Standard library · Machine learning

matrix

Linear algebra: matrices, systems of equations, decompositions, eigenvalues and statistics.

import matrix

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Types
Constants and variables
Functions

This package is complete linear algebra over 64 bit floats: from making a matrix and multiplying through to solving systems, decompositions, eigenvalues and singular values, and statistical tools such as principal component analysis. To solve one system, call Solve; for several systems with one matrix, Decompose once and LUSolve many times.

Matrices hold floats and the results carry a little rounding error; compare with AllClose and display with Chop. For neural networks see tensor.

Types

`PCA`

struct PCA

Principal component analysis.

Name	Type	Description
`components`	`Matrix`	The components in the rows.
`variances`	`Matrix`	The variance of each component.
`center`	`Matrix`	The column means.

matrix-Stats.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func show(name: str, m &: matrix.Matrix):
    println name, matrix.ToRepr(matrix.Chop(m, 0.000001), 3)
end

func r(value: f64): str:
    ret matrix.ToStringPrec(matrix.Full(1, 1, value), 3)
end

func main:
    show("Hilbert", matrix.Hilbert(2))
    show("Hilbert times its inverse", matrix.MatMul(matrix.Hilbert(3), matrix.HilbertInverse(3)))
    show("Vandermonde", matrix.Vandermonde(build(3, 1, "1 2 3"), 3))
    show("Toeplitz", matrix.Toeplitz(build(3, 1, "1 2 3"), build(1, 3, "1 4 5")))
    show("Hankel", matrix.Hankel(build(3, 1, "1 2 3"), build(1, 3, "3 4 5")))
    show("Circulant", matrix.Circulant(build(1, 3, "1 2 3")))
    show("Companion", matrix.Companion(build(1, 3, "1 -3 2")))
    show("Pascal", matrix.Pascal(3))
    show("Rotation2D", matrix.Rotation2D(0.0))
    show("Rotation3D", matrix.Rotation3D(2, 0.0))
    show("Householder", matrix.Householder(build(2, 1, "1 0")))
    show("Givens", matrix.Givens(2, 0, 1, 0.0))
    show("Permutation", matrix.Permutation(build(1, 3, "2 0 1")))

    mut rng := matrix.NewRng(7)
    println rng.next_bits() > 0, rng.next_float() < 1.0, rng.next_normal() < 10.0
    mut random := matrix.Random(2, 3, -1.0, 1.0, 5)
    println matrix.ShapeString(random), matrix.Min(random) >= -1.0, matrix.Max(random) < 1.0, matrix.AllClose(random, matrix.Random(2, 3, -1.0, 1.0, 5), 0.0)
    println matrix.ShapeString(matrix.RandomNormal(2, 2, 1))
    println matrix.IsSymmetric(matrix.RandomSymmetric(3, 1), 0.000001), matrix.IsPositiveDefinite(matrix.RandomSPD(3, 1), 0.000001), matrix.IsOrthogonal(matrix.RandomOrthogonal(3, 1), 0.00001)

    mut data := build(4, 2, "1 2 2 4 3 6 4 8")
    show("CenterColumns", matrix.CenterColumns(data))
    show("ColumnMeans", matrix.ColumnMeans(data))
    show("ColumnStds", matrix.ColumnStds(data, false))
    show("Standardize", matrix.Standardize(data, false))
    show("Covariance", matrix.Covariance(data, true))
    show("Correlation", matrix.Correlation(data))
    mut pca := matrix.PCAFit(data)
    println matrix.ShapeString(pca.components), matrix.ShapeString(pca.variances), matrix.ShapeString(pca.center)
    show("ExplainedVariance", matrix.ExplainedVariance(pca))
    show("PCATransform", matrix.Abs(matrix.PCATransform(pca, data, 1)))
    pca.free()
    mut pca2 := matrix.PCAFromSVD(data)
    show("from the SVD", matrix.ExplainedVariance(pca2))
    pca2.free()

    mut x := build(4, 1, "1 2 3 4")
    mut y := build(4, 1, "3 5 7 9")
    coefficients := matrix.LinearRegression(x, y)
    show("LinearRegression", coefficients)
    println r(matrix.RSquared(y, y))
end

OutputHilbert [1.000,0.500; 0.500,0.333] Hilbert times its inverse [1.000,0.000,0.000; 0.000,1.000,0.000; 0.000,0.000,1.000] Vandermonde [1.000,1.000,1.000; 1.000,2.000,4.000; 1.000,3.000,9.000] Toeplitz [1.000,4.000,5.000; 2.000,1.000,4.000; 3.000,2.000,1.000] Hankel [1.000,2.000,3.000; 2.000,3.000,4.000; 3.000,4.000,5.000] Circulant [1.000,2.000,3.000; 3.000,1.000,2.000; 2.000,3.000,1.000] Companion [0.000,-0.500; 1.000,1.500] Pascal [1.000,1.000,1.000; 1.000,2.000,3.000; 1.000,3.000,6.000] Rotation2D [1.000,0.000; 0.000,1.000] Rotation3D [1.000,0.000,0.000; 0.000,1.000,0.000; 0.000,0.000,1.000] Householder [-1.000,0.000; 0.000,1.000] Givens [1.000,0.000; 0.000,1.000] Permutation [0.000,0.000,1.000; 1.000,0.000,0.000; 0.000,1.000,0.000] true true true 2x3 true true true 2x2 true true true CenterColumns [-1.500,-3.000; -0.500,-1.000; 0.500,1.000; 1.500,3.000] ColumnMeans [2.500,5.000] ColumnStds [1.118,2.236] Standardize [-1.342,-1.342; -0.447,-0.447; 0.447,0.447; 1.342,1.342] Covariance [1.667,3.333; 3.333,6.667] Correlation [1.000,1.000; 1.000,1.000] 2x2 1x2 1x2 ExplainedVariance [1.000,0.000] PCATransform [3.354; 1.118; 1.118; 3.354] from the SVD [1.000,0.000] LinearRegression [1.000; 2.000] 1.000

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Stats.salam

`Eigen`

struct Eigen

The eigenvalues of a general matrix.

Name	Type	Description
`n`	`int = 0`	The size.
`re`	`Matrix`	The real part.
`im`	`Matrix`	The imaginary part.
`iterations`	`int = 0`	The number of iterations.
`ok`	`bool = false`	It converged.

matrix-Eigen.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func show(name: str, m &: matrix.Matrix):
    println name, matrix.ToRepr(matrix.Chop(m, 0.000001), 3)
end

func r(value: f64): str:
    ret matrix.ToStringPrec(matrix.Full(1, 1, value), 3)
end

func main:
    mut symmetric := build(2, 2, "2 1 1 2")
    mut decomposition := matrix.EigSym(symmetric)
    println decomposition.ok, decomposition.n, decomposition.sweeps >= 0
    show("values", decomposition.values)
    show("EigReconstruct", matrix.EigReconstruct(decomposition))
    decomposition.free()
    show("EigValsSym", matrix.EigValsSym(symmetric))
    println r(matrix.SpectralRadiusSym(symmetric))
    show("PowSym", matrix.PowSym(symmetric, 2.0))
    show("SqrtSym squared", matrix.MatMul(matrix.SqrtSym(symmetric), matrix.SqrtSym(symmetric)))

    mut general := build(2, 2, "0 1 -2 -3")
    mut eigen := matrix.EigVals(general)
    println eigen.ok, eigen.n, matrix.ToRepr(eigen.re, 3), matrix.ToRepr(eigen.im, 3), eigen.iterations >= 0
    eigen.free()
    mut rotation := build(2, 2, "0 -1 1 0")
    mut complex := matrix.EigValsAuto(rotation)
    show("imaginary", complex.im)
    println r(matrix.SpectralRadius(general))
    show("Hessenberg", matrix.Hessenberg(build(3, 3, "1 2 3 4 5 6 7 8 10")))
    show("CharPoly", matrix.CharPoly(general))
    mut vector := matrix.EigVecFor(symmetric, 3.0, 50)
    show("EigVecFor", matrix.Abs(vector))
    mut power := matrix.PowerIteration(symmetric, 100)
    mut dominant := matrix.SubMatrix(power, 0, 0, 2, 1)
    println r(power.at(2, 0)), r(matrix.Rayleigh(symmetric, dominant)), r(matrix.ShiftedInverseIteration(symmetric, 0.9, 50))

    mut m := build(3, 2, "3 0 0 4 0 0")
    mut svd := matrix.SVDecompose(m)
    println svd.ok, matrix.ShapeString(svd.u), matrix.ShapeString(svd.v)
    show("singular values", svd.s)
    svd.free()
    show("SingularValues", matrix.SingularValues(m))
    println matrix.MatrixRank(m, 0.000001)
    show("PseudoInverse", matrix.PseudoInverse(m, 0.000001))
    show("NullSpace", matrix.NullSpace(build(2, 2, "1 1 1 1"), 0.000001))
    show("ColumnSpace", matrix.Abs(matrix.ColumnSpace(build(2, 2, "1 1 1 1"), 0.000001)))
    show("LowRankApprox", matrix.LowRankApprox(build(2, 2, "3 0 0 1"), 1))

    tol := 0.000001
    identity := matrix.Eye(3)
    println matrix.IsSquare(identity), matrix.AllClose(identity, identity, tol), matrix.IsZero(matrix.Zeros(2, 2), tol), matrix.IsIdentity(identity, tol)
    println matrix.IsSymmetric(symmetric, tol), matrix.IsSkewSymmetric(rotation, tol), matrix.IsDiagonal(identity, tol)
    println matrix.IsUpperTriangular(build(2, 2, "1 2 0 3"), tol), matrix.IsLowerTriangular(build(2, 2, "1 0 2 3"), tol)
    tridiagonal := build(3, 3, "2 1 0 1 2 1 0 1 2")
    println matrix.IsBanded(tridiagonal, 1, 1, tol), matrix.LowerBandwidth(tridiagonal, tol), matrix.UpperBandwidth(tridiagonal, tol)
    println matrix.IsOrthogonal(rotation, tol), matrix.IsNormal(symmetric, tol), matrix.IsIdempotent(build(2, 2, "1 0 0 0"), tol)
    println matrix.IsInvolutory(build(2, 2, "0 1 1 0"), tol), matrix.IsNilpotent(build(2, 2, "0 1 0 0"), tol)
    println matrix.IsPositiveDefinite(symmetric, tol), matrix.IsPositiveSemiDefinite(build(2, 2, "1 1 1 1"), tol)
    println matrix.IsSingular(build(2, 2, "1 2 2 4")), matrix.IsInvertible(symmetric)
    probability := build(2, 2, "0.5 0.5 0.5 0.5")
    println matrix.IsRowStochastic(probability, tol), matrix.IsColStochastic(probability, tol), matrix.IsDoublyStochastic(probability, tol)
    println matrix.IsPermutation(build(2, 2, "0 1 1 0"), tol), matrix.IsToeplitz(tridiagonal, tol), matrix.IsHankel(build(2, 2, "1 2 2 3"), tol)
    println r(matrix.Density(tridiagonal))
end

Outputtrue 2 true values [3.000,1.000] EigReconstruct [2.000,1.000; 1.000,2.000] EigValsSym [3.000,1.000] 3.000 PowSym [5.000,4.000; 4.000,5.000] SqrtSym squared [2.000,1.000; 1.000,2.000] true 2 [-1.000,-2.000] [0.000,0.000] true imaginary [1.000,-1.000] 2.000 Hessenberg [1.000,-3.597,-0.248; -8.062,14.800,2.400; 0.000,0.400,0.200] CharPoly [2.000,3.000,1.000] EigVecFor [0.707; 0.707] 3.000 3.000 1.000 true 3x2 2x2 singular values [4.000,3.000] SingularValues [4.000,3.000] 2 PseudoInverse [0.333,0.000,0.000; 0.000,0.250,0.000] NullSpace [0.707; -0.707] ColumnSpace [0.707; 0.707] LowRankApprox [3.000,0.000; 0.000,0.000] true true true true true true true true true true 1 1 true true true true true true true true true true true true true true true 0.778

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Eigen.salam

`LU`

struct LU

The lower upper decomposition with row swaps.

