SciPy-Linear Algebra
The SciPy library is one of the core packages for scientific computing that provides mathematical algorithms and convenience functions built on the NumPy extension of Python.
Interacting With NumPy
>>> import numpy as np
>>> a = np.array([1,2,3])
>>> b = np.array([(1+5j,2j,3j), (4j,5j,6j)])
>>> c = np.array([[(1.5,2,3), (4,5,6)], [(3,2,1), (4,5,6)]])
Index Tricks
>>> np.mgrid[0:5,0:5] Create a dense meshgrid
>>> np.ogrid[0:2,0:2] Create an open meshgrid
>>> np.r_[[3,[0]*5,-1:1:10j] Stack arrays vertically (row-wise)
>>> np.c_[b,c] Create stacked column-wise arrays
Shape Manupulation
>>> np.transpose(b) Permute array dimensions
>>> b.flatten() Flatten the array
>>> np.hstack((b,c)) Stack arrays horizontally (column-wise)
>>> np.vstack((a,b)) Stack arrays vertically (row-wise)
>>> np.hsplit(c,2) Split the array horizontally at the 2nd index
>>> np.vpslit(d,2) Split the array vertically at the 2nd index
Polynomials
>>> from numpy import poly1d
>>> p = poly1d([3,4,5]) Create a polynomial object
Vectorizing Functions
>>> def myfunc(a):
if a < 0:
return a*2
else:
return a/2
>>> np.vectorize(myfunc) Vectorize functions
Type Handling
>>> np.real(c) Return the real part of the array elements
>>> np.imag(c) Return the imaginary part of the array elements
>>> np.real_if_close(c,tol=1000) Return a real array if complex parts close to 0
>>> np.cast['f'](np.pi) Cast object to a data type
Other Useful Functions
>>> np.angle(b,deg=True) Return the angle of the complex argument
>>> g = np.linspace(0,np.pi,num=5) Create an array of evenly spaced values
>>> g [3:] += np.pi number of samples
>>> np.unwrap(g) Unwrap
>>> np.logspace(0,10,3) Create an array of evenly spaced values (log scale)
>>> np.select([c<4],[c*2]) Return values from a list of arrays depending on conditions
>>> misc.factorial(a) Factorial
>>> misc.comb(10,3,exact=True) Combine N things taken at k time
>>> misc.central_diff_weights(3) Weights for Np-point central derivative
>>> misc.derivative(myfunc,1.0) Find the n-th derivative of a function at a point
Linear Algebra
You’ll use the linalg and sparse modules. Note that scipy.linalg contains and expands on numpy.linalg.
>>> from scipy import linalg, sparse
Creating Matrices
>>> A = np.matrix(np.random.random((2,2)))
>>> B = np.asmatrix(b)
>>> C = np.mat(np.random.random((10,5)))
>>> D = np.mat([[3,4], [5,6]])
Basic Matrix Routines
Inverse
>>> A.I Inverse
>>> linalg.inv(A) Inverse
>>> A.T Tranpose matrix
>>> A.H Conjugate transposition
>>> np.trace(A) Trace
Norm
>>> linalg.norm(A) Frobenius norm
>>> linalg.norm(A,1) L1 norm (max column sum)
>>> linalg.norm(A,np.inf) L inf norm (max row sum)
Rank
>>> np.linalg.matrix_rank(C) Matrix rank
Determinant
>>> linalg.det(A) Determinant
Solving linear problems
>>> linalg.solve(A,b) Solver for dense matrices
>>> E = np.mat(a).T Solver for dense matrices
>>> linalg.lstsq(D,E) Least-squares solution to linear matrix equation
Generalized inverse
>>> linalg.pinv(C) Compute the pseudo-inverse of a matrix (least-squares solver)
>>> linalg.pinv2(C) Compute the pseudo-inverse of a matrix (SVD)
Creating Sparse Matrices
>>> F = np.eye(3, k=1) Create a 2X2 identity matrix
>>> G = np.mat(np.identity(2)) Create a 2x2 identity matrix
>>> C[C > 0.5] = 0
>>> H = sparse.csr_matrix(C) Compressed Sparse Row matrix
>>> I = sparse.csc_matrix(D) Compressed Sparse Column matrix
>>> J = sparse.dok_matrix(A) Dictionary Of Keys matrix
>>> E.todense() Sparse matrix to full matrix
>>> sparse.isspmatrix_csc(A) Identify sparse matrix
Inverse
>>> sparse.linalg.inv(I) Inverse
Norm
>>> sparse.linalg.norm(I) Norm
Solving linear problems
>>> sparse.linalg.spsolve(H,I) Solver for sparse matrices
Sparse Matrix Functions
>>> sparse.linalg.expm(I) Sparse matrix exponential
Matrix Functions
Addition
>>> np.add(A,D) Addition
Subtraction
>>> np.subtract(A,D) Subtraction
Division
>>> np.divide(A,D) Division
Multiplication
>>> np.multiply(D,A) Multiplication
>>> np.dot(A,D) Dot product
>>> np.vdot(A,D) Vector dot product
>>> np.inner(A,D) Inner product
>>> np.outer(A,D) Outer product
>>> np.tensordot(A,D) Tensor dot product
>>> np.kron(A,D) Kronecker product
Exponential Functions
>>> linalg.expm(A) Matrix exponential
>>> linalg.expm2(A) Matrix exponential (Taylor Series)
>>> linalg.expm3(D) Matrix exponential (eigenvalue decomposition)
Logarithm Function
>>> linalg.logm(A) Matrix logarithm
Trigonometric Tunctions
>>> linalg.sinm(D) Matrix sine
>>> linalg.cosm(D) Matrix cosine
>>> linalg.tanm(A) Matrix tangent
Hyperbolic Trigonometric Functions
>>> linalg.sinhm(D) Hypberbolic matrix sine
>>> linalg.coshm(D) Hyperbolic matrix cosine
>>> linalg.tanhm(A) Hyperbolic matrix tangent
Matrix Sign Function
>>> np.sigm(A) Matrix sign function
Matrix Square Root
>>> linalg.sqrtm(A) Matrix square root
Arbitrary Functions
>>> linalg.funm(A, lambda x: x*x) Evaluate matrix function
Decompositions
Eigenvalues and Eigenvectors
>>> la, v = linalg.eig(A) Solve ordinary or generalized eigenvalue problem for square matrix
>>> l1, l2 = la Unpack eigenvalues
>>> v[:,0] First eigenvector
>>> v[:,1] Second eigenvector
>>> linalg.eigvals(A) Unpack eigenvalues
Singular Value Decomposition
>>> U,s,Vh = linalg.svd(B) Singular Value Decomposition (SVD)
>>> M,N = B.shape
>>> Sig = linalg.diagsvd(s,M,N) Construct sigma matrix in SVD LU Decomposition
>>> P,L,U = linalg.lu(C) LU Decomposition
Sparse Matrix Decompositions
>>> la, v = sparse.linalg.eigs(F,1) Eigenvalues and eigenvectors
>>> sparse.linalg.svds(H, 2) SVD
