operatortheory

package
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 Go to latest Published: Jul 19, 2026 License: MIT Imports: 7 Imported by: 0

Documentation

Overview

Package operatortheory implements finite-dimensional linear operator and spectral theory using only the Go standard library.

The central type is Matrix, a dense complex matrix that is interpreted as a bounded linear operator on the finite-dimensional Hilbert space C^n endowed with the standard inner product <x,y> = sum conj(x_i) y_i. On top of the usual matrix arithmetic (addition, multiplication, adjoint, Kronecker product, powers) the package provides the machinery of operator theory:

Numerical methods. Symmetric/Hermitian eigenproblems are solved with the cyclic Jacobi method applied to the real 2n-by-2n symmetric embedding of a Hermitian matrix, which is backward stable and returns a full orthonormal set of eigenvectors. General (non-normal) spectra are computed with the explicitly shifted QR algorithm on the upper-Hessenberg form. The singular value decomposition is obtained from the Hermitian eigendecomposition of the Gram matrix. Every routine depends only on math, math/cmplx and sort.

Randomised constructors (RandomMatrix, RandomHermitian, RandomUnitary) take a caller-supplied seed and use math/rand so that results are fully reproducible.

Index

Examples

Constants

This section is empty.

Variables

View Source 
var (
	// ErrDimensionMismatch reports that two operands have incompatible shapes.
	ErrDimensionMismatch = errors.New("operatortheory: dimension mismatch")
	// ErrNotSquare reports that a square matrix was required but not provided.
	ErrNotSquare = errors.New("operatortheory: matrix is not square")
	// ErrNotHermitian reports that a Hermitian matrix was required but not
	// provided.
	ErrNotHermitian = errors.New("operatortheory: matrix is not Hermitian")
	// ErrNotNormal reports that a normal matrix was required but not provided.
	ErrNotNormal = errors.New("operatortheory: matrix is not normal")
	// ErrSingular reports that a matrix is singular (or numerically so).
	ErrSingular = errors.New("operatortheory: matrix is singular")
	// ErrEmpty reports that an empty matrix or vector was supplied.
	ErrEmpty = errors.New("operatortheory: empty input")
	// ErrOutOfRange reports an index outside the valid range.
	ErrOutOfRange = errors.New("operatortheory: index out of range")
	// ErrInvalidArgument reports a value outside its permitted domain.
	ErrInvalidArgument = errors.New("operatortheory: invalid argument")
	// ErrNoConvergence reports that an iterative method failed to converge
	// within its iteration budget.
	ErrNoConvergence = errors.New("operatortheory: iteration did not converge")
)

Sentinel errors returned throughout the package. Callers may test for these with errors.Is.

Functions

This section is empty.

Types

type Eigenpair 

type Eigenpair struct {
	// Value is the eigenvalue.
	Value complex128
	// Vector is a unit eigenvector for Value.
	Vector Vector
}

Eigenpair bundles an eigenvalue with a corresponding unit eigenvector.

type Matrix 

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

Matrix is a dense complex matrix stored in row-major order. It is interpreted throughout the package as a bounded linear operator on C^n (when square).

func Companion 

func Companion(coeffs []complex128) (*Matrix, error)

Companion returns the companion matrix of the monic polynomial whose coefficients (excluding the leading 1) are given from the constant term up to the degree-(n-1) term: p(x) = x^n + c[n-1] x^(n-1) + ... + c[1] x + c[0]. Its eigenvalues are the roots of p. It returns ErrEmpty for no coefficients.

func Diagonal 

func Diagonal(d []complex128) *Matrix

Diagonal returns the square diagonal matrix with the given diagonal entries.

func FromReal 

func FromReal(rows, cols int, data []float64) (*Matrix, error)

FromReal returns a complex matrix whose entries are the given real numbers (in row-major order) with zero imaginary part.

func FromRows 

func FromRows(rows [][]complex128) (*Matrix, error)

FromRows builds a matrix from a slice of rows. All rows must have equal length. It returns ErrDimensionMismatch on ragged input and ErrEmpty when no rows are given.

func Identity 

func Identity(n int) *Matrix

Identity returns the n-by-n identity operator.

func JordanBlock 

func JordanBlock(lambda complex128, n int) *Matrix

JordanBlock returns the n-by-n Jordan block with eigenvalue lambda: lambda on the diagonal and 1 on the first superdiagonal.

func NewMatrix 

func NewMatrix(rows, cols int) *Matrix

NewMatrix returns a rows-by-cols zero matrix. It panics if either dimension is negative.

func NewMatrixFromData 

func NewMatrixFromData(rows, cols int, data []complex128) (*Matrix, error)

NewMatrixFromData returns a rows-by-cols matrix filled from data in row-major order. It returns ErrDimensionMismatch if len(data) != rows*cols.

func OrthogonalProjector 

func OrthogonalProjector(vectors []Vector) (*Matrix, error)