Name	Type	Description
`n`	`int = 0`	The size.
`lu`	`Matrix`	L and U in one matrix.
`piv`	`Vector`	The row swaps.
`sign`	`int = 1`	The sign of the permutation.
`singular`	`bool = false`	The matrix is singular.

matrix-Solve.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func show(name: str, m &: matrix.Matrix):
    println name, matrix.ToRepr(matrix.Chop(m, 0.000001), 3)
end

func r(value: f64): str:
    ret matrix.ToStringPrec(matrix.Full(1, 1, value), 3)
end

func main:
    mut a := build(3, 3, "4 1 0 1 3 1 0 1 2")
    mut b := build(3, 1, "1 2 3")
    println r(matrix.Det(a)), r(matrix.LogAbsDet(a))
    show("Solve", matrix.Solve(a, b))
    show("SolveVec", matrix.SolveVec(a, b))
    mut lu := matrix.Decompose(a)
    println lu.n, lu.piv.len(), lu.sign, lu.singular, matrix.ShapeString(lu.lu)
    show("LUSolve", matrix.LUSolve(lu, b))
    lu.free()
    mut inverse := matrix.Inverse(a)
    show("a times its inverse", matrix.MatMul(a, inverse))
    show("Adjugate", matrix.Adjugate(build(2, 2, "1 2 3 4")))
    show("Residual", matrix.Residual(a, matrix.Solve(a, b), b))

    mut qr := matrix.QRDecompose(a)
    println qr.m, qr.n, qr.full_rank, matrix.ShapeString(qr.qr), matrix.ShapeString(qr.rdiag)
    show("Q times R", matrix.MatMul(matrix.QRGetQ(qr), matrix.QRGetR(qr)))
    show("QRSolve", matrix.QRSolve(qr, b))
    qr.free()
    mut points := build(4, 2, "1 0 1 1 1 2 1 3")
    mut values := build(4, 1, "1 3 5 7")
    show("LeastSquares", matrix.LeastSquares(points, values))
    show("Orthonormalize", matrix.Gram(matrix.Orthonormalize(points)))

    show("Cholesky", matrix.Cholesky(a))
    show("SolveSPD", matrix.SolveSPD(a, b))
    show("CholeskySolve", matrix.CholeskySolve(matrix.Cholesky(a), b))
    show("InverseSPD", matrix.MatMul(a, matrix.InverseSPD(a)))
    println r(matrix.DetSPD(a)), r(matrix.LogDetSPD(a))
    mut ldl := matrix.LDLDecompose(a)
    println ldl.ok, ldl.n, matrix.ShapeString(ldl.l), matrix.ShapeString(ldl.d)
    show("LDLSolve", matrix.LDLSolve(ldl, b))
    ldl.free()
    show("Inertia", matrix.Inertia(build(2, 2, "1 0 0 -1")))

    mut singular := build(3, 3, "1 2 3 2 4 6 1 1 1")
    show("RowEchelon", matrix.RowEchelon(singular))
    show("RREF", matrix.RREF(singular))
    show("PivotColumns", matrix.PivotColumns(singular))
    println matrix.RankExact(singular), matrix.Nullity(singular)
    show("NullSpaceExact", matrix.NullSpaceExact(singular))
    show("ColumnSpaceExact", matrix.ColumnSpaceExact(singular))
    mut right := build(3, 1, "6 12 3")
    println matrix.IsConsistent(singular, right), matrix.IsConsistent(singular, build(3, 1, "1 3 3"))
    show("SolveGeneral", matrix.SolveGeneral(singular, right))

    println matrix.IsDiagonallyDominant(a)
    mut jacobi := matrix.SolveJacobi(a, b, 200, 0.00000001)
    println jacobi.converged, jacobi.iterations > 0, jacobi.residual < 0.0001
    show("Jacobi", jacobi.x)
    jacobi.free()
    each method in [1, 2, 3]:
        mut result := matrix.SolveGaussSeidel(a, b, 200, 0.00000001)
        if method == 2:
            result = matrix.SolveSOR(a, b, 1.2, 200, 0.00000001)
        else method == 3:
            result = matrix.SolveCG(a, b, 200, 0.00000001)
        end
        show("iterative", result.x)
        result.free()
    end

    mut d := build(2, 2, "1 0 0 2")
    show("Expm", matrix.Expm(matrix.Zeros(2, 2)))
    show("Sqrtm", matrix.Sqrtm(build(2, 2, "4 0 0 9")))
    show("MatrixSign", matrix.MatrixSign(build(2, 2, "2 0 0 -3")))
    show("PolyEval", matrix.PolyEval(d, build(1, 3, "1 0 1")))
    show("CayleyHamilton", matrix.CayleyHamilton(d))
    show("LogmSym", matrix.LogmSym(matrix.Eye(2)))
end

Output18.000 2.890 Solve [0.222; 0.111; 1.444] SolveVec [0.222; 0.111; 1.444] 3 3 1 false 3x3 LUSolve [0.222; 0.111; 1.444] a times its inverse [1.000,0.000,0.000; 0.000,1.000,0.000; 0.000,0.000,1.000] Adjugate [4.000,-2.000; -3.000,1.000] Residual [0.000; 0.000; 0.000] 3 3 true 3x3 1x3 Q times R [4.000,1.000,0.000; 1.000,3.000,1.000; 0.000,1.000,2.000] QRSolve [0.222; 0.111; 1.444] LeastSquares [1.000; 2.000] Orthonormalize [1.000,0.000; 0.000,1.000] Cholesky [2.000,0.000,0.000; 0.500,1.658,0.000; 0.000,0.603,1.279] SolveSPD [0.222; 0.111; 1.444] CholeskySolve [0.222; 0.111; 1.444] InverseSPD [1.000,0.000,0.000; 0.000,1.000,0.000; 0.000,0.000,1.000] 18.000 2.890 true 3 3x3 1x3 LDLSolve [0.222; 0.111; 1.444] Inertia [1.000,1.000,0.000] RowEchelon [2.000,4.000,6.000; 0.000,-1.000,-2.000; 0.000,0.000,0.000] RREF [1.000,0.000,-1.000; 0.000,1.000,2.000; 0.000,0.000,0.000] PivotColumns [0.000,1.000] 2 1 NullSpaceExact [1.000; -2.000; 1.000] ColumnSpaceExact [1.000,2.000; 2.000,4.000; 1.000,1.000] true false SolveGeneral [0.000; 3.000; 0.000] true true true true Jacobi [0.222; 0.111; 1.444] iterative [0.222; 0.111; 1.444] iterative [0.222; 0.111; 1.444] iterative [0.222; 0.111; 1.444] Expm [1.000,0.000; 0.000,1.000] Sqrtm [2.000,0.000; 0.000,3.000] MatrixSign [1.000,0.000; 0.000,-1.000] PolyEval [2.000,0.000; 0.000,5.000] CayleyHamilton [0.000,0.000; 0.000,0.000] LogmSym [0.000,0.000; 0.000,0.000]

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Solve.salam

`IterResult`

struct IterResult

The result of an iterative method.

Name	Type	Description
`x`	`Matrix`	The solution.
`iterations`	`int = 0`	The number of iterations.
`residual`	`f64 = 0`	The norm of the residual.
`converged`	`bool = false`	It converged.

The example for this item is shown with the ones above: matrix-Solve.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Solve.salam

`LDL`

struct LDL

The LDL decomposition.

Name	Type	Description
`n`	`int = 0`	The size.
`l`	`Matrix`	The unit lower triangular matrix.
`d`	`Matrix`	The diagonal D.
`ok`	`bool = false`	The decomposition succeeded.

The example for this item is shown with the ones above: matrix-Solve.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Solve.salam

`Matrix`

struct Matrix

A 64 bit float matrix with its elements in row order.

Name	Type	Description
`rows`	`int = 0`	The number of rows.
`cols`	`int = 0`	The number of columns.
`a`	`Vector`	The elements.

matrix-Create.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func main:
    mut a := build(2, 3, "1 2 3 4 5 6")
    matrix.Print(a)
    println matrix.ShapeString(a), a.rows, a.cols, a.a.len(), a.count(), a.min_dim(), a.is_square(), a.is_empty()
    println a.at(1, 2), a.flat(4)
    a.set(0, 0, 10.0)
    a.add_at(0, 1, 0.5)
    a.mul_at(0, 2, 2.0)
    a.set_flat(5, -1.0)
    matrix.PrintPrec(a, 1)
    matrix.PrintLabeled("a", a)

    println matrix.ToString(matrix.Zeros(1, 2)), matrix.ToString(matrix.Ones(1, 2)), matrix.ToString(matrix.Full(1, 2, 7.0))
    println matrix.ToString(matrix.Eye(2))
    println matrix.ToString(matrix.EyeOffset(2, 3, 1))
    mut data := Vector {} as Vector<f64>
    data.push(1.0)
    data.push(2.0)
    println matrix.ShapeString(matrix.RowVector(data)), matrix.ShapeString(matrix.ColVector(data)), matrix.ToString(matrix.Diag(data))
    mut b := matrix.Clone(a)
    matrix.SetAll(b, 0.0)
    matrix.CopyInto(b, a)
    println b.same_shape(a), b.at(0, 0)
    println matrix.ToStringPrec(matrix.Arange(0.0, 1.0, 0.25), 2)
    println matrix.ToStringPrec(matrix.Linspace(0.0, 1.0, 5), 2)
    println matrix.ToStringPrec(matrix.Logspace(0.0, 2.0, 3, 10.0), 1)
    println matrix.ToCSV(a, 1)
    println matrix.ToRepr(a, 1)
    println matrix.MAT_EPS > 0.0, matrix.MAT_TOL > 0.0, matrix.MAT_INF > 1000.0
    matrix.Free(b)
    a.free()
end

Output1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 2x3 2 3 6 6 2 false false 6 5 10.0 2.5 6.0 4.0 5.0 -1.0 a (2x3) 10.0000 2.5000 6.0000 4.0000 5.0000 -1.0000 0.0000 0.0000 1.0000 1.0000 7.0000 7.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1x2 2x1 1.0000 0.0000 0.0000 2.0000 true 10 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 1.00 1.0 10.0 100.0 10.0,2.5,6.0 4.0,5.0,-1.0 [10.0,2.5,6.0; 4.0,5.0,-1.0] true true true

`at`

func at(i: int, j: int): f64

The element at row i and column j.

