The concept of matrix decompositions is what makes Numerical Linear Algebra an efficient tool in Scientific Computing. If the matrix representing a problem is simple enough, any basic generic algorithm can find the solutions optimally (that is, fast, with minimal storage of data, and without a significant roundoff error). But, in real life, this situation seldom occurs. What we do in the general case is finding a suitable matrix factorization and tailoring an algorithm that is optimal on each factor, thus gaining on each step an obvious advantage. In this section, we explore the different factorizations included in the modules scipy.linalg
and scipy.sparse.linalg
that help us achieve a robust solution to matrix equations.
We have the following factorizations in this category:
It is always possible to perform a factorization of a square matrix A
as a product A = P ● L ● U of a permutation matrix P
(which performs a permutation of the rows of A
), a lower triangular matrix L
, and an upper triangular matrix U
:
Constructor 
Description 


Pivoted LU decomposition 

Pivoted LU decomposition 

Pivoted LU decomposition 

Incomplete pivoted LU decomposition 
For a square, symmetric, and positive definite matrix A
, we can realize the matrix as the product A = U^{T} ● U of an upper triangular matrix U
with its transpose, or as the product A = L^{T} ● L of a lower triangular matrix L
with its transpose. All the diagonal entries of U
or L
are strictly positive numbers:
Constructor 
Description 


Cholesky decomposition 

Cholesky decomposition for Hermitian positivedefinite banded matrices 
We can realize any matrix of size m × n as the product A=Q ● R of a square orthogonal matrix Q
of size m × m, with an upper triangular matrix R
of the same size as A
.
Constructor 
Description 


QR decomposition of a matrix 
We can realize any matrix A
as the product A = U ● D ● V^{H} of a unitary matrix U
with a diagonal matrix D
(where all entries in the diagonal are positive numbers), and the Hermitian transpose of another unitary matrix V
. The values on the diagonal of D
are called the singular values of A
.
Constructor 
Description 


