Matrices¶
Creating matrices¶
Basic methods¶
Matrices in mpmath are implemented using dictionaries. Only non-zero values are stored, so it is cheap to represent sparse matrices.
The most basic way to create one is to use the matrix class directly. You can create an empty matrix specifying the dimensions:
>>> from mpmath import *
>>> mp.dps = 15
>>> matrix(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
['0.0', '0.0', '0.0']])
Calling matrix with one dimension will create a square matrix.
To access the dimensions of a matrix, use the rows or cols keyword:
>>> A = matrix(3, 2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0'],
['0.0', '0.0']])
>>> A.rows
3
>>> A.cols
2
You can also change the dimension of an existing matrix. This will set the new elements to 0. If the new dimension is smaller than before, the concerning elements are discarded:
>>> A.rows = 2
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Internally convert_lossless is used every time an element is set. This is done using the syntax A[row,column], counting from 0:
>>> A = matrix(2)
>>> A[1,1] = 1 + 1j
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '(1.0 + 1.0j)']])
You can use the keyword force_type to change the function which is called on every new element:
>>> matrix(2, 5, force_type=int)
matrix(
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]])
A more comfortable way to create a matrix lets you use nested lists:
>>> matrix([[1, 2], [3, 4]])
matrix(
[['1.0', '2.0'],
['3.0', '4.0']])
If you want to preserve the type of the elements you can use force_type=None:
>>> matrix([[1, 2.5], [1j, mpf(2)]], force_type=None)
matrix(
[[1, 2.5],
[1j, '2.0']])
Advanced methods¶
Convenient functions are available for creating various standard matrices:
>>> zeros(2)
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> ones(2)
matrix(
[['1.0', '1.0'],
['1.0', '1.0']])
>>> diag([1, 2, 3]) # diagonal matrix
matrix(
[['1.0', '0.0', '0.0'],
['0.0', '2.0', '0.0'],
['0.0', '0.0', '3.0']])
>>> eye(2) # identity matrix
matrix(
[['1.0', '0.0'],
['0.0', '1.0']])
You can even create random matrices:
>>> randmatrix(2) # doctest:+SKIP
matrix(
[['0.53491598236191806', '0.57195669543302752'],
['0.85589992269513615', '0.82444367501382143']])
Vectors¶
Vectors may also be represented by the matrix class (with rows = 1 or cols = 1). For vectors there are some things which make life easier. A column vector can be created using a flat list, a row vectors using an almost flat nested list:
>>> matrix([1, 2, 3])
matrix(
[['1.0'],
['2.0'],
['3.0']])
>>> matrix([[1, 2, 3]])
matrix(
[['1.0', '2.0', '3.0']])
Optionally vectors can be accessed like lists, using only a single index:
>>> x = matrix([1, 2, 3])
>>> x[1]
mpf('2.0')
>>> x[1,0]
mpf('2.0')
Other¶
Like you probably expected, matrices can be printed:
>>> print randmatrix(3) # doctest:+SKIP
[ 0.782963853573023 0.802057689719883 0.427895717335467]
[0.0541876859348597 0.708243266653103 0.615134039977379]
[ 0.856151514955773 0.544759264818486 0.686210904770947]
Use nstr or nprint to specify the number of digits to print:
>>> nprint(randmatrix(5), 3) # doctest:+SKIP
[2.07e-1 1.66e-1 5.06e-1 1.89e-1 8.29e-1]
[6.62e-1 6.55e-1 4.47e-1 4.82e-1 2.06e-2]
[4.33e-1 7.75e-1 6.93e-2 2.86e-1 5.71e-1]
[1.01e-1 2.53e-1 6.13e-1 3.32e-1 2.59e-1]
[1.56e-1 7.27e-2 6.05e-1 6.67e-2 2.79e-1]
As matrices are mutable, you will need to copy them sometimes:
>>> A = matrix(2)
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
>>> B = A.copy()
>>> B[0,0] = 1
>>> B
matrix(
[['1.0', '0.0'],
['0.0', '0.0']])
>>> A
matrix(
[['0.0', '0.0'],
['0.0', '0.0']])
Finally, it is possible to convert a matrix to a nested list. This is very useful, as most Python libraries involving matrices or arrays (namely NumPy or SymPy) support this format:
>>> B.tolist()
[[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]]
Matrix operations¶
You can add and substract matrices of compatible dimensions:
>>> A = matrix([[1, 2], [3, 4]])
