Linear Algebra

Linear algebra module for internal usage: ezdxf.math.linalg

Functions

ezdxf.math.linalg.tridiagonal_vector_solver(A: List[List[float]], B: Iterable[float]) → list[float]

Solves the linear equation system given by a tri-diagonal nxn Matrix A . x = B, right-hand side quantities as vector B. Matrix A is diagonal matrix defined by 3 diagonals [-1 (a), 0 (b), +1 (c)].

Note: a0 is not used but has to be present, cn-1 is also not used and must not be present.

If an ZeroDivisionError exception occurs, the equation system can possibly be solved by BandedMatrixLU(A, 1, 1).solve_vector(B)

Parameters:

A –

diagonal matrix [[a0..an-1], [b0..bn-1], [c0..cn-1]]

[[b0, c0, 0, 0, ...],
[a1, b1, c1, 0, ...],
[0, a2, b2, c2, ...],
... ]

B – iterable of floats [[b1, b1, …, bn]

Returns:

list of floats

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.tridiagonal_matrix_solver(A: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]], B: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]]) → Matrix

Solves the linear equation system given by a tri-diagonal nxn Matrix A . x = B, right-hand side quantities as nxm Matrix B. Matrix A is diagonal matrix defined by 3 diagonals [-1 (a), 0 (b), +1 (c)].

Note: a0 is not used but has to be present, cn-1 is also not used and must not be present.

If an ZeroDivisionError exception occurs, the equation system can possibly be solved by BandedMatrixLU(A, 1, 1).solve_vector(B)

Parameters:

A –

diagonal matrix [[a0..an-1], [b0..bn-1], [c0..cn-1]]

[[b0, c0, 0, 0, ...],
[a1, b1, c1, 0, ...],
[0, a2, b2, c2, ...],
... ]

B – matrix [[b11, b12, …, b1m], [b21, b22, …, b2m], … [bn1, bn2, …, bnm]]

Returns:

matrix as Matrix object

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.banded_matrix(A: Matrix, check_all=True) → tuple[Matrix, int, int]

Transform matrix A into a compact banded matrix representation. Returns compact representation as Matrix object and lower- and upper band count m1 and m2.

Parameters:

A – input Matrix
check_all – check all diagonals if True or abort testing after first all zero diagonal if False.

ezdxf.math.linalg.detect_banded_matrix(A: Matrix, check_all=True) → tuple[int, int]

Returns lower- and upper band count m1 and m2.

Parameters:

A – input Matrix
check_all – check all diagonals if True or abort testing after first all zero diagonal if False.

ezdxf.math.linalg.compact_banded_matrix(A: Matrix, m1: int, m2: int) → Matrix

Returns compact banded matrix representation as Matrix object.

Parameters:

A – matrix to transform
m1 – lower band count, excluding main matrix diagonal
m2 – upper band count, excluding main matrix diagonal

Matrix Class

class ezdxf.math.linalg.Matrix(items: Any = None, shape: Tuple[int, int] | None = None, matrix: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]] | None = None)

Basic matrix implementation based numpy.ndarray. Matrix data is stored in row major order, this means in a list of rows, where each row is a list of floats.

Initialization:

Matrix(shape=(rows, cols)) … new matrix filled with zeros

Matrix(matrix[, shape=(rows, cols)]) … from copy of matrix and optional reshape

Matrix([[row_0], [row_1], …, [row_n]]) … from Iterable[Iterable[float]]

Matrix([a1, a2, …, an], shape=(rows, cols)) … from Iterable[float] and shape

Changed in version 1.2: Implementation based on numpy.ndarray.

matrix: matrix data as numpy.ndarray

nrows: Count of matrix rows.

ncols: Count of matrix columns.

shape: Shape of matrix as (n, m) tuple for n rows and m columns.

append_col(items: Sequence[float]) → None: Append a column to the matrix.

append_row(items: Sequence[float]) → None: Append a row to the matrix.

col(index: int) → list[float]: Return column index as list of floats.

cols() → list[list[float]]: Return a list of all columns.

determinant() → float: Returns determinant of matrix, raises ZeroDivisionError if matrix is singular.

diag(index: int) → list[float]

Returns diagonal index as list of floats.