OrthogonalProjector returns the orthogonal projection operator onto the subspace spanned by the given vectors of common length n. Linearly dependent vectors are handled correctly. It returns ErrEmpty when no vectors are given and ErrDimensionMismatch on unequal lengths.

func RandomHermitian 

func RandomHermitian(n int, seed int64) *Matrix

RandomHermitian returns a random n-by-n Hermitian matrix built as (A + A^H)/2 from a random matrix A seeded by seed.

func RandomMatrix 

func RandomMatrix(rows, cols int, seed int64) *Matrix

RandomMatrix returns an m-by-n matrix whose entries have real and imaginary parts drawn independently from the standard normal distribution, using a deterministic generator seeded by seed.

func RandomUnitary 

func RandomUnitary(n int, seed int64) *Matrix

RandomUnitary returns a random n-by-n unitary matrix obtained from the QR factorisation of a random complex matrix seeded by seed. The construction follows the standard recipe that yields a Haar-like distribution.

func RealDiagonal 

func RealDiagonal(d []float64) *Matrix

RealDiagonal returns the square diagonal matrix with the given real diagonal entries.

func Reflection 

func Reflection(v Vector) (*Matrix, error)

Reflection returns the Householder reflection I - 2 P, where P is the orthogonal projector onto the line spanned by the nonzero unit-normalisable vector v. The result is a unitary involution.

func Zero 

func Zero(rows, cols int) *Matrix

Zero returns the rows-by-cols zero matrix. It is a synonym for NewMatrix.

func (*Matrix) Add 

func (m *Matrix) Add(b *Matrix) (*Matrix, error)

Add returns m + b. It returns ErrDimensionMismatch on a shape mismatch.

func (*Matrix) Adjoint 

func (m *Matrix) Adjoint() *Matrix

Adjoint returns the conjugate transpose (Hermitian adjoint) m^H.

func (*Matrix) AntiCommutator 

func (m *Matrix) AntiCommutator(b *Matrix) (*Matrix, error)

AntiCommutator returns the anticommutator {m,b} = m*b + b*m.

func (*Matrix) Apply 

func (m *Matrix) Apply(v Vector) Vector

Apply returns the image m*v of the vector v under the operator, panicking on a dimension mismatch. It is a convenience wrapper around MulVec.

func (*Matrix) ApplyFunction 

func (m *Matrix) ApplyFunction(f ScalarFunction) (*Matrix, error)

ApplyFunction returns f(m) defined by the holomorphic functional calculus for a diagonalisable (in particular, normal) operator: if m = V diag(lambda) V^{-1} then f(m) = V diag(f(lambda)) V^{-1}. Hermitian matrices use the exact unitary eigendecomposition; other matrices use the eigenvectors from inverse iteration. It returns ErrNotSquare for a non-square matrix and ErrNotNormal when the eigenvectors are too ill-conditioned to invert (a defective matrix).

func (*Matrix) At 

func (m *Matrix) At(i, j int) complex128

At returns the entry in row i and column j. It panics if the indices are out of range.

func (*Matrix) CayleyTransform 

func (m *Matrix) CayleyTransform() (*Matrix, error)

CayleyTransform returns (m - i*I)(m + i*I)^{-1}, which maps a Hermitian operator to a unitary one. It returns ErrSingular if m + i*I is singular.

func (*Matrix) CharacteristicPolynomial 

func (m *Matrix) CharacteristicPolynomial() ([]complex128, error)

CharacteristicPolynomial returns the coefficients of the characteristic polynomial det(x*I - m) in ascending order, so the result has length n+1 with a leading coefficient of 1 at index n. It uses the Faddeev-LeVerrier algorithm and returns ErrNotSquare for a non-square matrix.

func (*Matrix) Clone 

func (m *Matrix) Clone() *Matrix

Clone returns an independent copy of the matrix.

func (*Matrix) Col 

func (m *Matrix) Col(j int) Vector

Col returns a copy of column j as a Vector.

func (*Matrix) Cols 

func (m *Matrix) Cols() int

Cols returns the number of columns.

func (*Matrix) Commutator 

func (m *Matrix) Commutator(b *Matrix) (*Matrix, error)

Commutator returns the commutator [m,b] = m*b - b*m. It returns an error if the products are not defined.

func (*Matrix) ConditionNumber 

func (m *Matrix) ConditionNumber() float64

ConditionNumber returns the 2-norm condition number, the ratio of the largest to the smallest singular value. It returns +Inf for a singular matrix.

func (*Matrix) Conjugate 

func (m *Matrix) Conjugate() *Matrix

Conjugate returns the entrywise complex conjugate of m.

func (*Matrix) Cos 

func (m *Matrix) Cos() (*Matrix, error)

Cos returns the matrix cosine cos(m) via the functional calculus.

func (*Matrix) Cosh 

func (m *Matrix) Cosh() (*Matrix, error)

Cosh returns the matrix hyperbolic cosine cosh(m) via the functional calculus.