>>> B = matrix([[-2, 4], [5, 9]])
>>> A + B
matrix(
[['-1.0', '6.0'],
['8.0', '13.0']])
>>> A - B
matrix(
[['3.0', '-2.0'],
['-2.0', '-5.0']])
>>> A + ones(3) # doctest:+ELLIPSIS
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "...", line 238, in __add__
raise ValueError('incompatible dimensions for addition')
ValueError: incompatible dimensions for addition
It is possible to multiply or add matrices and scalars. In the latter case the operation will be done element-wise:
>>> A * 2
matrix(
[['2.0', '4.0'],
['6.0', '8.0']])
>>> A / 4
matrix(
[['0.25', '0.5'],
['0.75', '1.0']])
>>> A - 1
matrix(
[['0.0', '1.0'],
['2.0', '3.0']])
Of course you can perform matrix multiplication, if the dimensions are compatible:
>>> A * B
matrix(
[['8.0', '22.0'],
['14.0', '48.0']])
>>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]])
matrix(
[['2.0']])
You can raise powers of square matrices:
>>> A**2
matrix(
[['7.0', '10.0'],
['15.0', '22.0']])
Negative powers will calculate the inverse:
>>> A**-1
matrix(
[['-2.0', '1.0'],
['1.5', '-0.5']])
>>> nprint(A * A**-1, 3)
[ 1.0 1.08e-19]
[-2.17e-19 1.0]
Matrix transposition is straightforward:
>>> A = ones(2, 3)
>>> A
matrix(
[['1.0', '1.0', '1.0'],
['1.0', '1.0', '1.0']])
>>> A.T
matrix(
[['1.0', '1.0'],
['1.0', '1.0'],
['1.0', '1.0']])
Norms¶
Sometimes you need to know how “large” a matrix or vector is. Due to their multidimensional nature it’s not possible to compare them, but there are several functions to map a matrix or a vector to a positive real number, the so called norms.
Linear algebra¶
Linear equations¶
Basic linear algebra is implemented; you can for example solve the linear equation system:
x + 2*y = -10
3*x + 4*y = 10
using lu_solve:
>>> A = matrix([[1, 2], [3, 4]])
>>> b = matrix([-10, 10])
>>> x = lu_solve(A, b)
>>> x
matrix(
[['30.0'],
['-20.0']])
If you don’t trust the result, use residual to calculate the residual ||A*x-b||:
>>> residual(A, x, b)
matrix(
[['3.46944695195361e-18'],
['3.46944695195361e-18']])
>>> str(eps)
'2.22044604925031e-16'
As you can see, the solution is quite accurate. The error is caused by the inaccuracy of the internal floating point arithmetic. Though, it’s even smaller than the current machine epsilon, which basically means you can trust the result.
If you need more speed, use NumPy. Or choose a faster data type using the keyword force_type:
>>> lu_solve(A, b, force_type=float)
matrix(
[[29.999999999999996],
[-19.999999999999996]])
lu_solve accepts overdetermined systems. It is usually not possible to solve such systems, so the residual is minimized instead. Internally this is done using Cholesky decomposition to compute a least squares approximation. This means that that lu_solve will square the errors. If you can’t afford this, use qr_solve instead. It is twice as slow but more accurate, and it calculates the residual automatically.
Matrix factorization¶
The function lu computes an explicit LU factorization of a matrix:
>>> P, L, U = lu(matrix([[0,2,3],[4,5,6],[7,8,9]]))
>>> print P
[0.0 0.0 1.0]
[1.0 0.0 0.0]
[0.0 1.0 0.0]
>>> print L
[ 1.0 0.0 0.0]
[ 0.0 1.0 0.0]
[0.571428571428571 0.214285714285714 1.0]
>>> print U
[7.0 8.0 9.0]
[0.0 2.0 3.0]
[0.0 0.0 0.214285714285714]
>>> print P.T*L*U
[0.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Interval matrices¶
Matrices may contain interval elements. This allows for performing basic linear algebra operations such as matrix multiplication and equation solving with rigorous error bounds:
>>> a = matrix([['0.1','0.3','1.0'],
... ['7.1','5.5','4.8'],
... ['3.2','4.4','5.6']], force_type=mpi)
>>>
>>> b = matrix(['4','0.6','0.5'], force_type=mpi)
>>> c = lu_solve(a, b)
>>> print c
[ [5.2582327113062393041, 5.2582327113062749951]]
[[-13.155049396267856583, -13.155049396267821167]]
[ [7.4206915477497212555, 7.4206915477497310922]]
>>> print a*c
[ [3.9999999999999866773, 4.0000000000000133227]]
[[0.59999999999972430942, 0.60000000000027142733]]
[[0.49999999999982236432, 0.50000000000018474111]]
