Solve matrix equation python

Solve a linear matrix equation or system of linear scalar equations in Python

To solve a linear matrix equation, use the numpy.linalg.solve() method in Python. The method computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b. Returns a solution to the system a x = b. Returned shape is identical to b. The 1st parameter a is the Coefficient matrix. The 2nd parameter b is the Ordinate or “dependent variable” values.

Steps

At first, import the required libraries —

Creating two 2D numpy arrays using the array() method. Consider the system of equations x0 + 2 * x1 = 1 and 3 * x0 + 5 * x1 = 2 −

arr1 = np.array([[1, 2], [3, 5]]) arr2 = np.array([1, 2])

print("Array1. \n",arr1) print("\nArray2. \n",arr2)

Check the Dimensions of both the arrays −

print("\nDimensions of Array1. \n",arr1.ndim) print("\nDimensions of Array2. \n",arr2.ndim)

Check the Shape of both the arrays −

Print(“\nShape of Array1…\n”,arr1.shape) print("\nShape of Array2. \n",arr2.shape)

To solve a linear matrix equation, use the numpy.linalg.solve() method −

print("\nResult. \n",np.linalg.solve(arr1, arr2))

Example

import numpy as np # Creating two 2D numpy arrays using the array() method # Consider the system of equations x0 + 2 * x1 = 1 and 3 * x0 + 5 * x1 = 2 arr1 = np.array([[1, 2], [3, 5]]) arr2 = np.array([1, 2]) # Display the arrays print("Array1. \n",arr1) print("\nArray2. \n",arr2) # Check the Dimensions of both the arrays print("\nDimensions of Array1. \n",arr1.ndim) print("\nDimensions of Array2. \n",arr2.ndim) # Check the Shape of both the arrays print("\nShape of Array1. \n",arr1.shape) print("\nShape of Array2. \n",arr2.shape) # To solve a linear matrix equation, use the numpy.linalg.solve() method in Python. print("\nResult. \n",np.linalg.solve(arr1, arr2))

Output

Array1. [[1 2] [3 5]] Array2. [1 2] Dimensions of Array1. 2 Dimensions of Array2. 1 Shape of Array1. (2, 2) Shape of Array2. (2,) Result. [-1. 1.]

Источник

numpy.linalg.solve#

Solve a linear matrix equation, or system of linear scalar equations.

Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b.

Parameters : a (…, M, M) array_like

b <(…, M,), (…, M, K)>, array_like

Ordinate or “dependent variable” values.

Returns : x <(…, M,), (…, M, K)>ndarray

Solution to the system a x = b. Returned shape is identical to b.

If a is singular or not square.

Similar function in SciPy.

Broadcasting rules apply, see the numpy.linalg documentation for details.

The solutions are computed using LAPACK routine _gesv .

a must be square and of full-rank, i.e., all rows (or, equivalently, columns) must be linearly independent; if either is not true, use lstsq for the least-squares best “solution” of the system/equation.

G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 22.

Solve the system of equations x0 + 2 * x1 = 1 and 3 * x0 + 5 * x1 = 2 :

>>> a = np.array([[1, 2], [3, 5]]) >>> b = np.array([1, 2]) >>> x = np.linalg.solve(a, b) >>> x array([-1., 1.]) 

Check that the solution is correct:

>>> np.allclose(np.dot(a, x), b) True 

Источник

Linear algebra ( numpy.linalg )#

The NumPy linear algebra functions rely on BLAS and LAPACK to provide efficient low level implementations of standard linear algebra algorithms. Those libraries may be provided by NumPy itself using C versions of a subset of their reference implementations but, when possible, highly optimized libraries that take advantage of specialized processor functionality are preferred. Examples of such libraries are OpenBLAS, MKL (TM), and ATLAS. Because those libraries are multithreaded and processor dependent, environmental variables and external packages such as threadpoolctl may be needed to control the number of threads or specify the processor architecture.

The SciPy library also contains a linalg submodule, and there is overlap in the functionality provided by the SciPy and NumPy submodules. SciPy contains functions not found in numpy.linalg , such as functions related to LU decomposition and the Schur decomposition, multiple ways of calculating the pseudoinverse, and matrix transcendentals such as the matrix logarithm. Some functions that exist in both have augmented functionality in scipy.linalg . For example, scipy.linalg.eig can take a second matrix argument for solving generalized eigenvalue problems. Some functions in NumPy, however, have more flexible broadcasting options. For example, numpy.linalg.solve can handle “stacked” arrays, while scipy.linalg.solve accepts only a single square array as its first argument.