Singular value decomposition 

Singular values 

Diagonal matrix of an SVD, from singular values `s` and prescribed size 

Largest 
In SciPy, we have robust algorithms to solve any matrix equation based on the following cases:
Given a square matrix
A
, and a righthand sideb
(which can be a onedimensional vector or another matrix with the same number of rows asA
), the basic systems are as follows:A ● x = b
A^{T} ● x = b
A^{H} ● x = b
Given any matrix
A
(not necessarily square) and a righthand side vector/matrixb
of an appropriate size, the least squares solution to the equation A ● x = b. This is, finding a vectorx
that minimizes the Frobenius norm of the expression A ● x  b.For the same case as before, and an extra damping coefficient
d
, the regularized least squares solution to the equation A ● x = b that minimizes the functionalnorm(A * x  b, 'f')**2 + d**2 * norm(x, 'f')**2
.Given square matrices
A
andB
, and a righthand side matrixQ
with appropriate sizes, the Sylvester system is A ● X + X ● B = Q.For a square matrix
A
and matrixQ
of an appropriate size, the continuous Lyapunov equation is A ● X + X ● A^{H} = Q.For matrices
A
andQ
, as in the previous case, the discrete Lyapunov equation is X  A ● X ● A^{H} = Q.Given square matrices
A
,Q
, andR
, and another matrixB
with an appropriate size, the continuous algebraic Riccati equation is A^{T} ● X + X ● A  X ● B ● R^{1} ● BT ● X + Q = 0.For matrices as in the previous case, the Discrete Algebraic Riccati equation is X = A^{T} ● X ● A  (A^{T} ● X ● B) ● (R+BT ● X ● B)^{1} ● (BT ● X ● A) + Q.
In any case, mastering matrix equations with SciPy basically means identifying the matrices involved and choosing the most adequate algorithm in the libraries to perform the requested operations. Besides being able to compute a solution with the least possible amount of roundoff error, we need to do so in the fastest possible way, and by using as few memory resources as possible.
Let us start with the easiest possible case: The basic system of linear equations A ● x = b (or the other two variants), where A
is a generic lower or upper triangular square matrix. In theory, these systems are easily solved by forward substitution (for lower triangular matrices) or back substitution (for upper triangular matrices). In SciPy, we accomplish this task with the function solve_triangular
in the module scipy.linalg
.
For this initial example, we will construct A
as a lower triangular Pascal matrix of size 1024 × 1024, where the nonzero values have been filtered: odd values are turned into ones, while even values are turned into zeros. The righthand side b
is a vector with 1024
ones.
In [1]: import numpy as np, \ ...: scipy.linalg as spla, scipy.sparse as spsp, \ ...: scipy.sparse.linalg as spspla In [2]: A = (spla.pascal(1024, kind='lower')%2 != 0) In [3]: %timeit spla.solve_triangular(A, np.ones(1024)) 10 loops, best of 3: 6.64 ms per loop
To solve the other related systems that involve the matrix A
, we employ the optional parameter trans
(by default set to 0
or N
, giving the basic system A ● x = b). If trans
is set to T
or 1
, we solve the system A^{T} ● x = b instead. If trans
is set to C
or 2
, we solve A^{H} ● x = b instead.
The next cases in terms of algorithm simplicity are those of basic systems A ● x = b, where A
is a square banded matrix. We use the routines solve_banded
(for a generic banded matrix) or solveh_banded
(for a generic real symmetric of complex Hermitian banded matrix). Both of them belong to the module scipy.linalg
.
Note
The functions solve_banded
and solveh_banded
are wrappers for the LAPACK
functions GBSV
, and PBSV
, respectively.
Neither function accepts a matrix in the usual format. For example, since solveh_banded
expects a symmetric banded matrix, the function requires as input only the elements of the diagonals on and under/over the main diagonal, stored sequentially from the top to the bottom.
This input method is best explained through a concrete example. Take the following symmetric banded matrix:
2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2 1 0 0 0 0 1 2
The size of the matrix is 6 × 6, and there are only three nonzero diagonals, two of which are identical due to symmetry. We collect the two relevant nonzero diagonals in ndarray
of size 2 × 6 in one of two ways, as follows:
If we decide to input the entries from the upper triangular matrix, we collect first the diagonals from the top to the bottom (ending in the main diagonal), right justified:
* 1 1 1 1 1 2 2 2 2 2 2
If we decide to input the entries from the lower triangular matrix, we collect the diagonals from the top to the bottom (starting from the main diagonal), left justified:
2 2 2 2 2 2 1 1 1 1 1 * In [4]: B_banded = np.zeros((2,6)); \ ...: B_banded[0,1:] = 1; \ ...: B_banded[1,:] = 2 In [5]: spla.solveh_banded(B_banded, np.ones(6)) Out[5]: array([ 3., 5., 6., 6., 5., 3.])
For a nonsymmetric banded square matrix, we use solve_banded
instead, and the input matrix also needs to be stored in this special way:
Count the number of nonzero diagonals under the main diagonal (set that to
l
). Count the number of nonzero diagonals over the main diagonal (set that tou
). Setr = l + u + 1
.If the matrix has size n × n, create
ndarray
with n columns and r rows. We refer to this storage as a matrix in theAB
form, or anAB
matrix, for short.Store in the AB matrix only the relevant nonzero diagonals, from the top to the bottom, in order. Diagonals over the main diagonal are right justified; diagonals under the main diagonal are left justified.
Let us illustrate this process with another example. We input the following matrix:
2 1 0 0 0 0 1 2 1 0 0 0 3 1 2 1 0 0 0 3 1 2 1 0 0 0 3 1 2 1 0 0 0 3 1 2 In [6]: C_banded = np.zeros((4,6)); \ ...: C_banded[0,1:] = 1; \ ...: C_banded[1,:] = 2; \ ...: C_banded[2,:1] = 1; \ ...: C_banded[3,:2] = 3; \ ...: print C_banded [[ 0. 1. 1. 1. 1. 1.] [ 2. 2. 2. 2. 2. 2.] [1. 1. 1. 1. 1. 0.] [ 3. 3. 3. 3. 0. 0.]]
To call the solver, we need to input manually the number of diagonals over and under the diagonal, together with the AB
matrix and the righthand side of the system:
In [7]: spla.solve_banded((2,1), C_banded, np.ones(6)) Out[7]: array([ 0.86842105, 0.73684211, 0.39473684, 0.07894737, 1.76315789, 1.26315789])
Let us examine the optional parameters that we can include in the call of these two functions:
Parameter 
Default values 
Description 