The example for this item is shown with the ones above: matrix-Create.salam

`set`

func set(i: int, j: int, v: f64)

Sets the element.

The example for this item is shown with the ones above: matrix-Create.salam

`add_at`

func add_at(i: int, j: int, v: f64)

Adds to the element.

The example for this item is shown with the ones above: matrix-Create.salam

`mul_at`

func mul_at(i: int, j: int, v: f64)

Multiplies the element by a number.

The example for this item is shown with the ones above: matrix-Create.salam

`flat`

func flat(k: int): f64

Element number k in row order.

The example for this item is shown with the ones above: matrix-Create.salam

`set_flat`

func set_flat(k: int, v: f64)

Sets element k.

The example for this item is shown with the ones above: matrix-Create.salam

`count`

func count(): int

The number of elements.

The example for this item is shown with the ones above: matrix-Create.salam

`is_square`

func is_square(): bool

It is square.

The example for this item is shown with the ones above: matrix-Create.salam

`is_empty`

func is_empty(): bool

It has no elements.

The example for this item is shown with the ones above: matrix-Create.salam

`same_shape`

func same_shape(o &: Matrix): bool

It has the same shape as another matrix.

The example for this item is shown with the ones above: matrix-Create.salam

`min_dim`

func min_dim(): int

The smaller of the row and column counts.

The example for this item is shown with the ones above: matrix-Create.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Create.salam

`EigenSym`

struct EigenSym

The eigendecomposition of a symmetric matrix.

Name	Type	Description
`n`	`int = 0`	The size.
`values`	`Matrix`	The eigenvalues, descending.
`vectors`	`Matrix`	The eigenvectors in the columns.
`sweeps`	`int = 0`	The number of Jacobi sweeps.
`ok`	`bool = false`	It converged.

The example for this item is shown with the ones above: matrix-Eigen.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Rng`

struct Rng

A deterministic random generator.

Name	Type	Description
`s`	`u64 = 0`	The state.

The example for this item is shown with the ones above: matrix-Stats.salam

`next_bits`

func next_bits(): u64

64 random bits.

The example for this item is shown with the ones above: matrix-Stats.salam

`next_float`

func next_float(): f64

A number between zero and one.

The example for this item is shown with the ones above: matrix-Stats.salam

`next_normal`

func next_normal(): f64

A standard normal number.

The example for this item is shown with the ones above: matrix-Stats.salam

`SVD`

struct SVD

The singular value decomposition.

Name	Type	Description
`u`	`Matrix`	The left singular vectors.
`s`	`Matrix`	The singular values, descending.
`v`	`Matrix`	The right singular vectors.
`ok`	`bool = false`	It converged.

The example for this item is shown with the ones above: matrix-Eigen.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Eigen.salam

`QR`

struct QR

The QR decomposition.

Name	Type	Description
`m`	`int = 0`	The number of rows.
`n`	`int = 0`	The number of columns.
`qr`	`Matrix`	The reflections and R, packed.
`rdiag`	`Matrix`	The diagonal of R.
`full_rank`	`bool = true`	The columns are independent.

The example for this item is shown with the ones above: matrix-Solve.salam

`free`

func free()

Frees it.

The example for this item is shown with the ones above: matrix-Solve.salam

Constants and variables

`MAT_EPS`

const MAT_EPS: f64 = 2.22045e-16

The machine precision of 64 bit floats.

The example for this item is shown with the ones above: matrix-Create.salam

`MAT_TOL`

const MAT_TOL: f64 = 1e-09

The default tolerance for comparisons.

The example for this item is shown with the ones above: matrix-Create.salam

`MAT_INF`

const MAT_INF: f64 = 1e+300

Infinity.

The example for this item is shown with the ones above: matrix-Create.salam

Functions

`CenterColumns`

func CenterColumns(x &: Matrix): Matrix

Subtracts the mean of each column.

The example for this item is shown with the ones above: matrix-Stats.salam

`ColumnMeans`

func ColumnMeans(x &: Matrix): Matrix

The mean of each column.

The example for this item is shown with the ones above: matrix-Stats.salam

`ColumnStds`

func ColumnStds(x &: Matrix, sample: bool): Matrix

The standard deviation of each column.

The example for this item is shown with the ones above: matrix-Stats.salam

`Standardize`

func Standardize(x &: Matrix, sample: bool): Matrix

Every column with mean zero and standard deviation one.

The example for this item is shown with the ones above: matrix-Stats.salam

`Covariance`

func Covariance(x &: Matrix, sample: bool): Matrix

The covariance matrix of the columns.

The example for this item is shown with the ones above: matrix-Stats.salam

`Correlation`

func Correlation(x &: Matrix): Matrix

The Pearson correlation matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`PCAFit`

func PCAFit(x &: Matrix): PCA

Principal component analysis through the covariance.

The example for this item is shown with the ones above: matrix-Stats.salam

`PCAFromSVD`

func PCAFromSVD(x &: Matrix): PCA

The same, through the singular value decomposition; more stable.

The example for this item is shown with the ones above: matrix-Stats.salam

`PCATransform`

func PCATransform(p &: PCA, x &: Matrix, k: int): Matrix

Projects the data onto the first k components.

The example for this item is shown with the ones above: matrix-Stats.salam

`ExplainedVariance`

func ExplainedVariance(p &: PCA): Matrix

The share each component takes of the total variance.

The example for this item is shown with the ones above: matrix-Stats.salam

`LinearRegression`

func LinearRegression(x &: Matrix, y &: Matrix): Matrix

The coefficients of a linear regression; a column of ones is added for the intercept.

The example for this item is shown with the ones above: matrix-Stats.salam

`RSquared`

func RSquared(y &: Matrix, fitted &: Matrix): f64

The coefficient of determination.

The example for this item is shown with the ones above: matrix-Stats.salam

`Expm`

func Expm(x &: Matrix): Matrix

The matrix exponential.

The example for this item is shown with the ones above: matrix-Solve.salam

`Sqrtm`

func Sqrtm(x &: Matrix): Matrix

The matrix square root.

The example for this item is shown with the ones above: matrix-Solve.salam

`MatrixSign`

func MatrixSign(x &: Matrix): Matrix

The matrix sign function.

The example for this item is shown with the ones above: matrix-Solve.salam

`PolyEval`

func PolyEval(x &: Matrix, coeffs &: Matrix): Matrix

A polynomial with coefficients from degree zero, over a matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`CayleyHamilton`

func CayleyHamilton(x &: Matrix): Matrix

The characteristic polynomial over the matrix itself; by the theorem it is zero.

The example for this item is shown with the ones above: matrix-Solve.salam

`LogmSym`

func LogmSym(x &: Matrix): Matrix

The matrix logarithm for a symmetric positive definite matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`Hessenberg`

func Hessenberg(x &: Matrix): Matrix

The Hessenberg form.

The example for this item is shown with the ones above: matrix-Eigen.salam

`EigVals`

func EigVals(x &: Matrix): Eigen

The eigenvalues by the QR method.

The example for this item is shown with the ones above: matrix-Eigen.salam

`EigValsAuto`

func EigValsAuto(x &: Matrix): Eigen

The eigenvalues by whichever method suits the matrix, symmetric or general.

The example for this item is shown with the ones above: matrix-Eigen.salam

`SpectralRadius`

func SpectralRadius(x &: Matrix): f64

The greatest absolute eigenvalue.

The example for this item is shown with the ones above: matrix-Eigen.salam

`CharPoly`

func CharPoly(x &: Matrix): Matrix

The coefficients of the characteristic polynomial from degree zero.

The example for this item is shown with the ones above: matrix-Eigen.salam

`EigVecFor`

func EigVecFor(x &: Matrix, lambda: f64, iters: int): Matrix

The eigenvector for one eigenvalue.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Decompose`

func Decompose(x &: Matrix): LU

The lower upper decomposition; for solving several systems with one matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`Det`

func Det(x &: Matrix): f64

The determinant.

The example for this item is shown with the ones above: matrix-Solve.salam

`LogAbsDet`

func LogAbsDet(x &: Matrix): f64

The logarithm of the absolute determinant; without overflow.

The example for this item is shown with the ones above: matrix-Solve.salam

`Solve`

func Solve(a &: Matrix, b &: Matrix): Matrix

Solves the system AX = B.

The example for this item is shown with the ones above: matrix-Solve.salam

`LUSolve`

func LUSolve(f &: LU, b &: Matrix): Matrix

Solves it with a ready decomposition.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveVec`

func SolveVec(a &: Matrix, b &: Matrix): Matrix

The system with one vector.

