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]]) tuple[ezdxf.math.linalg.Matrix, ezdxf.math.linalg.Matrix] ¶
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]]) Matrix ¶
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]]) Matrix ¶
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 byBandedMatrixLU(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]]) 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 byBandedMatrixLU(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 ifFalse
.
- 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 ifFalse
.
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 methodMatrix.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]) Matrix ¶
Returns a new matrix for iterable items in the configuration of shape.
- classmethod identity(shape: Tuple[int, int]) Matrix ¶
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]
- 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.
- determinant() float ¶
Returns determinant of matrix, raises
ZeroDivisionError
if matrix is singular.
- lu_decomp() LUDecomposition ¶
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 attributeMatrix.abs_tol
.
- __add__(other: Matrix | float) Matrix ¶
Matrix addition 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 theMatrix
class, and theLUDecomposition.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]]) 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]]
- Returns:
matrix as
Matrix
object
- inverse() Matrix ¶
Returns the inverse of matrix as
Matrix
object, raiseZeroDivisionError
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]]) 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
- determinant() float ¶
Returns the determinant of matrix.