Number of nonzero lower/upper diagonals 

Matrix in 
A banded square matrix 


Righthand side 

Boolean 
Discard data in 

Boolean 
Discard data in 

Boolean 
Whether to check that input matrices contain finite numbers 
Tip
All the functions in the scipy.linalg
module that require matrices as input and output either a solution to a system of equations, or a factorization, have two optional parameters with which we need to familiarize: overwrite_x
(for each matrix/vector in the input) and check_finite
. They are both Boolean.
The overwrite
options are set to False
by default. If we do not care about retaining the values of the input matrices, we may use the same object in the memory to perform operations, rather than creating another object with the same size in the memory. We gain speed and use fewer resources in such a case.
The check_finite
option is set to True
by default. In the algorithms where it is present, there are optional checks for the integrity of the data. If at any given moment, any of the values is (+/)numpy.inf
or NaN
, the process is halted, and an exception is raised. We may turn this option off, thus resulting in much faster solutions, but the code might crash if the data is corrupted at any point in the computations.
The function solveh_banded
has an extra optional Boolean parameter, lower
, which is initially set to False
. If set to True
, we must input the lower triangular matrix of the target AB matrix instead of the upper one (with the same input convention as before).
For solutions of basic systems where A
is a generic square matrix, it is a good idea to factorize A
so that some (or all) of the factors are triangular and then apply back and forward substitution, where appropriate. This is the idea behind pivoted LU
and Cholesky decompositions.
If matrix A
is real symmetric (or complex Hermitian) and positive definite, the optimal strategy goes through applying any of the two possible Cholesky decompositions A = U^{H} ● U or A = L ● L^{H} with the U
and L
upper/lower triangular matrices.
For example, if we use the form with the upper triangular matrices, the solution of the basic system of equations A ● x = b turns into U^{H} ● U ● x = b. Set y = U ● x and solve the system U^{H} ● y = b for y
by forward substitution. We have now a new triangular system U ● x = y that we solve for x
, by back substitution.
To perform the solution of such a system with this technique, we first compute the factorization by using either the functions cholesky
, cho_factor
or cholesky_banded
. The output is then used in the solver cho_solve
.
For Cholesky decompositions, the three relevant functions called cholesky
, cho_factor
, and cholesky_banded
have a set of options similar to those of solveh_banded
. They admit an extra Boolean option lower (set by default to False
) that decides whether to output a lower or an upper triangular factorization. The function cholesky_banded
requires a matrix in the AB
format as input.
Let us now test the Cholesky decomposition of matrix B
with all three methods:
In [8]: B = spsp.diags([[1]*5, [2]*6, [1]*5], [1,0,1]).todense() ...: print B [[ 2. 1. 0. 0. 0. 0.] [1. 2. 1. 0. 0. 0.] [ 0. 1. 2. 1. 0. 0.] [ 0. 0. 1. 2. 1. 0.] [ 0. 0. 0. 1. 2. 1.] [ 0. 0. 0. 0. 1. 2.]] In [9]: np.set_printoptions(suppress=True, precision=3) In [10]: print spla.cholesky(B) [[ 1.414 0.707 0. 0. 0. 0. ] [ 0. 1.225 0.816 0. 0. 0. ] [ 0. 0. 1.155 0.866 0. 0. ] [ 0. 0. 0. 1.118 0.894 0. ] [ 0. 0. 0. 0. 1.095 0.913] [ 0. 0. 0. 0. 0. 1.08 ]] In [11]: print spla.cho_factor(B)[0] [[ 1.414 0.707 0. 0. 0. 0. ] [1. 1.225 0.816 0. 0. 0. ] [ 0. 1. 1.155 0.866 0. 0. ] [ 0. 0. 1. 1.118 0.894 0. ] [ 0. 0. 0. 1. 1.095 0.913] [ 0. 0. 0. 0. 1. 1.08 ]] In [12]: print spla.cholesky_banded(B_banded) [[ 0. 0.707 0.816 0.866 0.894 0.913] [ 1.414 1.225 1.155 1.118 1.095 1.08 ]]
The output of cho_factor
is a tuple: the second element is the Boolean lower. The first element is ndarray
representing a square matrix. If lower