func (*Matrix) Determinant 

func (m *Matrix) Determinant() (complex128, error)

Determinant returns the determinant of a square matrix. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) Diag 

func (m *Matrix) Diag() Vector

Diag returns the main diagonal of the matrix as a Vector of length min(rows,cols).

func (*Matrix) DiagonalizeHermitian 

func (m *Matrix) DiagonalizeHermitian() (u, d *Matrix, err error)

DiagonalizeHermitian returns a unitary U and a real diagonal matrix D such that m = U*D*U^H, for a Hermitian matrix. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) Dims 

func (m *Matrix) Dims() (int, int)

Dims returns the number of rows and columns.

func (*Matrix) DirectSum 

func (m *Matrix) DirectSum(b *Matrix) *Matrix

DirectSum returns the block-diagonal matrix diag(m, b).

func (*Matrix) DistanceFrobenius 

func (m *Matrix) DistanceFrobenius(b *Matrix) (float64, error)

DistanceFrobenius returns the Frobenius norm of m - b, a metric on the space of matrices of equal shape. It returns ErrDimensionMismatch on a shape mismatch.

func (*Matrix) DistanceToSingularity 

func (m *Matrix) DistanceToSingularity() float64

DistanceToSingularity returns the 2-norm distance from a square matrix to the nearest singular matrix, which equals the smallest singular value. It is the backward-error interpretation of the resolvent norm evaluated at 0.

func (*Matrix) Eigen 

func (m *Matrix) Eigen() ([]Eigenpair, error)

Eigen returns eigenvalue/eigenvector pairs for a square matrix. Eigenvalues are found with the QR algorithm and eigenvectors by inverse iteration. The eigenvectors are reliable for matrices with well-separated eigenvalues; for Hermitian matrices prefer HermitianEigen. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) EigenvalueMultiplicities 

func (m *Matrix) EigenvalueMultiplicities(tol float64) (values []complex128, mult []int, err error)

EigenvalueMultiplicities groups the eigenvalues of a square matrix into clusters that agree to within tol and returns the distinct representatives together with their algebraic multiplicities. Representatives are sorted lexicographically.

func (*Matrix) Eigenvalues 

func (m *Matrix) Eigenvalues() ([]complex128, error)

Eigenvalues returns all eigenvalues of a square matrix, counted with multiplicity, computed with the shifted QR algorithm. The order is unspecified. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) Equal 

func (m *Matrix) Equal(other *Matrix, tol float64) bool

Equal reports whether m and other have the same shape and agree entrywise to within tol.

func (*Matrix) Exp 

func (m *Matrix) Exp() (*Matrix, error)

Exp returns the matrix exponential exp(m) computed with the scaling and squaring method combined with a truncated Taylor series. It is valid for any square matrix. It returns ErrNotSquare for a non-square matrix.

Example 
// exp of a 90-degree generator is a quarter-turn rotation.
g, _ := FromReal(2, 2, []float64{0, -math.Pi / 2, math.Pi / 2, 0})
e, _ := g.Exp()
fmt.Printf("%.3f %.3f\n%.3f %.3f\n",
	real(e.At(0, 0)), real(e.At(0, 1)),
	real(e.At(1, 0)), real(e.At(1, 1)))

Output:
0.000 -1.000
1.000 0.000

func (*Matrix) FrobeniusNorm 

func (m *Matrix) FrobeniusNorm() float64

FrobeniusNorm returns the Frobenius norm, the square root of the sum of the squared moduli of the entries.

func (*Matrix) HadamardProduct 

func (m *Matrix) HadamardProduct(b *Matrix) (*Matrix, error)

HadamardProduct returns the entrywise (Schur) product of m and b.

func (*Matrix) HermitianEigen 

func (m *Matrix) HermitianEigen() (values []float64, vectors *Matrix, err error)

HermitianEigen returns the real eigenvalues (in ascending order) and the orthonormal eigenvectors (as the columns of the returned matrix) of a Hermitian matrix. The matrix is not required to be exactly Hermitian; its Hermitian part is used, so callers should ensure the input truly is Hermitian for meaningful results. It returns ErrNotSquare for a non-square matrix.

Example 
// The Hermitian matrix [[2, i], [-i, 2]] has eigenvalues 1 and 3.
m, _ := NewMatrixFromData(2, 2, []complex128{
	2, complex(0, 1),
	complex(0, -1), 2,
})
vals, _, _ := m.HermitianEigen()
fmt.Printf("%.4f %.4f\n", vals[0], vals[1])

Output:
1.0000 3.0000

func (*Matrix) HermitianPart 

func (m *Matrix) HermitianPart() *Matrix

HermitianPart returns (m + m^H)/2, the Hermitian part of a square matrix.

func (*Matrix) Hessenberg 

func (m *Matrix) Hessenberg() (h, q *Matrix, err error)

Hessenberg returns the upper-Hessenberg reduction H together with the unitary Q such that m = Q*H*Q^H. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) ImagPart 

func (m *Matrix) ImagPart() *Matrix

ImagPart returns the entrywise imaginary part of m as a real-valued complex matrix.