An index of 0 specifies the main diagonal, negative values specifies diagonals below the main diagonal and positive values specifies diagonals above the main diagonal.

e.g. given a 4x4 matrix:

index 0 is [00, 11, 22, 33],
index -1 is [10, 21, 32] and
index +1 is [01, 12, 23]

freeze() → Matrix: Returns a frozen matrix, all data is stored in immutable tuples.

classmethod identity(shape: Tuple[int, int]) → Matrix: Returns the identity matrix for configuration shape.

inverse() → Matrix: Returns inverse of matrix as new object.

isclose(other: object) → bool: Returns True if matrices are close to equal, tolerance value for comparison is adjustable by the attribute Matrix.abs_tol.

static reshape(items: Iterable[float], shape: Tuple[int, int]) → Matrix: Returns a new matrix for iterable items in the configuration of shape.

row(index: int) → list[float]: Returns row index as list of floats.

rows() → list[list[float]]: Return a list of all rows.

set_col(index: int, items: float | Iterable[float] = 1.0) → None: Set column values to a fixed value or from an iterable of floats.

set_diag(index: int = 0, items: float | Iterable[float] = 1.0) → None

Set diagonal values to a fixed value or from an iterable of floats.

An index of 0 specifies the main diagonal, negative values specifies diagonals below the main diagonal and positive values specifies diagonals above the main diagonal.

e.g. given a 4x4 matrix: index 0 is [00, 11, 22, 33], index -1 is [10, 21, 32] and index +1 is [01, 12, 23]

set_row(index: int, items: float | Iterable[float] = 1.0) → None: Set row values to a fixed value or from an iterable of floats.

transpose() → Matrix: Returns a new transposed matrix.

__getitem__(item: tuple[int, int]) → float: Get value by (row, col) index tuple, fancy slicing as known from numpy is not supported.

__setitem__(item: tuple[int, int], value: float): Set value by (row, col) index tuple, fancy slicing as known from numpy is not supported.

__eq__(other: object) → bool: Returns True if matrices are equal.

__add__(other: Matrix | float) → Matrix: Matrix addition by another matrix or a float, returns a new matrix.

__sub__(other: Matrix | float) → Matrix: Matrix subtraction by another matrix or a float, returns a new matrix.

__mul__(other: Matrix | float) → Matrix: Matrix multiplication by another matrix or a float, returns a new matrix.

NumpySolver

class ezdxf.math.linalg.NumpySolver(A: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]])

Replaces in v1.2 the LUDecomposition solver.

solve_vector(B: Iterable[float]) → list[float]

Solves the linear equation system given by the nxn Matrix A . x = B, right-hand side quantities as vector B with n elements.

Parameters:: B – vector [b1, b2, …, bn]
Raises:: numpy.linalg.LinAlgError – singular matrix

solve_matrix(B: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]]) → Matrix

Solves the linear equation system given by the nxn Matrix A . x = B, right-hand side quantities as nxm Matrix B.

Parameters:: B – matrix [[b11, b12, …, b1m], [b21, b22, …, b2m], … [bn1, bn2, …, bnm]]
Raises:: numpy.linalg.LinAlgError – singular matrix

BandedMatrixLU Class

class ezdxf.math.linalg.BandedMatrixLU(A: Matrix, m1: int, m2: int)

Represents a LU decomposition of a compact banded matrix.

upper: Upper triangle

lower: Lower triangle

m1: Lower band count, excluding main matrix diagonal

m2: Upper band count, excluding main matrix diagonal

index: Swapped indices

nrows: Count of matrix rows.

solve_vector(B: Iterable[float]) → list[float]

Solves the linear equation system given by the banded nxn Matrix A . x = B, right-hand side quantities as vector B with n elements.

Parameters:: B – vector [b1, b2, …, bn]
Returns:: vector as list of floats

solve_matrix(B: List[List[float]] | ndarray[tuple[Any, ...], dtype[float64]]) → Matrix

Solves the linear equation system given by the banded nxn Matrix A . x = B, right-hand side quantities as nxm Matrix B.

Parameters:: B – matrix [[b11, b12, …, b1m], [b21, b22, …, b2m], … [bn1, bn2, …, bnm]]
Returns:: matrix as Matrix object