The example for this item is shown with the ones above: matrix-Solve.salam

`Inverse`

func Inverse(x &: Matrix): Matrix

The inverse.

matrix-Math.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func show(name: str, m &: matrix.Matrix):
    println name, matrix.ToRepr(m, 1)
end

func r(value: f64): str:
    ret matrix.ToStringPrec(matrix.Full(1, 1, value), 3)
end

func main:
    mut a := build(2, 2, "1 2 3 4")
    mut b := build(2, 2, "0 1 1 0")
    mut row := build(1, 2, "10 20")
    mut col := build(2, 1, "1 -1")
    show("Transpose", matrix.Transpose(a))
    show("MatMul", matrix.MatMul(a, b))
    mut out := matrix.Zeros(2, 2)
    println matrix.TransposeInto(a, out), matrix.MatMulInto(a, b, out)
    show("into", out)
    show("TransposeMul", matrix.TransposeMul(a, b))
    show("MulTranspose", matrix.MulTranspose(a, b))
    show("Gram", matrix.Gram(a))
    show("MatVec", matrix.MatVec(a, col))
    show("VecMat", matrix.VecMat(row, a))
    println matrix.Dot(col, col), matrix.Trace(a)
    show("Outer", matrix.Outer(col, col))
    show("Kron", matrix.Kron(matrix.Eye(2), matrix.Full(1, 2, 3.0)))
    show("MatPow", matrix.MatPow(a, 2))
    show("Cross3", matrix.Cross3(build(3, 1, "1 0 0"), build(3, 1, "0 1 0")))

    show("Add", matrix.Add(a, b))
    show("Sub", matrix.Sub(a, b))
    show("MulElem", matrix.MulElem(a, b))
    show("DivElem", matrix.DivElem(a, a))
    show("Neg", matrix.Neg(a))
    show("with scalars", matrix.AddScalar(matrix.SubScalar(matrix.Scale(matrix.DivScalar(a, 2.0), 4.0), 1.0), 1.0))
    mut c := matrix.Clone(a)
    matrix.AddInPlace(c, b)
    matrix.SubInPlace(c, b)
    matrix.MulElemInPlace(c, a)
    matrix.ScaleInPlace(c, 0.5)
    matrix.AddScalarInPlace(c, 1.0)
    matrix.NegInPlace(c)
    matrix.AddScaled(c, a, 2.0)
    show("in place", c)
    show("Combine", matrix.Combine(a, 2.0, b, -1.0))
    show("MaxElem", matrix.MaxElem(a, matrix.Full(2, 2, 2.5)))
    show("MinElem", matrix.MinElem(a, matrix.Full(2, 2, 2.5)))
    show("AddRowVector", matrix.AddRowVector(a, row))
    show("AddColVector", matrix.AddColVector(a, col))
    show("MulRowVector", matrix.MulRowVector(a, row))
    show("MulColVector", matrix.MulColVector(a, col))

    mut m := build(1, 4, "-2.25 0 0.5 9")
    show("Abs", matrix.Abs(m))
    show("Sqrt", matrix.Sqrt(matrix.Abs(m)))
    show("Cbrt", matrix.Cbrt(build(1, 2, "8 -27")))
    show("Floor and Ceil", matrix.HStack(matrix.Floor(m), matrix.Ceil(m)))
    show("Round and Trunc", matrix.HStack(matrix.Round(m), matrix.Trunc(m)))
    show("Sign", matrix.Sign(m))
    show("Square", matrix.Square(m))
    show("Reciprocal", matrix.Reciprocal(build(1, 2, "2 4")))
    show("PowElem", matrix.PowElem(build(1, 2, "2 3"), 3.0))
    show("Clip", matrix.Clip(m, 0.0, 1.0))
    show("MaxScalar and MinScalar", matrix.HStack(matrix.MaxScalar(m, 0.0), matrix.MinScalar(m, 0.0)))
    show("Chop", matrix.Chop(build(1, 2, "0.0000001 1"), 0.0001))
    mut zero := matrix.Zeros(1, 1)
    println matrix.ToStringPrec(matrix.HStack(matrix.Exp(zero), matrix.Log(matrix.Ones(1, 1))), 3), matrix.ToStringPrec(matrix.Log10(matrix.Full(1, 1, 100.0)), 3), matrix.ToStringPrec(matrix.Log2(matrix.Full(1, 1, 8.0)), 3)
    println matrix.ToStringPrec(matrix.HStack(matrix.Sin(zero), matrix.Cos(zero)), 3), matrix.ToStringPrec(matrix.Tan(zero), 3)
    println matrix.ToStringPrec(matrix.HStack(matrix.Asin(zero), matrix.Acos(matrix.Ones(1, 1))), 3), matrix.ToStringPrec(matrix.Atan(zero), 3)
    println matrix.ToStringPrec(matrix.HStack(matrix.Sinh(zero), matrix.Cosh(zero)), 3), matrix.ToStringPrec(matrix.Tanh(zero), 3)

    println matrix.Sum(a), matrix.Prod(a), matrix.Mean(a), matrix.Min(a), matrix.Max(a), matrix.Range(a)
    k := matrix.ArgMax(a)
    println matrix.ArgMin(a), k, matrix.IndexRow(a, k), matrix.IndexCol(a, k), matrix.MaxAbs(m)
    println r(matrix.Var(a, true)), r(matrix.Std(a, false)), matrix.Median(a)
    show("RowSums", matrix.RowSums(a))
    show("ColSums", matrix.ColSums(a))
    show("RowMeans", matrix.RowMeans(a))
    show("ColMeans", matrix.ColMeans(a))
    show("RowMins and RowMaxs", matrix.HStack(matrix.RowMins(a), matrix.RowMaxs(a)))
    show("ColMins and ColMaxs", matrix.VStack(matrix.ColMins(a), matrix.ColMaxs(a)))
    show("cumulative", matrix.HStack(matrix.CumSumRows(a), matrix.CumSumCols(a)))
    show("CumProdRows", matrix.CumProdRows(a))
    println matrix.CountNonZero(b), matrix.Any(b), matrix.All(b), matrix.AllFinite(b)

    println r(matrix.NormFrobenius(a)), matrix.Norm1(a), matrix.NormInf(a), matrix.NormMax(a)
    println r(matrix.Norm2(a)), r(matrix.NormNuclear(a)), matrix.VecNorm(col, 1.0), r(matrix.Distance(col, matrix.Zeros(2, 1)))
    show("Normalize", matrix.Normalize(build(1, 2, "3 4")))
    println matrix.ToStringPrec(matrix.NormalizeRows(a), 2)
    println matrix.ToStringPrec(matrix.NormalizeCols(a), 2)
    println r(matrix.Cond1(a)), r(matrix.CondInf(a)), r(matrix.Cond2(a))
end

OutputTranspose [1.0,3.0; 2.0,4.0] MatMul [2.0,1.0; 4.0,3.0] true true into [2.0,1.0; 4.0,3.0] TransposeMul [3.0,1.0; 4.0,2.0] MulTranspose [2.0,1.0; 4.0,3.0] Gram [10.0,14.0; 14.0,20.0] MatVec [-1.0; -1.0] VecMat [70.0,100.0] 2 5 Outer [1.0,-1.0; -1.0,1.0] Kron [3.0,3.0,0.0,0.0; 0.0,0.0,3.0,3.0] MatPow [7.0,10.0; 15.0,22.0] Cross3 [0.0; 0.0; 1.0] Add [1.0,3.0; 4.0,4.0] Sub [1.0,1.0; 2.0,4.0] MulElem [0.0,2.0; 3.0,0.0] DivElem [1.0,1.0; 1.0,1.0] Neg [-1.0,-2.0; -3.0,-4.0] with scalars [2.0,4.0; 6.0,8.0] in place [0.5,1.0; 0.5,-1.0] Combine [2.0,3.0; 5.0,8.0] MaxElem [2.5,2.5; 3.0,4.0] MinElem [1.0,2.0; 2.5,2.5] AddRowVector [11.0,22.0; 13.0,24.0] AddColVector [2.0,3.0; 2.0,3.0] MulRowVector [10.0,40.0; 30.0,80.0] MulColVector [1.0,2.0; -3.0,-4.0] Abs [2.3,0.0,0.5,9.0] Sqrt [1.5,0.0,0.7,3.0] Cbrt [2.0,-3.0] Floor and Ceil [-3.0,0.0,0.0,9.0,-2.0,0.0,1.0,9.0] Round and Trunc [-2.0,0.0,1.0,9.0,-2.0,0.0,0.0,9.0] Sign [-1.0,0.0,1.0,1.0] Square [5.1,0.0,0.3,81.0] Reciprocal [0.5,0.3] PowElem [8.0,27.0] Clip [0.0,0.0,0.5,1.0] MaxScalar and MinScalar [0.0,0.0,0.5,9.0,-2.3,0.0,0.0,0.0] Chop [0.0,1.0] 1.000 0.000 2.000 3.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 10 24 2.5 1 4 3 0 3 1 1 9 1.667 1.118 2.5 RowSums [3.0; 7.0] ColSums [4.0,6.0] RowMeans [1.5; 3.5] ColMeans [2.0,3.0] RowMins and RowMaxs [1.0,2.0; 3.0,4.0] ColMins and ColMaxs [1.0,2.0; 3.0,4.0] cumulative [1.0,3.0,1.0,2.0; 3.0,7.0,4.0,6.0] CumProdRows [1.0,2.0; 3.0,12.0] 2 true false true 5.477 6 7 4 5.465 5.831 2 1.414 Normalize [0.6,0.8] 0.45 0.89 0.60 0.80 0.32 0.45 0.95 0.89 21.000 21.000 14.933

`Adjugate`

func Adjugate(x &: Matrix): Matrix

The adjugate matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`IsDiagonallyDominant`

func IsDiagonallyDominant(a &: Matrix): bool

It is diagonally dominant; the Jacobi and Gauss Seidel methods then converge.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveJacobi`

func SolveJacobi(a &: Matrix, b &: Matrix, iters: int, tol: f64): IterResult

The Jacobi method.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveGaussSeidel`

func SolveGaussSeidel(a &: Matrix, b &: Matrix, iters: int, tol: f64): IterResult

The Gauss Seidel method.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveSOR`

func SolveSOR(a &: Matrix, b &: Matrix, omega: f64, iters: int, tol: f64): IterResult

Successive over relaxation with a factor omega.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveCG`

func SolveCG(a &: Matrix, b &: Matrix, iters: int, tol: f64): IterResult

Conjugate gradient for a symmetric positive definite matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`PowerIteration`

func PowerIteration(a &: Matrix, iters: int): Matrix

Power iteration: the dominant eigenvector and, in the last element, its eigenvalue.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Rayleigh`

func Rayleigh(a &: Matrix, x &: Matrix): f64

The Rayleigh quotient; the eigenvalue estimate for a vector.

The example for this item is shown with the ones above: matrix-Eigen.salam

`ShiftedInverseIteration`

func ShiftedInverseIteration(a &: Matrix, shift: f64, iters: int): f64

The eigenvalue nearest the shift.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Cholesky`

func Cholesky(x &: Matrix): Matrix

The Cholesky decomposition of a symmetric positive definite matrix, the L whose product with its transpose is A.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveSPD`

func SolveSPD(a &: Matrix, b &: Matrix): Matrix

Solves a symmetric positive definite system.

The example for this item is shown with the ones above: matrix-Solve.salam

`CholeskySolve`

func CholeskySolve(l &: Matrix, b &: Matrix): Matrix

Solves it with a ready L.