is set to True
, the lower triangular submatrix of this ndarray
is L
in the Cholesky factorization of A
. If lower
is set to False
, the upper triangular submatrix is U
in the factorization of A
. The remaining elements in the matrix are random, instead of zeros, since they are not used by cho_solve
. In a similar way, we can call cho_solve_banded
with the output of cho_banded
to solve the appropriate system.
Note
Both cholesky
and cho_factor
are wrappers to the same LAPACK function called potrf
, with different output options. cholesky_banded
calls pbtrf
. The cho_solve
function is a wrapper for potrs
, and cho_solve_banded
calls pbtrs
.
We are then ready to solve the system, with either of the two options:
In [13]: spla.cho_solve((spla.cholesky(B), False), np.ones(6)) Out[13]: array([ 3., 5., 6., 6., 5., 3.]) In [13]: spla.cho_solve(spla.cho_factor(B), np.ones(6)) Out[13]: array([ 3., 5., 6., 6., 5., 3.])
For any other kind of generic square matrix A
, the next best method to solve the basic system A ● x = b is pivoted LU
factorization. This is equivalent to finding a permutation matrix P
, and triangular matrices U
(upper) and L
(lower) so that P ● A = L ● U. In such a case, a permutation of the rows in the system according to P
gives the equivalent equation (P ● A) ● x = P ● b. Set c = P ● b
and y = U ● x
, and solve for y
in the system L ● y = c using forward substitution. Then, solve for x
in the system U ● x = y with back substitution.
The relevant functions to perform this operation are lu
, lu_factor
(for factorization), and lu_solve
(for solution) in the module scipy.linalg
. For sparse matrices we have splu
, and spilu
, in the module scipy.sparse.linalg
.
Let us start experimenting with factorizations first. We use a large circulant matrix (nonsymmetric) for this example:
In [14]: D = spla.circulant(np.arange(4096)) In [15]: %timeit spla.lu(D) 1 loops, best of 3: 7.04 s per loop In [16]: %timeit spla.lu_factor(D) 1 loops, best of 3: 5.48 s per loop
Note
The lu_factor
function is a wrapper to all *getrf
routines from LAPACK. The lu_solve
function is a wrapper for getrs
.
The function lu
has an extra Boolean option: permute_l
(set to False
by default). If set to True
, the function outputs only two matrices PL = P ● L (the properly permuted lower triangular matrix), and U
. Otherwise, the output is the triple P
, L
, U
, in that order.
In [17]: P, L, U = spla.lu(D) In [17]: PL, U = spla.lu(D, permute_l=True)
The outputs of the function lu_factor
are resourceefficient. We obtain a matrix LU
, with upper triangle U
and lower triangle L
. We also obtain a onedimensional ndarray
class of integer dtype
, piv
, indicating the pivot indices representing the permutation matrix P
.
In [18]: LU, piv = spla.lu_factor(D)
The solver lu_solve
takes the two outputs from lu_factor
, a righthand side matrix b
, and the optional indicator trans
to the kind of basic system to solve:
In [19]: spla.lu_solve(spla.lu_factor(D), np.ones(4096)) Out[19]: array([ 0., 0., 0., ..., 0., 0., 0.])
Tip
At this point, we must comment on the general function solve
in the module scipy.linalg
. It is a wrapper to both LAPACK
functions POSV
and GESV
. It allows us to input matrix A
and righthand side matrix b
, and indicate whether A
is symmetric and positive definite. In any case, the routine internally decides which of the two factorizations to use (Cholesky or pivoted LU), and computes a solution accordingly.
For large sparse matrices, provided they are stored in the CSC format, the pivoted LU
decomposition is more efficiently performed with either functions splu
or spilu
from the module scipy.sparse.linalg
. Both functions use the SuperLU
library directly. Their output is not a set of matrices, but a Python object called scipy.sparse.linalg.dsolve._superlu.SciPyLUType
. This object has four attributes and one instance method:
shape
: 2tuple containing the shape of matrixA
nnz
: The number of nonzero entries in matrixA
perm_c, perm_r
