algolib.maths.algebra 包

子模块

algolib.maths.algebra.matrix_dense module

class algolib.maths.algebra.matrix_dense.MatrixDense(rows: Sequence[Sequence[int | float]])

基类：object

A simple dense matrix (row-major) without external dependencies.

Designed for correctness and clarity (teaching/learning oriented). Not optimized for large-scale numerical workloads.

参数:

rows (Sequence[Sequence[Number]]) -- Row-major data. Must be rectangular (all rows same length).

抛出:

• InvalidTypeError -- If rows are not sequences of numbers.

• InvalidValueError -- If matrix is empty or not rectangular.

示例

>>> A = MatrixDense([[1, 2], [3, 4]])
>>> B = MatrixDense.identity(2)
>>> C = (A * B).rows == A.rows
True

T() → MatrixDense

Transpose.

copy() → MatrixDense

Shallow copy (rows are tuples; safe).

det() → float

Determinant via Gaussian elimination with partial pivoting (O(n^3)).

抛出:

InvalidValueError -- If matrix is not square.

equals(other: MatrixDense, tol: float = 1e-12) → bool

Element-wise comparison with tolerance for floats.

static eye(n: int) → MatrixDense

Alias of identity().

static from_rows(rows: Sequence[Sequence[int | float]]) → MatrixDense

Construct from row-major data (alias of the constructor).

static identity(n: int) → MatrixDense

Return the \(n \times n\) identity matrix.

inv() → MatrixDense

Inverse via Gauss-Jordan elimination (O(n^3)).

抛出:

InvalidValueError -- If matrix is not square or singular.

matvec(v: Sequence[int | float]) → List[float]

Matrix-vector product (Ax).

参数:

v (Sequence[Number]) -- Vector of length equal to number of columns.

返回:

Resulting vector.

返回类型:

list[float]

rows: Tuple[Tuple[int | float, ...], ...]

property shape: Tuple[int, int]

(n_rows, n_cols)

static zeros(n_rows: int, n_cols: int) → MatrixDense

Return an \((n_{\mathrm{rows}} × n_{\mathrm{cols}})\) zero matrix.

algolib.maths.algebra.polynomial module

Lightweight univariate polynomial with real coefficients.

备注

• Coefficients are stored in ascending degree order: p(x) = c0 + c1*x + … + cn*x**n.

• The public API is immutable; the internal representation is a tuple of float.

• Supported operations include addition, subtraction, multiplication (convolution), evaluation (Horner scheme), derivative, and antiderivative (indefinite integral).

示例

>>> p = Polynomial([1, 2, 3])     # 1 + 2x + 3x^2
>>> q = Polynomial.identity()      # 1
>>> (p * q).coeffs == p.coeffs
True
>>> p.derivative().coeffs          # d/dx (1 + 2x + 3x^2) = 2 + 6x
(2.0, 6.0)

class algolib.maths.algebra.polynomial.Polynomial(coeffs: Iterable[int | float])

基类：object

Univariate polynomial with real coefficients.

参数:

coeffs (Sequence[Number]) -- Coefficients in ascending degree order: [c0, c1, ..., cn] represents \(\sum_{k=0}^n c_k x^k\).

抛出:

• InvalidTypeError -- If any coefficient is not a real number.

• InvalidValueError -- If coeffs is empty.

备注

• Construction enforces a canonical form by stripping trailing zeros; the zero polynomial is thus represented as (0.0,) and has degree 0.

• All operations return new Polynomial instances (immutable API).

coeffs: Tuple[float, ...]

static constant(c: int | float) → Polynomial

Return a constant polynomial p(x) = c.

property degree: int

Degree of the polynomial (len(coeffs) - 1). By convention, deg(0) = 0.

derivative() → Polynomial

Return the analytical derivative \(p'(x)\).

备注

If \(p(x) = a_0 + a_1 x + \cdots + a_n x^n\), then \(p'(x) = a_1 + 2 a_2 x + \cdots + n a_n x^{n-1}\).

static identity() → Polynomial

Return the multiplicative identity polynomial 1.

integral(c0: int | float = 0.0) → Polynomial

Return an antiderivative \(\int p(x)\,dx\) with constant term c0.

参数:

c0 (Number, default=0.0) -- Constant of integration (the coefficient of \(x^0\)).

