numerical

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 Go to latest Published: Jul 14, 2022 License: BSD-2-Clause Imports: 9 Imported by: 0

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Constants

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const DefaultGSSIters = 64

Variables

This section is empty.

Functions

func GSS 

func GSS(min, max float64, iters int, f func(float64) float64) float64

GSS performs golden section search to find the minimum of a unimodal function, given correct bounds on the minimum.

If iters is 0, DefaultGSSIters is used.

Types

type KMeans added in v0.3.3 

type KMeans struct {
	// Centers is the K-means cluster centers.
	Centers []Vec

	// Data stores all of the data points.
	Data []Vec
}

KMeans stores the (intermediate) results of K-means clustering.

This type can be created using NewKMeans() to create an initial clustering using K-means++. Then, Iterate() can be called repeatedly to refine the clustering as desired.

func NewKMeans added in v0.3.3 

func NewKMeans(data []Vec, numCenters int) *KMeans

NewKMeans creates a KMeans object with K-means++ initialization.

func (*KMeans) Assign added in v0.3.3 

func (k *KMeans) Assign(vecs []Vec) []int

Assign gets the closest center index for every vector.

func (*KMeans) Iterate added in v0.3.3 

func (k *KMeans) Iterate() float64

Iterate performs a step of k-means and returns the current MSE loss. If the MSE loss does not decrease, then the process has converged.

type Matrix2 added in v0.3.4 

type Matrix2 [4]float64

Matrix2 is a 2x2 matrix, stored in row-major order.

func NewMatrix2Columns added in v0.3.4 

func NewMatrix2Columns(c1, c2 Vec2) *Matrix2

NewMatrix2Columns creates a Matrix2 with the given coordinates as column entries.

func NewMatrix2Rotation added in v0.3.4 

func NewMatrix2Rotation(theta float64) *Matrix2

NewMatrix2Rotation creates a rotation matrix that rotates column vectors by theta.

func (*Matrix2) Add added in v0.3.4 

func (m *Matrix2) Add(m1 *Matrix2) *Matrix2

Add computes m+m1 and returns the sum.

func (*Matrix2) Det added in v0.3.4 

func (m *Matrix2) Det() float64

Det computes the determinant of the matrix.

func (*Matrix2) Eigenvalues added in v0.3.4 

func (m *Matrix2) Eigenvalues() [2]complex128

Eigenvalues computes the eigenvalues of the matrix.

There may be a repeated eigenvalue, but for numerical reasons two are always returned.

func (*Matrix2) Inverse added in v0.3.4 

func (m *Matrix2) Inverse() *Matrix2

Inverse computes the inverse matrix.

func (*Matrix2) InvertInPlace added in v0.3.4 

func (m *Matrix2) InvertInPlace()

InvertInPlace moves the inverse of m into m without causing any new allocations.

func (*Matrix2) InvertInPlaceDet added in v0.3.4 

func (m *Matrix2) InvertInPlaceDet(det float64)

InvertInPlaceDet is an optimization for InvertInPlace when the determinant has been pre-computed.

func (*Matrix2) Mul added in v0.3.4 

func (m *Matrix2) Mul(m1 *Matrix2) *Matrix2

Mul computes m*m1 and returns the product.

func (*Matrix2) MulColumn added in v0.3.4 

func (m *Matrix2) MulColumn(c Vec2) Vec2

MulColumn multiplies the matrix m by a column vector represented by c.

func (*Matrix2) MulColumnInv added in v0.3.4 

func (m *Matrix2) MulColumnInv(c Vec2, det float64) Vec2

MulColumnInv multiplies the inverse of m by the column c, given the determinant of m.

func (*Matrix2) SVD added in v0.3.4 

func (m *Matrix2) SVD(u, s, v *Matrix2)

SVD computes the singular value decomposition of the matrix.

