3))
false

iex> Galixir.Algebras.Complex1.scalar?(Galixir.Algebras.Complex1.new())
true

Returns the scalar coefficient of a multivector.

This is equivalent to retrieving the coefficient of the scalar blade.

Examples

iex> Elixir.Galixir.Algebras.Complex1.scalar_part(Elixir.Galixir.Algebras.Complex1.new(scalar: 5.0, e1: 2.0))
5.0

Computes the scalar product of two multivectors.

The scalar product is the grade-0 component of the geometric product:

<a b>₀

The result depends on the metric signature of the algebra. In particular, basis vectors with negative or null squares affect the result.

Examples

iex> Galixir.Algebras.Complex1.scalar_product(
...>   Galixir.Algebras.Complex1.new(e1: 2),
...>   Galixir.Algebras.Complex1.new(e1: 3)
...> )
-6.0

Returns the metric signature of the algebra.

Example:

{1, 1, 1, 0}

represents a projective geometric algebra with three Euclidean basis vectors and one null basis vector.

Returns the number of coefficients stored by the algebra.

A dimension n algebra contains 2^n basis blades.

Returns the squared norm of a multivector.

The squared norm is computed as:

scalar_part(a * reverse(a))

The result may be negative for algebras with indefinite metrics.

Example

iex> a = Galixir.Algebras.Complex1.new(scalar: 3)
iex> Galixir.Algebras.Complex1.squared_norm(a)
9.0

Subtracts two multivectors component-wise.

Examples

iex> a = Galixir.Algebras.Complex1.new(scalar: 5)
iex> b = Galixir.Algebras.Complex1.new(scalar: 2)
iex> Galixir.Algebras.Complex1.sub(a, b)
Galixir.Algebras.Complex1.new(scalar: 3)

Returns the multiplication table for the algebra.

The table contains precomputed geometric products between basis blades. Each entry maps {left_blade, right_blade} to {coefficient, result_blade}.

The blades are represented internally as bitmasks.

Example

iex> Galixir.Algebras.Complex1.table() |> Map.has_key?({1, 1})
true

Formats a multivector using standard geometric algebra notation.

Zero coefficients are omitted. Coefficients of 1 and -1 are elided for non-scalar basis blades.

Examples

iex> inspect(Galixir.Algebras.Complex1.new())
"0"

iex> inspect(Galixir.Algebras.Complex1.new(scalar: 2))
"2.0"

iex> inspect(Galixir.Algebras.Complex1.new(e1: 1))
"e1"

iex> inspect(Galixir.Algebras.Complex1.new(scalar: 1, e1: 2))
"1.0 + 2.0e1"

Computes the inverse dual operation.

undual/1 reverses the blade complement operation performed by dual/1.

For non-degenerate Euclidean algebras this corresponds to applying the dual operation twice with the appropriate pseudoscalar factor. In degenerate algebras the result depends on the implemented dual convention.

Examples

iex> Galixir.Algebras.Complex1.undual(Galixir.Algebras.Complex1.dual(Galixir.Algebras.Complex1.new(e1: 2)))
Galixir.Algebras.Complex1.new(e1: 2)

Computes the outer product (wedge product) of two multivectors.

The wedge product combines blades by joining their basis vectors. It is antisymmetric:

a ∧ b = -(b ∧ a)

and vanishes when the operands share a basis vector.

Examples

iex> Galixir.Algebras.Complex1.wedge(
...>   Galixir.Algebras.Complex1.new(e1: 1),
...>   Galixir.Algebras.Complex1.new(e1: 1)
...> )
Galixir.Algebras.Complex1.new()

Returns the zero complex number:

0 + 0i

Checks whether all coefficients of a multivector are zero.

Examples

iex> Galixir.Algebras.Complex1.zero?(Galixir.Algebras.Complex1.new())
true

iex> Galixir.Algebras.Complex1.zero?(Galixir.Algebras.Complex1.new(e1: 1))
false