The term matrix as it is used on this page indicates a 2d numpy.array object, and not a numpy.matrix object. The latter is no longer recommended, even for linear algebra. See the matrix object documentation for more information.

The @ operator#

Introduced in NumPy 1.10.0, the @ operator is preferable to other methods when computing the matrix product between 2d arrays. The numpy.matmul function implements the @ operator.

Matrix and vector products#

Dot product of two arrays.

Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order.

Return the dot product of two vectors.

Inner product of two arrays.

Compute the outer product of two vectors.

Matrix product of two arrays.

Compute tensor dot product along specified axes.

einsum (subscripts, *operands[, out, dtype, . ])

Evaluates the Einstein summation convention on the operands.

einsum_path (subscripts, *operands[, optimize])

Evaluates the lowest cost contraction order for an einsum expression by considering the creation of intermediate arrays.

Raise a square matrix to the (integer) power n.

Kronecker product of two arrays.

Decompositions#

Compute the qr factorization of a matrix.

linalg.svd (a[, full_matrices, compute_uv, . ])

Singular Value Decomposition.

Matrix eigenvalues#

Compute the eigenvalues and right eigenvectors of a square array.

Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.

Compute the eigenvalues of a general matrix.

Compute the eigenvalues of a complex Hermitian or real symmetric matrix.

Norms and other numbers#

Compute the condition number of a matrix.

Compute the determinant of an array.

Return matrix rank of array using SVD method

Compute the sign and (natural) logarithm of the determinant of an array.

trace (a[, offset, axis1, axis2, dtype, out])

Return the sum along diagonals of the array.

Solving equations and inverting matrices#

Solve a linear matrix equation, or system of linear scalar equations.

Solve the tensor equation a x = b for x.

Return the least-squares solution to a linear matrix equation.

Compute the (multiplicative) inverse of a matrix.

Compute the (Moore-Penrose) pseudo-inverse of a matrix.

Compute the ‘inverse’ of an N-dimensional array.

Exceptions#

Generic Python-exception-derived object raised by linalg functions.

Linear algebra on several matrices at once#

Several of the linear algebra routines listed above are able to compute results for several matrices at once, if they are stacked into the same array.

This is indicated in the documentation via input parameter specifications such as a : (. M, M) array_like . This means that if for instance given an input array a.shape == (N, M, M) , it is interpreted as a “stack” of N matrices, each of size M-by-M. Similar specification applies to return values, for instance the determinant has det : (. ) and will in this case return an array of shape det(a).shape == (N,) . This generalizes to linear algebra operations on higher-dimensional arrays: the last 1 or 2 dimensions of a multidimensional array are interpreted as vectors or matrices, as appropriate for each operation.

Источник

NumPy, часть 4: linalg

В прошлых частях мы разбирались с основными операциями над массивами и randomом в NumPy. Теперь же мы приступим к более серьёзным вещам, которые есть в NumPy. Первый на очереди у нас модуль numpy.linalg, позволяющий делать многие операции из линейной алгебры.

Возведение в степень

linalg.matrix_power(M, n) — возводит матрицу в степень n.

Разложения

linalg.cholesky(a) — разложение Холецкого.

linalg.qr(a[, mode]) — QR разложение.

linalg.svd(a[, full_matrices, compute_uv]) — сингулярное разложение.

Некоторые характеристики матриц

linalg.eig(a) — собственные значения и собственные векторы.

linalg.norm(x[, ord, axis]) — норма вектора или оператора.

linalg.cond(x[, p]) — число обусловленности.

linalg.det(a) — определитель.

linalg.slogdet(a) — знак и логарифм определителя (для избежания переполнения, если сам определитель очень маленький).

Системы уравнений

linalg.solve(a, b) — решает систему линейных уравнений Ax = b.

linalg.tensorsolve(a, b[, axes]) — решает тензорную систему линейных уравнений Ax = b.

linalg.lstsq(a, b[, rcond]) — метод наименьших квадратов.

linalg.inv(a) — обратная матрица.

• linalg.LinAlgError — исключение, вызываемое данными функциями в случае неудачи (например, при попытке взять обратную матрицу от вырожденной).
• Подробная документация, как всегда, на английском: https://docs.scipy.org/doc/numpy/reference/routines.linalg.html
• Массивы большей размерности в большинстве функций linalg интерпретируются как набор из нескольких массивов нужной размерности. Таким образом, можно одним вызовом функции проделывать операции над несколькими объектами.

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