It populates matrices u, s, and v, such that 

m = u*s*v.Transpose()

The singular values in s are sorted largest to smallest.

func (*Matrix2) Scale added in v0.3.4 

func (m *Matrix2) Scale(s float64)

Scale scales m by a factor s.

func (*Matrix2) Transpose added in v0.3.4 

func (m *Matrix2) Transpose() *Matrix2

Transpose computes the matrix transpose.

type Matrix3 added in v0.3.4 

type Matrix3 [9]float64

Matrix3 is a 3x3 matrix, stored in row-major order.

func NewMatrix3Columns added in v0.3.4 

func NewMatrix3Columns(c1, c2, c3 Vec3) *Matrix3

NewMatrix3Columns creates a Matrix3 with the given coordinates as column entries.

func NewMatrix3Rotation added in v0.3.4 

func NewMatrix3Rotation(axis Vec3, angle float64) *Matrix3

NewMatrix3Rotation creates a 3D rotation matrix. Points are rotated around the given vector in a right-handed direction.

The axis is assumed to be normalized. The angle is measured in radians.

func (*Matrix3) Add added in v0.3.4 

func (m *Matrix3) Add(m1 *Matrix3) *Matrix3

Add computes m+m1 and returns the sum.

func (*Matrix3) Cols added in v0.3.4 

func (m *Matrix3) Cols() [3]Vec3

Cols returns the columns of the matrix as Vec4.

func (*Matrix3) Det added in v0.3.4 

func (m *Matrix3) Det() float64

Det computes the determinant of the matrix.

func (*Matrix3) Eigenvalues added in v0.3.4 

func (m *Matrix3) Eigenvalues() [3]complex128

Eigenvalues computes the eigenvalues of the matrix.

There may be repeated eigenvalues, but for numerical reasons three are always returned.

func (*Matrix3) Inverse added in v0.3.4 

func (m *Matrix3) Inverse() *Matrix3

Inverse computes the inverse matrix.

func (*Matrix3) InvertInPlace added in v0.3.4 

func (m *Matrix3) InvertInPlace()

InvertInPlace moves the inverse of m into m without causing any new allocations.

func (*Matrix3) InvertInPlaceDet added in v0.3.4 

func (m *Matrix3) InvertInPlaceDet(det float64)

InvertInPlaceDet is an optimization for InvertInPlace when the determinant has been pre-computed.

func (*Matrix3) Mul added in v0.3.4 

func (m *Matrix3) Mul(m1 *Matrix3) *Matrix3

Mul computes m*m1 and returns the product.

func (*Matrix3) MulColumn added in v0.3.4 

func (m *Matrix3) MulColumn(c Vec3) Vec3

MulColumn multiplies the matrix m by a column vector represented by c.

func (*Matrix3) MulColumnInv added in v0.3.4 

func (m *Matrix3) MulColumnInv(c Vec3, det float64) Vec3

MulColumnInv multiplies the inverse of m by the column c, given the determinant of m.

func (*Matrix3) Rows added in v0.3.4 

func (m *Matrix3) Rows() [3]Vec3

Rows returns the rows of the matrix as Vec3.

func (*Matrix3) SVD added in v0.3.4 

func (m *Matrix3) SVD(u, s, v *Matrix3)

SVD computes the singular value decomposition of the matrix.

It populates matrices u, s, and v, such that 

m = u*s*v.Transpose()

The singular values in s are sorted largest to smallest.

func (*Matrix3) Scale added in v0.3.4 

func (m *Matrix3) Scale(s float64)

Scale scales m by a factor s.

func (*Matrix3) Transpose added in v0.3.4 

func (m *Matrix3) Transpose() *Matrix3

Transpose computes the matrix transpose.

type Matrix4 added in v0.3.4 

type Matrix4 [16]float64

func NewMatrix4Columns added in v0.3.4 

func NewMatrix4Columns(c1, c2, c3, c4 Vec4) *Matrix4

NewMatrix4Columns creates a Matrix4 from the columns.

func NewMatrix4Identity added in v0.3.4 

func NewMatrix4Identity() *Matrix4

NewMatrix4Identity creates an identity matrix.

func (*Matrix4) Add added in v0.3.4 

func (m *Matrix4) Add(m1 *Matrix4) *Matrix4

Add computes m+m1 and returns the sum.

func (*Matrix4) CharPoly added in v0.3.4 

func (m *Matrix4) CharPoly() Polynomial

CharPoly computes the characteristic polynomial of the matrix.

func (*Matrix4) Cols added in v0.3.4 

func (m *Matrix4) Cols() [4]Vec4

Cols returns the columns of the matrix as Vec4.