func (*Matrix) InPseudospectrum 

func (m *Matrix) InPseudospectrum(z complex128, eps float64) bool

InPseudospectrum reports whether z lies in the eps-pseudospectrum of m, i.e. whether ResolventNorm(z) >= 1/eps. It requires eps > 0.

func (*Matrix) Inertia 

func (m *Matrix) Inertia(tol float64) (neg, zero, pos int)

Inertia returns the numbers of negative, zero and positive eigenvalues of a Hermitian matrix, using tol to decide which eigenvalues count as zero.

func (*Matrix) InfNorm 

func (m *Matrix) InfNorm() float64

InfNorm returns the induced infinity-norm, the maximum absolute row sum.

func (*Matrix) Inverse 

func (m *Matrix) Inverse() (*Matrix, error)

Inverse returns the inverse of a square matrix. It returns ErrNotSquare for a non-square matrix and ErrSingular when the matrix is (numerically) singular.

func (*Matrix) IsContraction 

func (m *Matrix) IsContraction(tol float64) bool

IsContraction reports whether the operator norm of m is at most 1 + tol.

func (*Matrix) IsDiagonal 

func (m *Matrix) IsDiagonal(tol float64) bool

IsDiagonal reports whether all off-diagonal entries vanish to within tol.

func (*Matrix) IsHermitian 

func (m *Matrix) IsHermitian(tol float64) bool

IsHermitian reports whether the matrix equals its conjugate transpose to within tol. A non-positive tol selects a default tolerance.

func (*Matrix) IsIdempotent 

func (m *Matrix) IsIdempotent(tol float64) bool

IsIdempotent is a synonym for IsProjection.

func (*Matrix) IsInvolution 

func (m *Matrix) IsInvolution(tol float64) bool

IsInvolution reports whether m^2 = I to within tol.

func (*Matrix) IsIsometry 

func (m *Matrix) IsIsometry(tol float64) bool

IsIsometry reports whether m preserves norms, i.e. m^H m = I. For square matrices this coincides with unitarity; for tall matrices it means the columns are orthonormal.

func (*Matrix) IsLowerTriangular 

func (m *Matrix) IsLowerTriangular(tol float64) bool

IsLowerTriangular reports whether all entries above the main diagonal vanish to within tol.

func (*Matrix) IsNilpotent 

func (m *Matrix) IsNilpotent(tol float64) bool

IsNilpotent reports whether some power m^k (k <= n) is the zero matrix to within tol.

func (*Matrix) IsNormal 

func (m *Matrix) IsNormal(tol float64) bool

IsNormal reports whether m commutes with its adjoint, m^H m = m m^H, to within tol.

func (*Matrix) IsOrthogonal 

func (m *Matrix) IsOrthogonal(tol float64) bool

IsOrthogonal reports whether a real matrix satisfies m^T m = I to within tol. If the matrix has non-negligible imaginary part it is not orthogonal in this sense.

func (*Matrix) IsOrthogonalProjection 

func (m *Matrix) IsOrthogonalProjection(tol float64) bool

IsOrthogonalProjection reports whether m is both Hermitian and idempotent, so that it projects orthogonally onto its range.

func (*Matrix) IsPartialIsometry 

func (m *Matrix) IsPartialIsometry(tol float64) bool

IsPartialIsometry reports whether m^H m is an orthogonal projection, the defining property of a partial isometry.

func (*Matrix) IsPositiveDefinite 

func (m *Matrix) IsPositiveDefinite(tol float64) bool

IsPositiveDefinite reports whether m is Hermitian with strictly positive eigenvalues, all exceeding tol.

func (*Matrix) IsPositiveSemidefinite 

func (m *Matrix) IsPositiveSemidefinite(tol float64) bool

IsPositiveSemidefinite reports whether m is Hermitian with eigenvalues no smaller than -tol.

func (*Matrix) IsProjection 

func (m *Matrix) IsProjection(tol float64) bool

IsProjection reports whether m is idempotent, m^2 = m, to within tol.

func (*Matrix) IsSchurStable 

func (m *Matrix) IsSchurStable(tol float64) bool

IsSchurStable reports whether every eigenvalue of m lies strictly inside the unit disc (a discrete-time stable operator).

func (*Matrix) IsSelfAdjoint 

func (m *Matrix) IsSelfAdjoint(tol float64) bool

IsSelfAdjoint is a synonym for IsHermitian.

func (*Matrix) IsSkewHermitian 

func (m *Matrix) IsSkewHermitian(tol float64) bool

IsSkewHermitian reports whether m^H = -m to within tol.

func (*Matrix) IsSquare 

func (m *Matrix) IsSquare() bool

IsSquare reports whether the matrix has equal numbers of rows and columns.

func (*Matrix) IsStable 

func (m *Matrix) IsStable(tol float64) bool

IsStable reports whether every eigenvalue of m has strictly negative real part (a continuous-time stable, or Hurwitz, operator).