: The permutations applied to the columns and rows (respectively) to the matrixA
to obtain the computedLU
decompositionsolve
: instance method that converts the object into a functionobject.solve(b,trans)
acceptingndarray b
, and the optional description stringtrans
.
The big idea is that, dealing with large amounts of data, the actual matrices in the LU
decomposition are not as important as the main application behind the factorization: the solution of the system. All the relevant information to perform this operation is optimally stored in the object's method solve
.
The main difference between splu
and spilu
is that the latter computes an incomplete decomposition. With it, we can obtain really good approximations to the inverse of matrix A
, and use matrix multiplication to compute the solution of large systems in a fraction of the time that it would take to calculate the actual solution.
Note
The usage of these two functions is rather complex. The purpose is to compute a factorization of the form Pr*Dr*A*Dc*Pc = L*U with diagonal matrices Dr
and Dc
and permutation matrices Pr
and Pc
. The idea is to equilibrate matrix A
manually so that the product B = Dr*A*Dc is better conditioned than A
. In case of the possibility of solving this problem in a parallel architecture, we are allowed to help by rearranging the rows and columns optimally. The permutation matrices Pr
and Pc
are then manually input to preorder the rows and columns of B
. All of these options can be fed to either splu
or spilu
.
The algorithm exploits the idea of relaxing supernodes to reduce inefficient indirect addressing and symbolic time (besides permitting the use of higherlevel BLAS operations). We are given the option to determine the degree of these objects, to tailor the algorithm to the matrix at hand.
For a complete explanation of the algorithms and all the different options, the best reference is SuperLU User Guide, which can be found online at crdlegacy.lbl.gov/~xiaoye/SuperLU/superlu_ug.pdf.
Let us illustrate this with a simple example, where the permutation of rows or columns is not needed. In a large lower triangular Pascal matrix, turn into zero all the evenvalued entries and into ones all the oddvalued entries. Use this as matrix A
. For the righthand side, use a vector of ones:
In [20]: A_csc = spsp.csc_matrix(A, dtype=np.float64) In [21]: invA = spspla.splu(A_csc) In [22]: %time invA.solve(np.ones(1024)) CPU times: user: 4.32 ms, sys: 105 µs, total: 4.42 ms Wall time: 4.44 ms Out[22]: array([ 1., 0., 0., ..., 0., 0., 0.]) In [23]: invA = spspla.spilu(A_csc) In [24]: %time invA.solve(np.ones(1024)) CPU times: user 656 µs, sys: 22 µs, total: 678 µs Wall time: 678 µs Out[24]: array([ 1., 0., 0., ..., 0., 0., 0.])
Note
Compare the time of execution of the procedures on sparse matrices, with the initial solve_triangular
procedure on the corresponding matrix A
at the beginning of the section. Which process is faster?
However, in general, if a basic system must be solved and matrix A
is large and sparse, we prefer to use iterative methods with fast convergence to the actual solutions. When they converge, they are consistently less sensitive to roundingoff errors and thus more suitable when the number of computations is extremely high.
In the module scipy.sparse.linalg
, we have eight different iterative methods, all of which accept the following as parameters:
Matrix
A
in any format (matrix, ndarray, sparse matrix, or even a linear operator!), and righthand side vector/matrixb
asndarray
.Initial guess
x0
, asndarray
.Tolerance to
l
, a floating point number. If the difference of successive iterations is less than this value, the code stops and the last computed values are output as the solution.Maximum number of iterations allowed, maxiter, an integer.
A Preconditioner sparse matrix
M
that should approximate the inverse ofA
.A
callback
function of the current solution vectorxk
, called after each iteration.
Constructor 
Description 