func (*Matrix4) Det added in v0.3.4 

func (m *Matrix4) Det() float64

Det returns the determinant of m.

func (*Matrix4) Mul added in v0.3.4 

func (m *Matrix4) Mul(m1 *Matrix4) *Matrix4

Mul computes m*m1 and returns the product.

func (*Matrix4) MulColumn added in v0.3.4 

func (m *Matrix4) MulColumn(v Vec4) Vec4

MulColumn computes a matrix-vector product.

func (*Matrix4) Rows added in v0.3.4 

func (m *Matrix4) Rows() [4]Vec4

Rows returns the rows of the matrix as Vec4.

func (*Matrix4) SVD added in v0.3.4 

func (m *Matrix4) SVD(u, s, v *Matrix4)

SVD computes the singular value decomposition of the matrix.

It populates matrices u, s, and v, such that 

m = u*s*v.Transpose()

The singular values in s are sorted largest to smallest.

func (*Matrix4) Scale added in v0.3.4 

func (m *Matrix4) Scale(s float64) *Matrix4

Scale returns the scalar-matrix product.

func (*Matrix4) Sub added in v0.3.4 

func (m *Matrix4) Sub(m1 *Matrix4) *Matrix4

Sub computes m-m1 and returns the result.

func (*Matrix4) Transpose added in v0.3.4 

func (m *Matrix4) Transpose() *Matrix4

Transpose computes the matrix transpose.

type Polynomial 

type Polynomial []float64

A Polynomial is an equation of the form 

a0 + a1*x + a2*x^2 + a3*x^3 + ...

Here, the polynomial is represented as an array of [a0, a1, ...].

func (Polynomial) Add 

func (p Polynomial) Add(p1 Polynomial) Polynomial

Add returns the sum of p and p1.

func (Polynomial) Derivative 

func (p Polynomial) Derivative() Polynomial

Derivative computes the derivative of the polynomial.

func (Polynomial) Eval 

func (p Polynomial) Eval(x float64) float64

Eval evaluates the polynomial at the given x value.

func (Polynomial) IterRealRoots 

func (p Polynomial) IterRealRoots(f func(x float64) bool)

IterRealRoots iterates over the real roots of p. This is similar to RealRoots(), but allows the caller to stop iteration early be returning false.

func (Polynomial) Mul 

func (p Polynomial) Mul(p1 Polynomial) Polynomial

Mul computes the product of two polynomials.

func (Polynomial) RealRoots 

func (p Polynomial) RealRoots() []float64

RealRoots computes the real roots of p, i.e. values of X such that p(x) = 0. The result may have duplicates since roots can be repeated.

If the polynomial has an infinite number of roots, one NaN root is returned.

func (Polynomial) Scale added in v0.3.0 

func (p Polynomial) Scale(c float64) Polynomial

Scale returns a new polynomial with the coefficients of p multiplied by c.

func (Polynomial) String 

func (p Polynomial) String() string

String returns a string representation of p.

type SparseCholesky 

type SparseCholesky struct {
	// contains filtered or unexported fields
}

SparseCholesky is a sparse LU decomposition of a symmetric matrix.

Once instantiated, this object can be used to quickly apply the inverse of a matrix to vectors.

func NewSparseCholesky 

func NewSparseCholesky(mat *SparseMatrix) *SparseCholesky

func (*SparseCholesky) ApplyInverseVec3 

func (a *SparseCholesky) ApplyInverseVec3(x []Vec3) []Vec3

ApplyInverseVec3 computes (A^-1*x, A^-1*y, A^-1*z).

func (*SparseCholesky) ApplyVec3 

func (a *SparseCholesky) ApplyVec3(x []Vec3) []Vec3

ApplyVec3 computes (A*x, A*y, A*z).

type SparseMatrix 

type SparseMatrix struct {
	// contains filtered or unexported fields
}

A SparseMatrix is a square matrix where entries can be set to non-zero values in any order, and not all entries must be set.

func NewSparseMatrix 

func NewSparseMatrix(size int) *SparseMatrix

func (*SparseMatrix) ApplyVec3 

func (a *SparseMatrix) ApplyVec3(x []Vec3) []Vec3

ApplyVec3 computes (A*x, A*y, A*z).

