# Linear Algebra¶

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

## Functions¶

ezdxf.math.linalg.gauss_jordan_solver(A: Iterable[Iterable[float]], B: Iterable[Iterable[float]])

Solves the linear equation system given by a nxn Matrix A . x = B, right-hand side quantities as nxm Matrix B by the Gauss-Jordan algorithm, which is the slowest of all, but it is very reliable. Returns a copy of the modified input matrix A and the result matrix x.

Internally used for matrix inverse calculation.

Parameters:
• A – matrix [[a11, a12, …, a1n], [a21, a22, …, a2n], [a21, a22, …, a2n], … [an1, an2, …, ann]]

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

Returns:

2-tuple of `Matrix` objects

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.gauss_jordan_inverse(A: Iterable[Iterable[float]])

Returns the inverse of matrix A as `Matrix` object.

Hint

For small matrices (n<10) is this function faster than LUDecomposition(m).inverse() and as fast even if the decomposition is already done.

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.gauss_vector_solver(A: Iterable[Iterable[float]], B: Iterable[float]) list[float]

Solves the linear equation system given by a nxn Matrix A . x = B, right-hand side quantities as vector B with n elements by the Gauss-Elimination algorithm, which is faster than the Gauss-Jordan algorithm. The speed improvement is more significant for solving multiple right-hand side quantities as matrix at once.

Reference implementation for error checking.

Parameters:
• A – matrix [[a11, a12, …, a1n], [a21, a22, …, a2n], [a21, a22, …, a2n], … [an1, an2, …, ann]]

• B – vector [b1, b2, …, bn]

Returns:

vector as list of floats

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.gauss_matrix_solver(A: Iterable[Iterable[float]], B: Iterable[Iterable[float]])

Solves the linear equation system given by a nxn Matrix A . x = B, right-hand side quantities as nxm Matrix B by the Gauss-Elimination algorithm, which is faster than the Gauss-Jordan algorithm.

Reference implementation for error checking.

Parameters:
• A – matrix [[a11, a12, …, a1n], [a21, a22, …, a2n], [a21, a22, …, a2n], … [an1, an2, …, ann]]

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

Returns:

matrix as `Matrix` object

Raises:

ZeroDivisionError – singular matrix

ezdxf.math.linalg.tridiagonal_vector_solver(A: Iterable[Iterable[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: Iterable[Iterable[float]], B: Iterable[Iterable[float]])

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[ezdxf.math.linalg.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)

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

ezdxf.math.linalg.freeze_matrix(A: Iterable[Iterable[float]] | Matrix)

Returns a frozen matrix, all data is stored in immutable tuples.

## Matrix Class¶

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

Basic matrix implementation without any optimization for speed or memory usage. Matrix data is stored in row major order, this means in a list of rows, where each row is a list of floats. Direct access to the data is accessible by the attribute `Matrix.matrix`.

The matrix can be frozen by function `freeze_matrix()` or method `Matrix.freeze()`, than the data is stored in immutable tuples.

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

nrows

Count of matrix rows.

ncols

Count of matrix columns.

shape

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

static reshape(items: Iterable[float], shape: Tuple[int, int])

Returns a new matrix for iterable items in the configuration of shape.

classmethod identity(shape: Tuple[int, int])

Returns the identity matrix for configuration shape.

row(index: int) list[float]

Returns row index as list of floats.

iter_row(index: int) Iterator[float]

Yield values of row index.

col(index: int) list[float]

Return column index as list of floats.

iter_col(index: int) Iterator[float]

Yield values of column index.

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]

iter_diag(index: int) Iterator[float]

Yield values of diagonal index, see also `diag()`.

rows() List[List[float]]

Return a list of all rows.

cols() List[List[float]]

Return a list of all columns.

set_row(index: int, items: float | Sequence[float] = 1.0) None

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

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]

append_row(items: Sequence[float]) None

Append a row to the matrix.

append_col(items: Sequence[float]) None

Append a column to the matrix.

swap_rows(a: int, b: int) None

Swap rows a and b inplace.

swap_cols(a: int, b: int) None

Swap columns a and b inplace.

transpose()

Returns a new transposed matrix.

inverse()

Returns inverse of matrix as new object.

determinant() float

Returns determinant of matrix, raises `ZeroDivisionError` if matrix is singular.

freeze()

Returns a frozen matrix, all data is stored in immutable tuples.

lu_decomp()

Returns the LU decomposition as `LUDecomposition` object, a faster linear equation solver.

__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, tolerance value for comparison is adjustable by the attribute `Matrix.abs_tol`.

Matrix addition by another matrix or a float, returns a new matrix.

__sub__(other: Matrix | float)

Matrix subtraction by another matrix or a float, returns a new matrix.

__mul__(other: Matrix | float)

Matrix multiplication by another matrix or a float, returns a new matrix.

## LUDecomposition Class¶

class ezdxf.math.linalg.LUDecomposition(A: Iterable[Iterable[float]])

Represents a LU decomposition matrix of A, raise `ZeroDivisionError` for a singular matrix.

This algorithm is a little bit faster than the Gauss-Elimination algorithm using CPython and much faster when using pypy.

The `LUDecomposition.matrix` attribute gives access to the matrix data as list of rows like in the `Matrix` class, and the `LUDecomposition.index` attribute gives access to the swapped row indices.

Parameters:

A – matrix [[a11, a12, …, a1n], [a21, a22, …, a2n], [a21, a22, …, a2n], … [an1, an2, …, ann]]

Raises:

ZeroDivisionError – singular matrix

nrows

Count of matrix rows (and cols).

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]

Returns:

vector as list of floats

solve_matrix(B: Iterable[Iterable[float]])

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]]

Returns:

matrix as `Matrix` object

inverse()

Returns the inverse of matrix as `Matrix` object, raise `ZeroDivisionError` for a singular matrix.

determinant() float

Returns the determinant of matrix, raises `ZeroDivisionError` if matrix is singular.

## 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: Iterable[Iterable[float]])

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

determinant() float

Returns the determinant of matrix.