func (*Matrix) IsStrictContraction 

func (m *Matrix) IsStrictContraction(tol float64) bool

IsStrictContraction reports whether the operator norm of m is strictly less than 1 - tol.

func (*Matrix) IsSymmetric 

func (m *Matrix) IsSymmetric(tol float64) bool

IsSymmetric reports whether m equals its transpose (no conjugation) to within tol.

func (*Matrix) IsUnitary 

func (m *Matrix) IsUnitary(tol float64) bool

IsUnitary reports whether m^H m equals the identity to within tol.

func (*Matrix) IsUpperTriangular 

func (m *Matrix) IsUpperTriangular(tol float64) bool

IsUpperTriangular reports whether all entries below the main diagonal vanish to within tol.

func (*Matrix) KernelBasis 

func (m *Matrix) KernelBasis(tol float64) []Vector

KernelBasis returns an orthonormal basis for the null space (kernel) of a square matrix, obtained from the right singular vectors with negligible singular values. tol sets the relative threshold below which a singular value is treated as zero.

func (*Matrix) Kron 

func (m *Matrix) Kron(b *Matrix) *Matrix

Kron returns the Kronecker product m (x) b.

func (*Matrix) KyFanNorm 

func (m *Matrix) KyFanNorm(k int) float64

KyFanNorm returns the Ky Fan k-norm, the sum of the k largest singular values. If k exceeds the number of singular values it uses all of them.

func (*Matrix) Log 

func (m *Matrix) Log() (*Matrix, error)

Log returns a matrix logarithm of m via the functional calculus, so that exp(Log(m)) = m for a diagonalisable m with eigenvalues off the negative real axis. It returns an error for a non-square or defective matrix.

func (*Matrix) MaxAbs 

func (m *Matrix) MaxAbs() float64

MaxAbs returns the largest modulus among the entries of m.

func (*Matrix) MaxEigenvalue 

func (m *Matrix) MaxEigenvalue() float64

MaxEigenvalue returns the largest eigenvalue of a Hermitian matrix.

func (*Matrix) MaxNorm 

func (m *Matrix) MaxNorm() float64

MaxNorm returns the largest modulus among the entries (the entrywise max-norm).

func (*Matrix) MinEigenvalue 

func (m *Matrix) MinEigenvalue() float64

MinEigenvalue returns the smallest eigenvalue of a Hermitian matrix.

func (*Matrix) Mul 

func (m *Matrix) Mul(b *Matrix) (*Matrix, error)

Mul returns the matrix product m*b. It returns ErrDimensionMismatch when the inner dimensions do not agree.

func (*Matrix) MulVec 

func (m *Matrix) MulVec(v Vector) (Vector, error)

MulVec returns the matrix-vector product m*v. It returns ErrDimensionMismatch when len(v) != cols.

func (*Matrix) Neg 

func (m *Matrix) Neg() *Matrix

Neg returns -m.

func (*Matrix) NuclearNorm 

func (m *Matrix) NuclearNorm() float64

NuclearNorm returns the nuclear (trace) norm, the sum of the singular values.

func (*Matrix) NumericalAbscissa 

func (m *Matrix) NumericalAbscissa() float64

NumericalAbscissa returns the maximum real part of the numerical range, equal to the largest eigenvalue of the Hermitian part (m + m^H)/2. It bounds the initial growth rate of exp(tA).

func (*Matrix) NumericalRadius 

func (m *Matrix) NumericalRadius(samples int) float64

NumericalRadius returns the numerical radius w(m) = max { |z| : z in W(m) }, estimated by sampling the field of values at the given number of angles.

func (*Matrix) NumericalRange 

func (m *Matrix) NumericalRange(samples int) []complex128

NumericalRange returns a convex polygon approximating the boundary of the numerical range (field of values) W(m) = { <x, m x> : ||x|| = 1 }, sampled at the given number of angles (a minimum of 8 is used). Consecutive points trace the boundary counter-clockwise.

func (*Matrix) OneNorm 

func (m *Matrix) OneNorm() float64

OneNorm returns the induced 1-norm, the maximum absolute column sum.

func (*Matrix) OperatorNorm 

func (m *Matrix) OperatorNorm() float64

OperatorNorm returns the induced 2-norm (spectral norm), the largest singular value. It is computed by power iteration on m^H m for efficiency.

func (*Matrix) PolarDecomposition 

func (m *Matrix) PolarDecomposition() (unitary, positive *Matrix, err error)

PolarDecomposition returns the polar factors of a square matrix, U unitary (or a partial isometry when singular) and P Hermitian positive semidefinite, such that m = U*P. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) Power 

func (m *Matrix) Power(k int) *Matrix

Power returns m raised to the non-negative integer power k using binary exponentiation. Power(0) is the identity. It panics if m is not square or k is negative.

func (*Matrix) PowerReal 

func (m *Matrix) PowerReal(p float64) (*Matrix, error)