Biconjugate Gradient Iteration 

Biconjugate Gradient Stabilized Iteration 

Conjugate Gradient Iteration 

Conjugate Gradient Squared Iteration 

Generalized Minimal Residual Iteration 

LGMRES Iteration 

Minimum Residual Iteration 

Quasiminimal Residual Iteration 
Choosing the right iterative method, a good initial guess, and especially a successful Preconditioner is an art in itself. It involves learning about topics such as operators in Functional Analysis, or Krylov subspace methods, which are far beyond the scope of this book. At this point, we are content with showing a few simple examples for the sake of comparison:
In [25]: spspla.cg(A_csc, np.ones(1024), x0=np.zeros(1024)) Out[25]: (array([ nan, nan, nan, ..., nan, nan, nan]), 1) In [26]: %time spspla.gmres(A_csc, np.ones(1024), x0=np.zeros(1024)) CPU times: user 4.26 ms, sys: 712 µs, total: 4.97 ms Wall time: 4.45 ms Out[26]: (array([ 1., 0., 0., ..., 0., 0., 0.]), 0) In [27]: Nsteps = 1 ....: def callbackF(xk): ....: global Nsteps ....: print'{0:4d} {1:3.6f} {2:3.6f}'.format(Nsteps, \ ....: xk[0],xk[1]) ....: Nsteps += 1 ....: In [28]: print '{0:4s} {1:9s} {1:9s}'.format('Iter', \ ....: 'X[0]','X[1]'); \ ....: spspla.bicg(A_csc, np.ones(1024), x0=np.zeros(1024), ....: callback=callbackF) ....: Iter X[0] X[1] 1 0.017342 0.017342 2 0.094680 0.090065 3 0.258063 0.217858 4 0.482973 0.328061 5 0.705223 0.337023 6 0.867614 0.242590 7 0.955244 0.121250 8 0.989338 0.040278 9 0.998409 0.008022 10 0.999888 0.000727 11 1.000000 0.000000 12 1.000000 0.000000 13 1.000000 0.000000 14 1.000000 0.000000 15 1.000000 0.000000 16 1.000000 0.000000 17 1.000000 0.000000 Out[28]: (array([ 1., 0., 0., ..., 0., 0., 0.]), 0)
Given a generic matrix A
(not necessarily square) and a righthand side vector/matrix b
, we look for a vector/matrix x
such that the Frobenius norm of the expression A ● x  b is minimized.
The main three methods to solve this problem numerically are contemplated in scipy
:
Normal equations
QR factorization
Singular value decomposition
Normal equations reduce the least square problem to solving a basic system of linear equations, with a symmetric (notnecessarily positivedefinite) matrix. It is very fast but can be inaccurate due to presence of roundoff errors. Basically, it amounts to solving the system (A^{H} ● A) ● x = A^{H} ● b. This is equivalent to solving x = (A^{H} ● A)^{1} ● A^{H} ● b = pinv(A) ● b.
Let us show by example:
In [29]: E = D[:512,:256]; b = np.ones(512) In [30]: sol1 = np.dot(spla.pinv2(E), b) In [31]: sol2 = spla.solve(np.dot(F.T, F), np.dot(F.T, b))
The QR factorization turns any matrix into the product A = Q ● R of an orthogonal/unitary matrix Q
with a square upper triangular matrix R
. This allows us to solve the system without the need to invert any matrix (since Q^{H} = Q^{1}), and thus, A ● x = b turns into R ● x = Q^{H} ● b, which is easily solvable by back substitution. Note that the two methods below are equivalent, since the mode economic
reports the submatrices of maximum rank:
In [32]: Q, R = spla.qr(E); \ ....: RR = R[:256, :256]; BB = np.dot(Q.T, b)[:256]; \ ....: sol3 = spla.solve_triangular(RR, BB) In [32]: Q, R = spla.qr(E, mode='economic'); \ ....: sol3 = spla.solve_triangular(R, np.dot(Q.T, b))
Both methods of normal equations and QR factorization work fast and are reliable only when the rank of A
is full. If this is not the case, we must use singular value decomposition A = U ● D ● V^{H} with unitary matrices U
and V
and a diagonal matrix D
, where all the entries in the diagonal are positive values. This allows for a fast solution x = V ● D^{1} ● U^{H} ● b.
Note that the two methods discussed below are equivalent, since the option full_matrices
set to False
reports the submatrices of the minimum possible size:
In [33]: U, s, Vh = spla.svd(E); \ ....: Uh = U.T; \ ....: Si = spla.diagsvd(1./s, 256, 256); \ ....: V = Vh.T; \ ....: sol4 = np.dot(V, Si).dot(np.dot(Uh, b)[:256]) In [33]: U, s, Vh = spla.svd(E, full_matrices=False); \ ....: Uh = U.T; \ ....: Si = spla.diagsvd(1./s, 256, 256); \ ....: V = Vh.T; \ ....: sol4 = np.dot(V, Si).dot(np.dot(Uh, b))
The module scipy.linalg
has one function that actually performs least squares with the SVD method: lstsq
. There is no need to manually transpose, invert, and multiply all the required matrices. It is a wrapper to the LAPACK
function GELSS
. It outputs the desired solution, together with the residues of computation, the effective rank, and the singular values of the input matrix A
.
In [34]: sol5, residue, rank, s = spla.lstsq(E, b)
Note how all the computations that we have carried out offer solutions that are very close to each other (if not equal!):
In [35]: map(lambda x: np.allclose(sol5,x), [sol1, sol2, sol3, sol4]) Out[35]: [True, True, True, True]
The module scipy.sparse.linalg
has two iterative methods for least squares in the context of large sparse matrices, lsqr
and lsmr
, which allow for a more generalized version with a damping factor d
for regularization. We seek to minimize the functional norm(A * x  b, 'f')**2 + d**2 * norm(x, 'f')**2
. The usage and parameters are very similar to the iterative functions we studied before.
The rest of the matrix equation solvers are summarized in the following table. None of these routines enjoy any parameters to play around with performance or memory management, or check for the integrity of data:
Constructor 
Description 


Sylvester equation 

continuous algebraic Riccati equation 

discrete algebraic Riccati equation 

continuous Lyapunov equation 

discrete Lyapunov equation 