返回:

An antiderivative whose derivative equals self.

返回类型:

Polynomial

static zeros(deg: int) → Polynomial

Return the (canonical) zero polynomial.

参数:

deg (int) -- Requested degree (non‑negative). This value is accepted for API symmetry, but the canonical zero polynomial always has degree 0 after construction.

返回:

The zero polynomial.

返回类型:

Polynomial

Module contents

class algolib.maths.algebra.MatrixDense(rows: Sequence[Sequence[int | float]])

基类：object

A simple dense matrix (row-major) without external dependencies.

Designed for correctness and clarity (teaching/learning oriented). Not optimized for large-scale numerical workloads.

参数:

rows (Sequence[Sequence[Number]]) -- Row-major data. Must be rectangular (all rows same length).

抛出:

• InvalidTypeError -- If rows are not sequences of numbers.

• InvalidValueError -- If matrix is empty or not rectangular.

示例

>>> A = MatrixDense([[1, 2], [3, 4]])
>>> B = MatrixDense.identity(2)
>>> C = (A * B).rows == A.rows
True

T() → MatrixDense

Transpose.

copy() → MatrixDense

Shallow copy (rows are tuples; safe).

det() → float

Determinant via Gaussian elimination with partial pivoting (O(n^3)).

抛出:

InvalidValueError -- If matrix is not square.

equals(other: MatrixDense, tol: float = 1e-12) → bool

Element-wise comparison with tolerance for floats.

static eye(n: int) → MatrixDense

Alias of identity().

static from_rows(rows: Sequence[Sequence[int | float]]) → MatrixDense

Construct from row-major data (alias of the constructor).

static identity(n: int) → MatrixDense

Return the \(n \times n\) identity matrix.

inv() → MatrixDense

Inverse via Gauss-Jordan elimination (O(n^3)).

抛出:

InvalidValueError -- If matrix is not square or singular.

matvec(v: Sequence[int | float]) → List[float]

Matrix-vector product (Ax).

参数:

v (Sequence[Number]) -- Vector of length equal to number of columns.

返回:

Resulting vector.

返回类型:

list[float]

rows: Tuple[Tuple[int | float, ...], ...]

property shape: Tuple[int, int]

(n_rows, n_cols)

static zeros(n_rows: int, n_cols: int) → MatrixDense

Return an \((n_{\mathrm{rows}} × n_{\mathrm{cols}})\) zero matrix.

class algolib.maths.algebra.Polynomial(coeffs: Iterable[int | float])

基类：object

Univariate polynomial with real coefficients.

参数:

coeffs (Sequence[Number]) -- Coefficients in ascending degree order: [c0, c1, ..., cn] represents \(\sum_{k=0}^n c_k x^k\).

抛出:

• InvalidTypeError -- If any coefficient is not a real number.

• InvalidValueError -- If coeffs is empty.

备注

• Construction enforces a canonical form by stripping trailing zeros; the zero polynomial is thus represented as (0.0,) and has degree 0.

• All operations return new Polynomial instances (immutable API).

coeffs: Tuple[float, ...]

static constant(c: int | float) → Polynomial

Return a constant polynomial p(x) = c.

property degree: int

Degree of the polynomial (len(coeffs) - 1). By convention, deg(0) = 0.

derivative() → Polynomial

Return the analytical derivative \(p'(x)\).

备注

If \(p(x) = a_0 + a_1 x + \cdots + a_n x^n\), then \(p'(x) = a_1 + 2 a_2 x + \cdots + n a_n x^{n-1}\).

static identity() → Polynomial

Return the multiplicative identity polynomial 1.

integral(c0: int | float = 0.0) → Polynomial

Return an antiderivative \(\int p(x)\,dx\) with constant term c0.

参数:

c0 (Number, default=0.0) -- Constant of integration (the coefficient of \(x^0\)).

返回:

An antiderivative whose derivative equals self.

返回类型:

Polynomial

static zeros(deg: int) → Polynomial

Return the (canonical) zero polynomial.

参数:

deg (int) -- Requested degree (non‑negative). This value is accepted for API symmetry, but the canonical zero polynomial always has degree 0 after construction.

返回:

The zero polynomial.

返回类型:

Polynomial