func (*SparseMatrix) Iterate 

func (a *SparseMatrix) Iterate(row int, f func(col int, x float64))

Iterate loops through the non-zero entries in a row.

func (*SparseMatrix) Permute 

func (a *SparseMatrix) Permute(perm []int) *SparseMatrix

Permute permutes the rows and columns by perm, where perm is the result of applying the permutation to the list [0...n-1].

func (*SparseMatrix) RCM 

func (a *SparseMatrix) RCM() []int

RCM computes the reverse Cuthill-McKee permutation for the matrix.

func (*SparseMatrix) Set 

func (a *SparseMatrix) Set(row, col int, x float64)

Set adds an entry to the matrix.

The entry should not already be set.

type Vec 

type Vec []float64

Vec is a vector of arbitrary dimension.

func (Vec) Add 

func (v Vec) Add(v1 Vec) Vec

Add returns v + v1.

func (Vec) Dist added in v0.3.3 

func (v Vec) Dist(v1 Vec) float64

Dist computes ||v-v1||.

func (Vec) DistSquared added in v0.3.3 

func (v Vec) DistSquared(v1 Vec) float64

DistSquared computes ||v-v1||^2.

func (Vec) Dot 

func (v Vec) Dot(v1 Vec) float64

Dot returns the dot product of v and v1.

func (Vec) Norm 

func (v Vec) Norm() float64

Norm returns ||v||.

func (Vec) NormSquared 

func (v Vec) NormSquared() float64

NormSquared returns ||v||^2.

func (Vec) Normalize 

func (v Vec) Normalize() Vec

Normalize returns v/||v||.

func (Vec) ProjectOut 

func (v Vec) ProjectOut(v1 Vec) Vec

ProjectOut projects the direction v1 out of v.

func (Vec) Scale 

func (v Vec) Scale(s float64) Vec

Scale returns v * s.

func (Vec) Sub 

func (v Vec) Sub(v1 Vec) Vec

Sub returns v - v1.

type Vec2 added in v0.3.4 

type Vec2 [2]float64

A Vec2 is a 2-dimensional tuple of floats.

func NewVec2RandomNormal added in v0.3.4 

func NewVec2RandomNormal() Vec2

NewVec2RandomNormal creates a random normal vector.

func NewVec2RandomUnit added in v0.3.4 

func NewVec2RandomUnit() Vec2

NewVec2RandomUnit creates a unit-length Vec2 in a random direction.

func (Vec2) Add added in v0.3.4 

func (v Vec2) Add(v1 Vec2) Vec2

Add returns v + v1.

func (Vec2) Dist added in v0.3.4 

func (v Vec2) Dist(v1 Vec2) float64

Dist gets the Euclidean distance ||v - v1||.

func (Vec2) Dot added in v0.3.4 

func (v Vec2) Dot(v1 Vec2) float64

Dot computes (v dot v1).

func (Vec2) Norm added in v0.3.4 

func (v Vec2) Norm() float64

Norm computes ||v||.

func (Vec2) Normalize added in v0.3.4 

func (v Vec2) Normalize() Vec2

Normalize gets a unit vector from v.

func (Vec2) ProjectOut added in v0.3.4 

func (v Vec2) ProjectOut(v1 Vec2) Vec2

ProjectOut projects the v1 direction out of v.

func (Vec2) Scale added in v0.3.4 

func (v Vec2) Scale(f float64) Vec2

Scale returns v * f.

func (Vec2) Sub added in v0.3.4 

func (v Vec2) Sub(v1 Vec2) Vec2

Sub returns v - v1.

func (Vec2) Sum added in v0.3.4 

func (v Vec2) Sum() float64

Sum gets the sum of the components.

type Vec3 

type Vec3 [3]float64

A Vec3 is a 3-dimensional tuple of floats.

func LeastSquares3 added in v0.3.4 

func LeastSquares3(a []Vec3, b []float64, epsilon float64) Vec3

LeastSquares3 computes the least squares solution to the equation Ax = b, where A is a matrix with rows vs, and b is a column matrix.