The example for this item is shown with the ones above: matrix-Solve.salam

`InverseSPD`

func InverseSPD(x &: Matrix): Matrix

The inverse of a positive definite matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`DetSPD`

func DetSPD(x &: Matrix): f64

The determinant of a positive definite matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`LogDetSPD`

func LogDetSPD(x &: Matrix): f64

The logarithm of the determinant; for the multivariate normal distribution.

The example for this item is shown with the ones above: matrix-Solve.salam

`LDLDecompose`

func LDLDecompose(x &: Matrix): LDL

The LDL decomposition of a symmetric matrix.

The example for this item is shown with the ones above: matrix-Solve.salam

`LDLSolve`

func LDLSolve(f &: LDL, b &: Matrix): Matrix

Solves it with the LDL decomposition.

The example for this item is shown with the ones above: matrix-Solve.salam

`Inertia`

func Inertia(x &: Matrix): Matrix

The number of positive, negative and zero eigenvalues.

The example for this item is shown with the ones above: matrix-Solve.salam

`Zeros`

func Zeros(rows: int, cols: int): Matrix

A matrix of zeros.

The example for this item is shown with the ones above: matrix-Create.salam

`Ones`

func Ones(rows: int, cols: int): Matrix

A matrix of ones.

The example for this item is shown with the ones above: matrix-Create.salam

`Full`

func Full(rows: int, cols: int, x: f64): Matrix

A matrix filled with one value.

The example for this item is shown with the ones above: matrix-Create.salam

`Eye`

func Eye(n: int): Matrix

The n by n identity matrix.

The example for this item is shown with the ones above: matrix-Create.salam

`EyeOffset`

func EyeOffset(rows: int, cols: int, k: int): Matrix

Ones on the kth diagonal.

The example for this item is shown with the ones above: matrix-Create.salam

`FromArray`

func FromArray(rows: int, cols: int, data &: Vector<f64>): Matrix

A matrix from a row count, a column count and the data in row order.

The example for this item is shown with the ones above: matrix-Create.salam

`RowVector`

func RowVector(data &: Vector<f64>): Matrix

A row vector.

The example for this item is shown with the ones above: matrix-Create.salam

`ColVector`

func ColVector(data &: Vector<f64>): Matrix

A column vector.

The example for this item is shown with the ones above: matrix-Create.salam

`Diag`

func Diag(data &: Vector<f64>): Matrix

A diagonal matrix from a set of values.

The example for this item is shown with the ones above: matrix-Create.salam

`Clone`

func Clone(x &: Matrix): Matrix

An independent copy.

The example for this item is shown with the ones above: matrix-Create.salam

`CopyInto`

func CopyInto(dst &: Matrix, src &: Matrix)

Copies the data into a matrix of the same shape.

The example for this item is shown with the ones above: matrix-Create.salam

`Free`

func Free(x &: Matrix)

Frees it.

The example for this item is shown with the ones above: matrix-Create.salam

`SetAll`

func SetAll(x &: Matrix, v: f64)

Sets everything to one value.

The example for this item is shown with the ones above: matrix-Create.salam

`Arange`

func Arange(start: f64, stop: f64, step: f64): Matrix

A row vector from the start up to but not including the end, by a step.

The example for this item is shown with the ones above: matrix-Create.salam

`Linspace`

func Linspace(start: f64, stop: f64, num: int): Matrix

n evenly spaced numbers from the start to the end.

The example for this item is shown with the ones above: matrix-Create.salam

`Logspace`

func Logspace(start: f64, stop: f64, num: int, base: f64): Matrix

n logarithmically spaced numbers; from the base to the power of the start to the base to the power of the end.

The example for this item is shown with the ones above: matrix-Create.salam

`RowEchelon`

func RowEchelon(x &: Matrix): Matrix

The row echelon form.

The example for this item is shown with the ones above: matrix-Solve.salam

`RREF`

func RREF(x &: Matrix): Matrix

The reduced row echelon form.

The example for this item is shown with the ones above: matrix-Solve.salam

`PivotColumns`

func PivotColumns(x &: Matrix): Matrix

The indices of the pivot columns.

The example for this item is shown with the ones above: matrix-Solve.salam

`RankExact`

func RankExact(x &: Matrix): int

The rank by Gaussian elimination.

The example for this item is shown with the ones above: matrix-Solve.salam

`Nullity`

func Nullity(x &: Matrix): int

The dimension of the null space.

The example for this item is shown with the ones above: matrix-Solve.salam

`NullSpaceExact`

func NullSpaceExact(x &: Matrix): Matrix

A basis of the null space from the echelon form.

The example for this item is shown with the ones above: matrix-Solve.salam

`ColumnSpaceExact`

func ColumnSpaceExact(x &: Matrix): Matrix

A basis of the column space from the pivot columns.

The example for this item is shown with the ones above: matrix-Solve.salam

`SolveGeneral`

func SolveGeneral(a &: Matrix, b &: Matrix): Matrix

One solution of the system, even when the matrix is singular.

The example for this item is shown with the ones above: matrix-Solve.salam

`IsConsistent`

func IsConsistent(a &: Matrix, b &: Matrix): bool

The system has a solution.

The example for this item is shown with the ones above: matrix-Solve.salam

`EigSym`

func EigSym(x &: Matrix): EigenSym

The eigendecomposition of a symmetric matrix by the Jacobi method.

The example for this item is shown with the ones above: matrix-Eigen.salam

`EigValsSym`

func EigValsSym(x &: Matrix): Matrix

The eigenvalues alone.

The example for this item is shown with the ones above: matrix-Eigen.salam

`EigReconstruct`

func EigReconstruct(f &: EigenSym): Matrix

Rebuilds the matrix from the eigenvectors and eigenvalues.

The example for this item is shown with the ones above: matrix-Eigen.salam

`SpectralRadiusSym`

func SpectralRadiusSym(x &: Matrix): f64

The spectral radius of a symmetric matrix.

The example for this item is shown with the ones above: matrix-Eigen.salam

`PowSym`

func PowSym(x &: Matrix, p: f64): Matrix

A float power of a symmetric matrix.

The example for this item is shown with the ones above: matrix-Eigen.salam

`SqrtSym`

func SqrtSym(x &: Matrix): Matrix

The square root of a symmetric positive definite matrix.

The example for this item is shown with the ones above: matrix-Eigen.salam

`HStack`

func HStack(x &: Matrix, y &: Matrix): Matrix

Two matrices side by side.

matrix-Shape.salam

import matrix

func build(rows: int, cols: int, numbers: str): matrix.Matrix:
    mut data := Vector {} as Vector<f64>
    each number in numbers.split(" "):
        data.push(number.to_float())
    end
    ret matrix.FromArray(rows, cols, data)
end

func show(name: str, m &: matrix.Matrix):
    println name, matrix.ToRepr(m, 1)
end

func main:
    mut a := build(2, 3, "1 2 3 4 5 6")
    mut square := build(3, 3, "1 2 3 4 5 6 7 8 9")
    show("Reshape", matrix.Reshape(a, 3, 2))
    show("Flatten", matrix.Flatten(a))
    show("FlattenCol", matrix.FlattenCol(a))
    mut row := matrix.Row(a, 1)
    mut col := matrix.Col(a, 2)
    show("Row", row)
    show("Col", col)
    mut b := matrix.Clone(a)
    matrix.SetRow(b, 0, row)
    matrix.SetCol(b, 0, col)
    show("SetRow and SetCol", b)
    show("SubMatrix", matrix.SubMatrix(square, 1, 1, 2, 2))
    show("Minor", matrix.Minor(square, 0, 0))
    show("DeleteRow", matrix.DeleteRow(square, 1))
    show("DeleteCol", matrix.DeleteCol(square, 1))
    show("InsertRow", matrix.InsertRow(a, 1, matrix.Row(a, 0)))
    show("InsertCol", matrix.InsertCol(a, 0, matrix.Col(a, 2)))
    matrix.SwapRows(b, 0, 1)
    matrix.SwapCols(b, 0, 2)
    show("swapped", b)
    show("Diagonal", matrix.Diagonal(square))
    show("DiagonalOffset", matrix.DiagonalOffset(square, 1))
    mut c := matrix.Clone(square)
    matrix.SetDiagonal(c, matrix.Ones(3, 1))
    matrix.ShiftDiagonal(c, 10.0)
    show("new diagonal", matrix.Diagonal(c))
    show("HStack", matrix.HStack(a, a))
    show("VStack", matrix.VStack(a, a))
    show("Augment", matrix.Augment(square, matrix.Eye(3)))
    show("BlockDiag", matrix.BlockDiag(matrix.Eye(1), matrix.Full(1, 1, 2.0)))
    show("Triu", matrix.Triu(square, 0))
    show("Tril", matrix.Tril(square, -1))
    show("SymmetrizeUpper", matrix.SymmetrizeUpper(square))
    show("SymPart", matrix.SymPart(square))
    show("SkewPart", matrix.SkewPart(square))
    show("FlipVertical", matrix.FlipVertical(a))
    show("FlipHorizontal", matrix.FlipHorizontal(a))
    show("Rot90", matrix.Rot90(a, 1))
    show("Roll", matrix.Roll(a, 0, 1))
    show("Tile", matrix.Tile(matrix.Eye(1), 2, 2))
    show("Pad", matrix.Pad(matrix.Eye(1), 1, 0, 0, 1, 9.0))
end

OutputReshape [1.0,2.0; 3.0,4.0; 5.0,6.0] Flatten [1.0,2.0,3.0,4.0,5.0,6.0] FlattenCol [1.0; 2.0; 3.0; 4.0; 5.0; 6.0] Row [4.0,5.0,6.0] Col [3.0; 6.0] SetRow and SetCol [3.0,5.0,6.0; 6.0,5.0,6.0] SubMatrix [5.0,6.0; 8.0,9.0] Minor [5.0,6.0; 8.0,9.0] DeleteRow [1.0,2.0,3.0; 7.0,8.0,9.0] DeleteCol [1.0,3.0; 4.0,6.0; 7.0,9.0] InsertRow [1.0,2.0,3.0; 1.0,2.0,3.0; 4.0,5.0,6.0] InsertCol [3.0,1.0,2.0,3.0; 6.0,4.0,5.0,6.0] swapped [6.0,5.0,6.0; 6.0,5.0,3.0] Diagonal [1.0,5.0,9.0] DiagonalOffset [2.0,6.0] new diagonal [11.0,11.0,11.0] HStack [1.0,2.0,3.0,1.0,2.0,3.0; 4.0,5.0,6.0,4.0,5.0,6.0] VStack [1.0,2.0,3.0; 4.0,5.0,6.0; 1.0,2.0,3.0; 4.0,5.0,6.0] Augment [1.0,2.0,3.0,1.0,0.0,0.0; 4.0,5.0,6.0,0.0,1.0,0.0; 7.0,8.0,9.0,0.0,0.0,1.0] BlockDiag [1.0,0.0; 0.0,2.0] Triu [1.0,2.0,3.0; 0.0,5.0,6.0; 0.0,0.0,9.0] Tril [0.0,0.0,0.0; 4.0,0.0,0.0; 7.0,8.0,0.0] SymmetrizeUpper [1.0,2.0,3.0; 2.0,5.0,6.0; 3.0,6.0,9.0] SymPart [1.0,3.0,5.0; 3.0,5.0,7.0; 5.0,7.0,9.0] SkewPart [0.0,-1.0,-2.0; 1.0,0.0,-1.0; 2.0,1.0,0.0] FlipVertical [4.0,5.0,6.0; 1.0,2.0,3.0] FlipHorizontal [3.0,2.0,1.0; 6.0,5.0,4.0] Rot90 [3.0,6.0; 2.0,5.0; 1.0,4.0] Roll [3.0,1.0,2.0; 6.0,4.0,5.0] Tile [1.0,1.0; 1.0,1.0] Pad [9.0,9.0; 1.0,9.0]

`VStack`

func VStack(x &: Matrix, y &: Matrix): Matrix

Two matrices one above the other.