PowerReal returns m raised to a real power p via the functional calculus, exp(p*Log(m)). For a Hermitian positive definite matrix and integer p it agrees with the ordinary matrix power.

func (*Matrix) PseudospectralAbscissa 

func (m *Matrix) PseudospectralAbscissa(eps float64, samples int) (float64, error)

PseudospectralAbscissa estimates the eps-pseudospectral abscissa, the maximum real part of any point in the eps-pseudospectrum. It scans a grid around the spectrum enlarged by a margin proportional to eps and refines the rightmost crossing. It requires eps > 0 and returns ErrInvalidArgument otherwise.

func (*Matrix) PseudospectralRadius 

func (m *Matrix) PseudospectralRadius(eps float64, samples int) (float64, error)

PseudospectralRadius estimates the eps-pseudospectral radius, the largest modulus of any point in the eps-pseudospectrum, by scanning a grid enclosing the spectrum enlarged by a margin proportional to eps.

func (*Matrix) PseudospectrumGrid 

func (m *Matrix) PseudospectrumGrid(reMin, reMax, imMin, imMax float64, nx, ny int) ([][]float64, error)

PseudospectrumGrid evaluates the resolvent norm on a rectangular grid in the complex plane spanning [reMin,reMax] x [imMin,imMax] with nx columns and ny rows. The result is indexed as grid[row][col], where row 0 corresponds to imMin. It returns ErrInvalidArgument for non-positive grid sizes.

func (*Matrix) QR 

func (m *Matrix) QR() (q, r *Matrix)

QR returns a reduced QR factorisation m = Q*R computed with complex Householder reflections. Q has orthonormal columns and R is upper triangular.

func (*Matrix) QuadraticForm 

func (m *Matrix) QuadraticForm(x, y Vector) complex128

QuadraticForm returns the sesquilinear form <x, m y>.

func (*Matrix) RangeProjector 

func (m *Matrix) RangeProjector() *Matrix

RangeProjector returns the orthogonal projection onto the column space (range) of m.

func (*Matrix) Rank 

func (m *Matrix) Rank(tol float64) int

Rank returns the numerical rank of the matrix, the number of singular values exceeding tol * (largest singular value). If tol <= 0 a default relative tolerance is used.

func (*Matrix) RayleighQuotient 

func (m *Matrix) RayleighQuotient(v Vector) complex128

RayleighQuotient returns <v, m v> / <v, v> for a nonzero vector v. For a Hermitian operator this is real and lies between the smallest and largest eigenvalues.

func (*Matrix) RealPart 

func (m *Matrix) RealPart() *Matrix

RealPart returns the entrywise real part of m as a complex matrix with zero imaginary part.

func (*Matrix) Resolvent 

func (m *Matrix) Resolvent(z complex128) (*Matrix, error)

Resolvent returns the resolvent operator (z*I - m)^{-1} at the complex point z. It returns ErrSingular when z is (numerically) an eigenvalue.

func (*Matrix) ResolventNorm 

func (m *Matrix) ResolventNorm(z complex128) float64

ResolventNorm returns the spectral norm of the resolvent at z, ||(z*I - m)^{-1}||_2 = 1 / sigma_min(z*I - m). It returns +Inf when z is (very close to) an eigenvalue. This quantity defines the pseudospectra: z belongs to the eps-pseudospectrum precisely when ResolventNorm(z) >= 1/eps.

func (*Matrix) Row 

func (m *Matrix) Row(i int) Vector

Row returns a copy of row i as a Vector.

func (*Matrix) Rows 

func (m *Matrix) Rows() int

Rows returns the number of rows.

func (*Matrix) SVD 

func (m *Matrix) SVD() (u *Matrix, s []float64, v *Matrix)

SVD returns a reduced singular value decomposition m = U * diag(s) * V^H, where U has orthonormal columns (m-by-k), s holds the k = min(rows,cols) singular values in descending order, and V has orthonormal columns (cols-by-k).

func (*Matrix) Scale 

func (m *Matrix) Scale(s complex128) *Matrix

Scale returns the matrix s*m.

func (*Matrix) ScaleReal 

func (m *Matrix) ScaleReal(s float64) *Matrix

ScaleReal returns the matrix s*m for a real scalar s.

func (*Matrix) SchattenNorm 

func (m *Matrix) SchattenNorm(p float64) float64

SchattenNorm returns the Schatten p-norm, the l^p norm of the vector of singular values, for p >= 1. SchattenNorm(1) is the nuclear norm, SchattenNorm(2) the Frobenius norm and the limit p -> infinity the operator norm.

func (*Matrix) Semigroup 

func (m *Matrix) Semigroup(t float64) (*Matrix, error)

Semigroup returns exp(t*m), the value at time t of the one-parameter operator semigroup generated by m.

func (*Matrix) Set 

func (m *Matrix) Set(i, j int, v complex128)

Set stores v in row i and column j. It panics if the indices are out of range.