The epsilon argument is a lower bound for singular values in the psuedoinverse.

func LeastSquaresReg3 added in v0.3.4 

func LeastSquaresReg3(a []Vec3, b []float64, lambda, epsilon float64) Vec3

LeastSquaresReg3 is like LeastSquares3, but uses ridge regression with a penalty equal to lambda.

func NewVec3RandomNormal 

func NewVec3RandomNormal() Vec3

NewVec3RandomNormal creates a random normal vector.

func NewVec3RandomUnit added in v0.3.4 

func NewVec3RandomUnit() Vec3

NewVec3RandomUnit creates a unit-length Vec3 in a random direction.

func (Vec3) Add 

func (v Vec3) Add(v1 Vec3) Vec3

Add returns v + v1.

func (Vec3) Cross added in v0.3.4 

func (v Vec3) Cross(v1 Vec3) Vec3

Cross computes the cross product of v and v1.

func (Vec3) Dist 

func (v Vec3) Dist(v1 Vec3) float64

Dist gets the Euclidean distance ||v - v1||.

func (Vec3) Dot added in v0.3.4 

func (v Vec3) Dot(v1 Vec3) float64

Dot computes (v dot v1).

func (Vec3) Norm added in v0.3.4 

func (v Vec3) Norm() float64

Norm computes ||v||.

func (Vec3) Normalize added in v0.3.4 

func (v Vec3) Normalize() Vec3

Normalize gets a unit vector from v.

func (Vec3) OrthoBasis added in v0.3.4 

func (v Vec3) OrthoBasis() (Vec3, Vec3)

OrthoBasis creates two unit vectors which are orthogonal to v and to each other.

If v is axis-aligned, the other vectors will be as well.

func (Vec3) ProjectOut added in v0.3.4 

func (v Vec3) ProjectOut(v1 Vec3) Vec3

ProjectOut projects the v1 direction out of v.

func (Vec3) Scale 

func (v Vec3) Scale(f float64) Vec3

Scale returns v * f.

func (Vec3) Sub 

func (v Vec3) Sub(v1 Vec3) Vec3

Sub returns v - v1.

func (Vec3) Sum 

func (v Vec3) Sum() float64

Sum gets the sum of the components.

type Vec4 added in v0.3.4 

type Vec4 [4]float64

A Vec4 is a 4-dimensional tuple of floats.

func NewVec4RandomNormal added in v0.3.4 

func NewVec4RandomNormal() Vec4

NewVec4RandomNormal creates a random normal vector.

func NewVec4RandomUnit added in v0.3.4 

func NewVec4RandomUnit() Vec4

NewVec4RandomUnit creates a unit-length Vec3 in a random direction.

func (Vec4) Add added in v0.3.4 

func (v Vec4) Add(v1 Vec4) Vec4

Add returns v + v1.

func (Vec4) Dist added in v0.3.4 

func (v Vec4) Dist(v1 Vec4) float64

Dist gets the Euclidean distance ||v - v1||.

func (Vec4) Dot added in v0.3.4 

func (v Vec4) Dot(v1 Vec4) float64

Dot computes (v dot v1).

func (Vec4) Norm added in v0.3.4 

func (v Vec4) Norm() float64

Norm computes ||v||.

func (Vec4) Normalize added in v0.3.4 

func (v Vec4) Normalize() Vec4

Normalize gets a unit vector from v.

func (Vec4) OrthoBasis added in v0.3.4 

func (v Vec4) OrthoBasis() (Vec4, Vec4, Vec4)

OrthoBasis creates three unit vectors which are orthogonal to v and to each other.

If v is axis-aligned, the other vectors will be as well.

The behavior of this method is undefined if v is zero.

func (Vec4) ProjectOut added in v0.3.4 

func (v Vec4) ProjectOut(v1 Vec4) Vec4

ProjectOut projects the v1 direction out of v.

func (Vec4) Scale added in v0.3.4 

func (v Vec4) Scale(f float64) Vec4

Scale returns v * f.

func (Vec4) Sub added in v0.3.4 

func (v Vec4) Sub(v1 Vec4) Vec4

Sub returns v - v1.

func (Vec4) Sum added in v0.3.4 

func (v Vec4) Sum() float64

Sum gets the sum of the components.

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