The example for this item is shown with the ones above: matrix-Shape.salam

`Augment`

func Augment(a &: Matrix, b &: Matrix): Matrix

The augmented matrix, for solving a system.

The example for this item is shown with the ones above: matrix-Shape.salam

`BlockDiag`

func BlockDiag(x &: Matrix, y &: Matrix): Matrix

A block diagonal matrix.

The example for this item is shown with the ones above: matrix-Shape.salam

`Triu`

func Triu(x &: Matrix, k: int): Matrix

The part above diagonal k.

The example for this item is shown with the ones above: matrix-Shape.salam

`Tril`

func Tril(x &: Matrix, k: int): Matrix

The part below diagonal k.

The example for this item is shown with the ones above: matrix-Shape.salam

`SymmetrizeUpper`

func SymmetrizeUpper(x &: Matrix): Matrix

Mirrors the upper half into the lower.

The example for this item is shown with the ones above: matrix-Shape.salam

`SymPart`

func SymPart(x &: Matrix): Matrix

The symmetric part, half of A and its transpose.

The example for this item is shown with the ones above: matrix-Shape.salam

`SkewPart`

func SkewPart(x &: Matrix): Matrix

The skew symmetric part.

The example for this item is shown with the ones above: matrix-Shape.salam

`FlipVertical`

func FlipVertical(x &: Matrix): Matrix

Reverses the rows.

The example for this item is shown with the ones above: matrix-Shape.salam

`FlipHorizontal`

func FlipHorizontal(x &: Matrix): Matrix

Reverses the columns.

The example for this item is shown with the ones above: matrix-Shape.salam

`Rot90`

func Rot90(x &: Matrix, k: int): Matrix

k ninety degree turns anticlockwise.

The example for this item is shown with the ones above: matrix-Shape.salam

`Roll`

func Roll(x &: Matrix, dr: int, dc: int): Matrix

Rotates the rows and columns around.

The example for this item is shown with the ones above: matrix-Shape.salam

`Tile`

func Tile(x &: Matrix, ry: int, rx: int): Matrix

Repeats the matrix across rows and columns.

The example for this item is shown with the ones above: matrix-Shape.salam

`Pad`

func Pad(x &: Matrix, top: int, bottom: int, left: int, right: int, v: f64): Matrix

Adds a border of a value around the matrix.

The example for this item is shown with the ones above: matrix-Shape.salam

`Sum`

func Sum(x &: Matrix): f64

The sum of everything.

The example for this item is shown with the ones above: matrix-Math.salam

`Prod`

func Prod(x &: Matrix): f64

The product of everything.

The example for this item is shown with the ones above: matrix-Math.salam

`Mean`

func Mean(x &: Matrix): f64

The mean.

The example for this item is shown with the ones above: matrix-Math.salam

`Min`

func Min(x &: Matrix): f64

The smallest.

The example for this item is shown with the ones above: matrix-Create.salam

`Max`

func Max(x &: Matrix): f64

The greatest.

The example for this item is shown with the ones above: matrix-Math.salam

`Range`

func Range(x &: Matrix): f64

The greatest minus the smallest.

The example for this item is shown with the ones above: matrix-Math.salam

`ArgMin`

func ArgMin(x &: Matrix): int

Where the minimum is, in row order.

The example for this item is shown with the ones above: matrix-Math.salam

`ArgMax`

func ArgMax(x &: Matrix): int

Where the maximum is.

The example for this item is shown with the ones above: matrix-Math.salam

`IndexRow`

func IndexRow(x &: Matrix, k: int): int

The row of place k in row order.

The example for this item is shown with the ones above: matrix-Math.salam

`IndexCol`

func IndexCol(x &: Matrix, k: int): int

The column of place k.

The example for this item is shown with the ones above: matrix-Math.salam

`MaxAbs`

func MaxAbs(x &: Matrix): f64

The greatest absolute value.

The example for this item is shown with the ones above: matrix-Math.salam

`Var`

func Var(x &: Matrix, sample: bool): f64

The variance; the sample one with sample.

The example for this item is shown with the ones above: matrix-Math.salam

`Std`

func Std(x &: Matrix, sample: bool): f64

The standard deviation.

The example for this item is shown with the ones above: matrix-Math.salam

`Median`

func Median(x &: Matrix): f64

The median.

The example for this item is shown with the ones above: matrix-Math.salam

`RowSums`

func RowSums(x &: Matrix): Matrix

The sum of each row.

The example for this item is shown with the ones above: matrix-Math.salam

`ColSums`

func ColSums(x &: Matrix): Matrix

The sum of each column.

The example for this item is shown with the ones above: matrix-Math.salam

`RowMeans`

func RowMeans(x &: Matrix): Matrix

The mean of each row.

The example for this item is shown with the ones above: matrix-Math.salam

`ColMeans`

func ColMeans(x &: Matrix): Matrix

The mean of each column.

The example for this item is shown with the ones above: matrix-Math.salam

`RowMins`

func RowMins(x &: Matrix): Matrix

The smallest of each row.

The example for this item is shown with the ones above: matrix-Math.salam

`RowMaxs`

func RowMaxs(x &: Matrix): Matrix

The greatest of each row.

The example for this item is shown with the ones above: matrix-Math.salam

`ColMins`

func ColMins(x &: Matrix): Matrix

The smallest of each column.

The example for this item is shown with the ones above: matrix-Math.salam

`ColMaxs`

func ColMaxs(x &: Matrix): Matrix

The greatest of each column.

The example for this item is shown with the ones above: matrix-Math.salam

`CumSumRows`

func CumSumRows(x &: Matrix): Matrix

The cumulative sum along each row.

The example for this item is shown with the ones above: matrix-Math.salam

`CumSumCols`

func CumSumCols(x &: Matrix): Matrix

The cumulative sum along each column.

The example for this item is shown with the ones above: matrix-Math.salam

`CumProdRows`

func CumProdRows(x &: Matrix): Matrix

The cumulative product along each row.

The example for this item is shown with the ones above: matrix-Math.salam

`CountNonZero`

func CountNonZero(x &: Matrix): int

The number of non zero elements.

The example for this item is shown with the ones above: matrix-Math.salam

`Any`

func Any(x &: Matrix): bool

At least one element is non zero.

The example for this item is shown with the ones above: matrix-Create.salam

`All`

func All(x &: Matrix): bool

Every element is non zero.

The example for this item is shown with the ones above: matrix-Create.salam

`AllFinite`

func AllFinite(x &: Matrix): bool

No element is NaN or infinite.

The example for this item is shown with the ones above: matrix-Math.salam

`NewRng`

func NewRng(seed: i64): Rng

A generator from a seed.

The example for this item is shown with the ones above: matrix-Stats.salam

`Hilbert`

func Hilbert(n: int): Matrix

The Hilbert matrix; the famous example of an ill conditioned matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`HilbertInverse`

func HilbertInverse(n: int): Matrix

The exact inverse of the Hilbert matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`Vandermonde`

func Vandermonde(x &: Matrix, cols: int): Matrix

The Vandermonde matrix; for polynomial interpolation.

The example for this item is shown with the ones above: matrix-Stats.salam

`Toeplitz`

func Toeplitz(first_col &: Matrix, first_row &: Matrix): Matrix

A Toeplitz matrix from its first column and row.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Hankel`

func Hankel(first_col &: Matrix, last_row &: Matrix): Matrix

A Hankel matrix from its first column and last row.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Circulant`

func Circulant(c &: Matrix): Matrix

A circulant matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`Companion`

func Companion(coeffs &: Matrix): Matrix

The companion matrix of a polynomial with coefficients from degree zero; its eigenvalues are the roots.

The example for this item is shown with the ones above: matrix-Stats.salam

`Pascal`

func Pascal(n: int): Matrix

The Pascal matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`Rotation2D`

func Rotation2D(theta: f64): Matrix

A two dimensional rotation by an angle in radians.

The example for this item is shown with the ones above: matrix-Stats.salam

`Rotation3D`

func Rotation3D(axis: int, theta: f64): Matrix

A three dimensional rotation about axis 0, 1 or 2.

The example for this item is shown with the ones above: matrix-Stats.salam

`Householder`

func Householder(v &: Matrix): Matrix

A Householder reflection.

The example for this item is shown with the ones above: matrix-Stats.salam

`Givens`

func Givens(n: int, p: int, q: int, theta: f64): Matrix

A Givens rotation in the p and q plane.

The example for this item is shown with the ones above: matrix-Stats.salam

`Permutation`

func Permutation(perm &: Matrix): Matrix

A permutation matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`Random`

func Random(rows: int, cols: int, lo: f64, hi: f64, seed: i64): Matrix

A uniform random matrix from a seed.

The example for this item is shown with the ones above: matrix-Eigen.salam

`RandomNormal`

func RandomNormal(rows: int, cols: int, seed: i64): Matrix

A normal random matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`RandomSymmetric`

func RandomSymmetric(n: int, seed: i64): Matrix

A random symmetric matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`RandomSPD`

func RandomSPD(n: int, seed: i64): Matrix

A random positive definite matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`RandomOrthogonal`

func RandomOrthogonal(n: int, seed: i64): Matrix

A random orthogonal matrix.

The example for this item is shown with the ones above: matrix-Stats.salam

`Add`

func Add(x &: Matrix, y &: Matrix): Matrix

Addition.

The example for this item is shown with the ones above: matrix-Math.salam

`Sub`

func Sub(x &: Matrix, y &: Matrix): Matrix

Subtraction.