func (*Matrix) SetCol 

func (m *Matrix) SetCol(j int, v Vector)

SetCol overwrites column j with the entries of v. It panics on a length mismatch.

func (*Matrix) SetRow 

func (m *Matrix) SetRow(i int, v Vector)

SetRow overwrites row i with the entries of v. It panics on a length mismatch.

func (*Matrix) Sign 

func (m *Matrix) Sign() (*Matrix, error)

Sign returns the matrix sign function, mapping each eigenvalue lambda to sign(Re lambda) (and 0 when Re lambda is 0). For a Hermitian matrix this is the difference of the projectors onto the positive and negative eigenspaces.

func (*Matrix) Sin 

func (m *Matrix) Sin() (*Matrix, error)

Sin returns the matrix sine sin(m) via the functional calculus.

func (*Matrix) SingularValues 

func (m *Matrix) SingularValues() []float64

SingularValues returns the singular values of the matrix in descending order.

func (*Matrix) Sinh 

func (m *Matrix) Sinh() (*Matrix, error)

Sinh returns the matrix hyperbolic sine sinh(m) via the functional calculus.

func (*Matrix) SkewHermitianPart 

func (m *Matrix) SkewHermitianPart() *Matrix

SkewHermitianPart returns (m - m^H)/2, the skew-Hermitian part of a square matrix.

func (*Matrix) SmallestSingularValue 

func (m *Matrix) SmallestSingularValue() float64

SmallestSingularValue returns the smallest singular value of the matrix.

func (*Matrix) Solve 

func (m *Matrix) Solve(b *Matrix) (*Matrix, error)

Solve returns the solution X of the linear system m*X = b for a matrix right-hand side. It returns ErrNotSquare, ErrDimensionMismatch or ErrSingular as appropriate.

func (*Matrix) SolveVec 

func (m *Matrix) SolveVec(b Vector) (Vector, error)

SolveVec returns the solution x of the linear system m*x = b for a vector right-hand side.

func (*Matrix) SpectralAbscissa 

func (m *Matrix) SpectralAbscissa() float64

SpectralAbscissa returns the maximum real part among the eigenvalues, which governs the growth rate of the operator semigroup exp(tA).

func (*Matrix) SpectralDecomposition 

func (m *Matrix) SpectralDecomposition(tol float64) ([]SpectralComponent, error)

SpectralDecomposition returns the spectral resolution of a Hermitian matrix: one SpectralComponent per distinct eigenvalue (eigenvalues within tol are merged), sorted in ascending order. The projectors sum to the identity and m = sum lambda_k P_k. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) SpectralGap 

func (m *Matrix) SpectralGap() float64

SpectralGap returns the difference between the two smallest distinct eigenvalues of a Hermitian matrix (the gap above the ground state). If there are fewer than two distinct eigenvalues it returns 0.

func (*Matrix) SpectralNorm 

func (m *Matrix) SpectralNorm() float64

SpectralNorm is an alias for OperatorNorm.

func (*Matrix) SpectralProjector 

func (m *Matrix) SpectralProjector(target, tol float64) *Matrix

SpectralProjector returns the orthogonal projector onto the eigenspace of a Hermitian matrix associated with the eigenvalue closest to target, provided it lies within tol; otherwise it returns the zero matrix. It is the operator obtained from functional calculus with the indicator of {target}.

func (*Matrix) SpectralRadius 

func (m *Matrix) SpectralRadius() float64

SpectralRadius returns the largest modulus among the eigenvalues. It returns 0 for a non-square or empty matrix.

Example 
// The rotation-by-scaling matrix has spectral radius equal to the scale.
m, _ := FromReal(2, 2, []float64{0, -2, 2, 0})
fmt.Printf("%.1f\n", m.SpectralRadius())

Output:
2.0

func (*Matrix) Spectrum 

func (m *Matrix) Spectrum() ([]complex128, error)

Spectrum returns the eigenvalues of a square matrix sorted lexicographically by real part and then imaginary part. It returns ErrNotSquare for a non-square matrix.

func (*Matrix) Sqrt 

func (m *Matrix) Sqrt() (*Matrix, error)

Sqrt returns the principal square root of m via the functional calculus, so that Sqrt(m)^2 = m for a diagonalisable m. For a Hermitian positive semidefinite matrix this is the unique positive semidefinite square root.

func (*Matrix) String 

func (m *Matrix) String() string

String renders the matrix with each entry formatted to three decimal places.

func (*Matrix) Sub 

func (m *Matrix) Sub(b *Matrix) (*Matrix, error)

Sub returns m - b. It returns ErrDimensionMismatch on a shape mismatch.

func (*Matrix) Submatrix 

func (m *Matrix) Submatrix(r1, r2, c1, c2 int) (*Matrix, error)

Submatrix returns the r1..r2-1 by c1..c2-1 block of m. It returns ErrOutOfRange for an invalid range.