The example for this item is shown with the ones above: matrix-Math.salam

`MulElem`

func MulElem(x &: Matrix, y &: Matrix): Matrix

Elementwise multiplication.

The example for this item is shown with the ones above: matrix-Math.salam

`DivElem`

func DivElem(x &: Matrix, y &: Matrix): Matrix

Elementwise division.

The example for this item is shown with the ones above: matrix-Math.salam

`Neg`

func Neg(x &: Matrix): Matrix

The negation.

The example for this item is shown with the ones above: matrix-Math.salam

`AddScalar`

func AddScalar(x &: Matrix, s: f64): Matrix

Addition of a number.

The example for this item is shown with the ones above: matrix-Math.salam

`SubScalar`

func SubScalar(x &: Matrix, s: f64): Matrix

Subtraction of a number.

The example for this item is shown with the ones above: matrix-Math.salam

`Scale`

func Scale(x &: Matrix, s: f64): Matrix

Multiplication by a number.

The example for this item is shown with the ones above: matrix-Math.salam

`DivScalar`

func DivScalar(x &: Matrix, s: f64): Matrix

Division by a number.

The example for this item is shown with the ones above: matrix-Math.salam

`AddInPlace`

func AddInPlace(dst &: Matrix, src &: Matrix)

Adds in place.

The example for this item is shown with the ones above: matrix-Math.salam

`SubInPlace`

func SubInPlace(dst &: Matrix, src &: Matrix)

Subtracts in place.

The example for this item is shown with the ones above: matrix-Math.salam

`MulElemInPlace`

func MulElemInPlace(dst &: Matrix, src &: Matrix)

Multiplies elementwise in place.

The example for this item is shown with the ones above: matrix-Math.salam

`ScaleInPlace`

func ScaleInPlace(dst &: Matrix, s: f64)

Multiplies by a number in place.

The example for this item is shown with the ones above: matrix-Math.salam

`AddScalarInPlace`

func AddScalarInPlace(dst &: Matrix, s: f64)

Adds a number in place.

The example for this item is shown with the ones above: matrix-Math.salam

`NegInPlace`

func NegInPlace(dst &: Matrix)

Negates in place.

The example for this item is shown with the ones above: matrix-Math.salam

`AddScaled`

func AddScaled(dst &: Matrix, src &: Matrix, alpha: f64)

Adds alpha times the source to the destination.

The example for this item is shown with the ones above: matrix-Math.salam

`Combine`

func Combine(x &: Matrix, a: f64, y &: Matrix, b: f64): Matrix

The linear combination a times x plus b times y.

The example for this item is shown with the ones above: matrix-Math.salam

`MaxElem`

func MaxElem(x &: Matrix, y &: Matrix): Matrix

The elementwise greater of two matrices.

The example for this item is shown with the ones above: matrix-Math.salam

`MinElem`

func MinElem(x &: Matrix, y &: Matrix): Matrix

The elementwise smaller.

The example for this item is shown with the ones above: matrix-Math.salam

`AddRowVector`

func AddRowVector(x &: Matrix, row &: Matrix): Matrix

Adds a row vector to every row.

The example for this item is shown with the ones above: matrix-Math.salam

`AddColVector`

func AddColVector(x &: Matrix, col &: Matrix): Matrix

Adds a column vector to every column.

The example for this item is shown with the ones above: matrix-Math.salam

`MulRowVector`

func MulRowVector(x &: Matrix, row &: Matrix): Matrix

Multiplies every row elementwise by a row vector.

The example for this item is shown with the ones above: matrix-Math.salam

`MulColVector`

func MulColVector(x &: Matrix, col &: Matrix): Matrix

Multiplies every column by a column vector.

The example for this item is shown with the ones above: matrix-Math.salam

`SVDecompose`

func SVDecompose(x &: Matrix): SVD

The singular value decomposition.

The example for this item is shown with the ones above: matrix-Eigen.salam

`SingularValues`

func SingularValues(x &: Matrix): Matrix

The singular values alone.

The example for this item is shown with the ones above: matrix-Eigen.salam

`MatrixRank`

func MatrixRank(x &: Matrix, tol: f64): int

The rank, within a tolerance.

The example for this item is shown with the ones above: matrix-Solve.salam

`PseudoInverse`

func PseudoInverse(x &: Matrix, tol: f64): Matrix

The Moore Penrose pseudoinverse.

The example for this item is shown with the ones above: matrix-Eigen.salam

`NullSpace`

func NullSpace(x &: Matrix, tol: f64): Matrix

An orthogonal basis of the null space.

The example for this item is shown with the ones above: matrix-Solve.salam

`ColumnSpace`

func ColumnSpace(x &: Matrix, tol: f64): Matrix

An orthogonal basis of the column space.

The example for this item is shown with the ones above: matrix-Solve.salam

`LowRankApprox`

func LowRankApprox(x &: Matrix, k: int): Matrix

The best rank k approximation; for compression.

The example for this item is shown with the ones above: matrix-Eigen.salam

`NormFrobenius`

func NormFrobenius(x &: Matrix): f64

The Frobenius norm, the square root of the sum of the squares.

The example for this item is shown with the ones above: matrix-Math.salam

`Norm1`

func Norm1(x &: Matrix): f64

The one norm, the greatest absolute column sum.

The example for this item is shown with the ones above: matrix-Math.salam

`NormInf`

func NormInf(x &: Matrix): f64

The infinity norm, the greatest absolute row sum.

The example for this item is shown with the ones above: matrix-Math.salam

`NormMax`

func NormMax(x &: Matrix): f64

The greatest absolute element.

The example for this item is shown with the ones above: matrix-Math.salam

`Norm2`

func Norm2(x &: Matrix): f64

The two norm, the largest singular value.

The example for this item is shown with the ones above: matrix-Math.salam

`NormNuclear`

func NormNuclear(x &: Matrix): f64

The nuclear norm, the sum of the singular values.

The example for this item is shown with the ones above: matrix-Math.salam

`VecNorm`

func VecNorm(x &: Matrix, p: f64): f64

The p norm of a vector.

The example for this item is shown with the ones above: matrix-Math.salam

`Distance`

func Distance(x &: Matrix, y &: Matrix): f64

The Euclidean distance.

The example for this item is shown with the ones above: matrix-Math.salam

`Normalize`

func Normalize(x &: Matrix): Matrix

The vector scaled to length one.

The example for this item is shown with the ones above: matrix-Math.salam

`NormalizeRows`

func NormalizeRows(x &: Matrix): Matrix

Every row scaled to length one.

The example for this item is shown with the ones above: matrix-Math.salam

`NormalizeCols`

func NormalizeCols(x &: Matrix): Matrix

Every column scaled to length one.

The example for this item is shown with the ones above: matrix-Math.salam

`Cond1`

func Cond1(x &: Matrix): f64

The condition number in the one norm.

The example for this item is shown with the ones above: matrix-Math.salam

`CondInf`

func CondInf(x &: Matrix): f64

The condition number in the infinity norm.

The example for this item is shown with the ones above: matrix-Math.salam

`Cond2`

func Cond2(x &: Matrix): f64

The condition number in the two norm; a large one means solving the system is unstable.

The example for this item is shown with the ones above: matrix-Math.salam

`QRDecompose`

func QRDecompose(x &: Matrix): QR

The QR decomposition by Householder reflection.

The example for this item is shown with the ones above: matrix-Solve.salam

`QRGetQ`

func QRGetQ(f &: QR): Matrix

The orthogonal matrix Q.

The example for this item is shown with the ones above: matrix-Solve.salam

`QRGetR`

func QRGetR(f &: QR): Matrix

The upper triangular matrix R.

The example for this item is shown with the ones above: matrix-Solve.salam

`LeastSquares`

func LeastSquares(a &: Matrix, b &: Matrix): Matrix

The least squares solution of an overdetermined system; fitting a line, for instance.

The example for this item is shown with the ones above: matrix-Solve.salam

`QRSolve`

func QRSolve(f &: QR, b &: Matrix): Matrix

Solves it with the QR decomposition.

The example for this item is shown with the ones above: matrix-Solve.salam

`Residual`

func Residual(a &: Matrix, x &: Matrix, b &: Matrix): Matrix

The residual, AX minus B.

The example for this item is shown with the ones above: matrix-Solve.salam

`Orthonormalize`

func Orthonormalize(x &: Matrix): Matrix

Makes the columns orthonormal.

The example for this item is shown with the ones above: matrix-Solve.salam

`ToStringPrec`

func ToStringPrec(x &: Matrix, prec: int): str

Multi line text with a number of decimal places.

The example for this item is shown with the ones above: matrix-Create.salam

`ToString`

func ToString(x &: Matrix): str

Multi line text.

The example for this item is shown with the ones above: matrix-Create.salam

`ShapeString`

func ShapeString(x &: Matrix): str

The shape, such as 2x3.

The example for this item is shown with the ones above: matrix-Create.salam

`Print`

func Print(x &: Matrix)

Prints the matrix.

The example for this item is shown with the ones above: matrix-Create.salam

`PrintPrec`

func PrintPrec(x &: Matrix, prec: int)

Prints it with a number of decimal places.

The example for this item is shown with the ones above: matrix-Create.salam

`PrintLabeled`

func PrintLabeled(label: str, x &: Matrix)

Prints it with a label and its shape.

The example for this item is shown with the ones above: matrix-Create.salam

`ToCSV`

func ToCSV(x &: Matrix, prec: int): str

CSV text.

The example for this item is shown with the ones above: matrix-Create.salam

`ToRepr`

func ToRepr(x &: Matrix, prec: int): str

One line text such as [1,2; 3,4].

The example for this item is shown with the ones above: matrix-Create.salam

`Transpose`

func Transpose(x &: Matrix): Matrix

The transpose.

The example for this item is shown with the ones above: matrix-Math.salam

`MatMul`

func MatMul(x &: Matrix, y &: Matrix): Matrix

Matrix multiplication.

The example for this item is shown with the ones above: matrix-Math.salam

`TransposeInto`

func TransposeInto(x &: Matrix, out &: Matrix): bool

The transpose into a ready matrix.

The example for this item is shown with the ones above: matrix-Math.salam

`MatMulInto`

func MatMulInto(x &: Matrix, y &: Matrix, out &: Matrix): bool

The product into a ready matrix.

The example for this item is shown with the ones above: matrix-Math.salam

`TransposeMul`

func TransposeMul(x &: Matrix, y &: Matrix): Matrix

The transpose of x times y.

The example for this item is shown with the ones above: matrix-Math.salam

`MulTranspose`

func MulTranspose(x &: Matrix, y &: Matrix): Matrix

x times the transpose of y.

The example for this item is shown with the ones above: matrix-Math.salam

`Gram`

func Gram(x &: Matrix): Matrix

The Gram matrix, the transpose of x times x.

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`MatVec`

func MatVec(a &: Matrix, x &: Matrix): Matrix

A matrix times a column vector.

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`VecMat`

func VecMat(x &: Matrix, a &: Matrix): Matrix

A row vector times a matrix.

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`Dot`

func Dot(x &: Matrix, y &: Matrix): f64

The dot product of two vectors.

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`Outer`

func Outer(u &: Matrix, v &: Matrix): Matrix

The outer product.

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`Kron`

func Kron(x &: Matrix, y &: Matrix): Matrix

The Kronecker product.

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`MatPow`

func MatPow(x &: Matrix, k: int): Matrix

An integer power of a square matrix.

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`Trace`

func Trace(x &: Matrix): f64

The trace, the sum of the diagonal.

The example for this item is shown with the ones above: matrix-Math.salam

`Cross3`

func Cross3(u &: Matrix, v &: Matrix): Matrix

The three dimensional cross product.

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`IsSquare`

func IsSquare(x &: Matrix): bool

The matrix is square.

The example for this item is shown with the ones above: matrix-Create.salam

`AllClose`

func AllClose(x &: Matrix, y &: Matrix, tol: f64): bool

The two matrices are equal within a tolerance.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsZero`

func IsZero(x &: Matrix, tol: f64): bool

The matrix is zero.

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`IsIdentity`

func IsIdentity(x &: Matrix, tol: f64): bool

The matrix is the identity.

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`IsSymmetric`

func IsSymmetric(x &: Matrix, tol: f64): bool

The matrix is symmetric.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsSkewSymmetric`

func IsSkewSymmetric(x &: Matrix, tol: f64): bool

The matrix is skew symmetric.

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`IsDiagonal`

func IsDiagonal(x &: Matrix, tol: f64): bool

The matrix is diagonal.

The example for this item is shown with the ones above: matrix-Solve.salam

`IsUpperTriangular`

func IsUpperTriangular(x &: Matrix, tol: f64): bool

The matrix is upper triangular.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsLowerTriangular`

func IsLowerTriangular(x &: Matrix, tol: f64): bool

The matrix is lower triangular.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsBanded`

func IsBanded(x &: Matrix, lower: int, upper: int, tol: f64): bool

Every element sits within a band below and above the diagonal.

The example for this item is shown with the ones above: matrix-Eigen.salam

`LowerBandwidth`

func LowerBandwidth(x &: Matrix, tol: f64): int

The bandwidth below the diagonal.

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`UpperBandwidth`

func UpperBandwidth(x &: Matrix, tol: f64): int

The bandwidth above the diagonal.

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`IsOrthogonal`

func IsOrthogonal(x &: Matrix, tol: f64): bool

The matrix is orthogonal.

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`IsNormal`

func IsNormal(x &: Matrix, tol: f64): bool

The matrix is normal, meaning it commutes with its transpose.

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`IsIdempotent`

func IsIdempotent(x &: Matrix, tol: f64): bool

The matrix is idempotent, meaning its square is itself.

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`IsInvolutory`

func IsInvolutory(x &: Matrix, tol: f64): bool

The matrix is its own inverse.

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`IsNilpotent`

func IsNilpotent(x &: Matrix, tol: f64): bool

The matrix is nilpotent.

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`IsPositiveDefinite`

func IsPositiveDefinite(x &: Matrix, tol: f64): bool

The matrix is positive definite.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsPositiveSemiDefinite`

func IsPositiveSemiDefinite(x &: Matrix, tol: f64): bool

The matrix is positive semidefinite.

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`IsSingular`

func IsSingular(x &: Matrix): bool

The matrix is singular.

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`IsInvertible`

func IsInvertible(x &: Matrix): bool

The matrix is invertible.

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`IsRowStochastic`

func IsRowStochastic(x &: Matrix, tol: f64): bool

The matrix is row stochastic, meaning every row sums to one.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsColStochastic`

func IsColStochastic(x &: Matrix, tol: f64): bool

The matrix is column stochastic.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsDoublyStochastic`

func IsDoublyStochastic(x &: Matrix, tol: f64): bool

The matrix is doubly stochastic.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsPermutation`

func IsPermutation(x &: Matrix, tol: f64): bool

The matrix is a permutation matrix.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsToeplitz`

func IsToeplitz(x &: Matrix, tol: f64): bool

The matrix is Toeplitz, meaning every diagonal is constant.

The example for this item is shown with the ones above: matrix-Eigen.salam

`IsHankel`

func IsHankel(x &: Matrix, tol: f64): bool

The matrix is Hankel, meaning every antidiagonal is constant.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Density`

func Density(x &: Matrix): f64

The fraction of non zero elements.

The example for this item is shown with the ones above: matrix-Eigen.salam

`Reshape`

func Reshape(x &: Matrix, rows: int, cols: int): Matrix

A copy with a new shape.

The example for this item is shown with the ones above: matrix-Shape.salam

`Flatten`

func Flatten(x &: Matrix): Matrix

A row vector of every element.

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`FlattenCol`

func FlattenCol(x &: Matrix): Matrix

A column vector, column by column.

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`Row`

func Row(x &: Matrix, i: int): Matrix

Row i as a row vector.

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`Col`

func Col(x &: Matrix, j: int): Matrix

Column j as a column vector.

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`SetRow`

func SetRow(x &: Matrix, i: int, v &: Matrix)

Sets the row.

The example for this item is shown with the ones above: matrix-Shape.salam

`SetCol`

func SetCol(x &: Matrix, j: int, v &: Matrix)

Sets the column.

The example for this item is shown with the ones above: matrix-Shape.salam

`SubMatrix`

func SubMatrix(x &: Matrix, r0: int, c0: int, dr: int, dc: int): Matrix

A submatrix from a starting row and column, with a size.

The example for this item is shown with the ones above: matrix-Shape.salam

`Minor`

func Minor(x &: Matrix, i: int, j: int): Matrix

The matrix without row i and column j.

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`DeleteRow`

func DeleteRow(x &: Matrix, i: int): Matrix

Without row i.

The example for this item is shown with the ones above: matrix-Shape.salam

`DeleteCol`

func DeleteCol(x &: Matrix, j: int): Matrix

Without column j.

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`InsertRow`

func InsertRow(x &: Matrix, i: int, v &: Matrix): Matrix

With a new row at place i.

The example for this item is shown with the ones above: matrix-Shape.salam

`InsertCol`

func InsertCol(x &: Matrix, j: int, v &: Matrix): Matrix

With a new column at place j.

The example for this item is shown with the ones above: matrix-Shape.salam

`SwapRows`

func SwapRows(x &: Matrix, i: int, j: int)

Swaps two rows in place.

The example for this item is shown with the ones above: matrix-Shape.salam

`SwapCols`

func SwapCols(x &: Matrix, i: int, j: int)

Swaps two columns in place.

The example for this item is shown with the ones above: matrix-Shape.salam

`Diagonal`

func Diagonal(x &: Matrix): Matrix

The main diagonal.

The example for this item is shown with the ones above: matrix-Shape.salam

`DiagonalOffset`

func DiagonalOffset(x &: Matrix, k: int): Matrix

The kth diagonal; positive is above the main one.

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`SetDiagonal`

func SetDiagonal(x &: Matrix, v &: Matrix)

Sets the diagonal.

The example for this item is shown with the ones above: matrix-Shape.salam

`ShiftDiagonal`

func ShiftDiagonal(x &: Matrix, c: f64)

Adds a number to the diagonal.

The example for this item is shown with the ones above: matrix-Shape.salam

`Abs`

func Abs(x &: Matrix): Matrix

The absolute value, element by element.

The example for this item is shown with the ones above: matrix-Math.salam

`Sqrt`

func Sqrt(x &: Matrix): Matrix

The square root, element by element.

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`Exp`

func Exp(x &: Matrix): Matrix

The exponential, element by element.

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`Log`

func Log(x &: Matrix): Matrix

The natural logarithm, element by element.

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`Log10`

func Log10(x &: Matrix): Matrix

The base ten logarithm, element by element.

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`Log2`

func Log2(x &: Matrix): Matrix

The base two logarithm, element by element.

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`Sin`

func Sin(x &: Matrix): Matrix

The sine, element by element.

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`Cos`

func Cos(x &: Matrix): Matrix

The cosine, element by element.

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`Tan`

func Tan(x &: Matrix): Matrix

The tangent, element by element.

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`Asin`

func Asin(x &: Matrix): Matrix

The inverse sine, element by element.

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`Acos`

func Acos(x &: Matrix): Matrix

The inverse cosine, element by element.

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`Atan`

func Atan(x &: Matrix): Matrix

The inverse tangent, element by element.

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`Sinh`

func Sinh(x &: Matrix): Matrix

The hyperbolic sine, element by element.

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`Cosh`

func Cosh(x &: Matrix): Matrix

The hyperbolic cosine, element by element.

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`Tanh`

func Tanh(x &: Matrix): Matrix

The hyperbolic tangent, element by element.

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`Floor`

func Floor(x &: Matrix): Matrix

The floor, element by element.

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`Ceil`

func Ceil(x &: Matrix): Matrix

The ceiling, element by element.

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`Round`

func Round(x &: Matrix): Matrix

Rounds to the nearest integer, element by element.

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`Trunc`

func Trunc(x &: Matrix): Matrix

Drops the fractional part, element by element.

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`Sign`

func Sign(x &: Matrix): Matrix

The sign, meaning -1, 0 or 1, element by element.

The example for this item is shown with the ones above: matrix-Math.salam

`Square`

func Square(x &: Matrix): Matrix

The square, element by element.

The example for this item is shown with the ones above: matrix-Create.salam

`Reciprocal`

func Reciprocal(x &: Matrix): Matrix

The reciprocal, one over the element, element by element.

The example for this item is shown with the ones above: matrix-Math.salam

`Cbrt`

func Cbrt(x &: Matrix): Matrix

The cube root, negative numbers included, element by element.

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`PowElem`

func PowElem(x &: Matrix, p: f64): Matrix

Raises the elements to a power.

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`Clip`

func Clip(x &: Matrix, lo: f64, hi: f64): Matrix

Holds the elements within a range.

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`MaxScalar`

func MaxScalar(x &: Matrix, s: f64): Matrix

The greater of each element and a number.

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`MinScalar`

func MinScalar(x &: Matrix, s: f64): Matrix

The smaller of each element and a number.

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`Chop`

func Chop(x &: Matrix, tol: f64): Matrix

Zeroes the elements smaller than a tolerance; for a clean display.

The example for this item is shown with the ones above: matrix-Math.salam