func (*Matrix) Trace 

func (m *Matrix) Trace() complex128

Trace returns the sum of the diagonal entries. It panics if the matrix is not square.

func (*Matrix) TraceNorm 

func (m *Matrix) TraceNorm() float64

TraceNorm is an alias for NuclearNorm.

func (*Matrix) Transpose 

func (m *Matrix) Transpose() *Matrix

Transpose returns the transpose of m (no conjugation).

type ScalarFunction 

type ScalarFunction func(complex128) complex128

ScalarFunction is a complex-valued function of a complex variable, the kind of analytic function to which the functional calculus applies.

type SpectralComponent 

type SpectralComponent struct {
	// Eigenvalue is the (real) eigenvalue.
	Eigenvalue float64
	// Multiplicity is the dimension of the eigenspace.
	Multiplicity int
	// Projector is the orthogonal projection onto the eigenspace.
	Projector *Matrix
}

SpectralComponent bundles a (real) eigenvalue of a Hermitian operator with the orthogonal projector onto its eigenspace.

type Vector 

type Vector []complex128

Vector is a finite-dimensional complex vector, an element of the Hilbert space C^n with inner product <x,y> = sum conj(x_i) y_i.

func BasisVector 

func BasisVector(n, i int) Vector

BasisVector returns the i-th standard basis vector of C^n. It panics if i is out of range.

func GramSchmidt 

func GramSchmidt(vectors []Vector, tol float64) []Vector

GramSchmidt orthonormalises the supplied vectors using the modified Gram-Schmidt process with the Hermitian inner product. Vectors that are (numerically) linearly dependent on the earlier ones are dropped, so the result is an orthonormal basis for the span of the inputs.

func NewVector 

func NewVector(n int) Vector

NewVector returns a zero vector of length n. It panics if n is negative.

func VectorFromReal 

func VectorFromReal(data []float64) Vector

VectorFromReal returns a complex vector whose entries are the given real numbers with zero imaginary part.

func VectorOf 

func VectorOf(entries ...complex128) Vector

VectorOf returns a Vector containing the supplied entries.

func (Vector) Add 

func (v Vector) Add(w Vector) Vector

Add returns v + w. It panics if the lengths differ.

func (Vector) Angle 

func (v Vector) Angle(w Vector) float64

Angle returns the angle in radians between the real directions of v and w, computed from |<v,w>| / (||v|| ||w||). It lies in [0, pi/2].

func (Vector) Clone 

func (v Vector) Clone() Vector

Clone returns an independent copy of v.

func (Vector) Conjugate 

func (v Vector) Conjugate() Vector

Conjugate returns the entrywise complex conjugate of v.

func (Vector) Dot 

func (v Vector) Dot(w Vector) complex128

Dot returns the Hermitian inner product <v,w> = sum conj(v_i) w_i. It is conjugate-linear in its first argument and linear in the second. It panics if the lengths differ.

func (Vector) Equal 

func (v Vector) Equal(w Vector, tol float64) bool

Equal reports whether v and w have the same length and agree entrywise to within tol.

func (Vector) InfNorm 

func (v Vector) InfNorm() float64

InfNorm returns the l-infinity norm, the largest modulus among the entries.

func (Vector) IsOrthogonal 

func (v Vector) IsOrthogonal(w Vector, tol float64) bool

IsOrthogonal reports whether v and w are orthogonal to within tol, i.e. |<v,w>| <= tol.

func (Vector) IsZero 

func (v Vector) IsZero(tol float64) bool

IsZero reports whether every entry of v has modulus at most tol.

func (Vector) Len 

func (v Vector) Len() int

Len returns the dimension of the vector.

func (Vector) Neg 

func (v Vector) Neg() Vector

Neg returns -v.

func (Vector) Norm 

func (v Vector) Norm() float64

Norm returns the Euclidean (l2) norm of v.

func (Vector) NormSquared 

func (v Vector) NormSquared() float64

NormSquared returns the squared Euclidean norm <v,v>.

func (Vector) Normalize 

func (v Vector) Normalize() (Vector, float64)

Normalize returns a unit vector in the direction of v together with the original norm. If v is the zero vector it returns a copy of v and a zero norm.

func (Vector) OneNorm 

func (v Vector) OneNorm() float64

OneNorm returns the l1 norm, the sum of the moduli of the entries.

func (Vector) OuterProduct 

func (v Vector) OuterProduct(w Vector) *Matrix

OuterProduct returns the rank-one operator v w^H, whose (i,j) entry is v_i conj(w_j).

func (Vector) ProjectOnto 

func (v Vector) ProjectOnto(u Vector) Vector

ProjectOnto returns the orthogonal projection of v onto the line spanned by the nonzero vector u, namely (<u,v>/<u,u>) u.

func (Vector) Scale 

func (v Vector) Scale(s complex128) Vector

Scale returns the vector s*v.

func (Vector) Sub 

func (v Vector) Sub(w Vector) Vector

Sub returns v - w. It panics if the lengths differ.

Source Files

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