Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
Date: May 24, 2018
Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
The aim of this set of lectures is to review some central linear algebra algorithms that we will need in our data analysis part and in the construction of Machine Learning algorithms (ML). This will allow us to introduce some central programming features of high-level languages like Python and compiled languages like C++ and/or Fortran.
As discussed in the introductory notes, these series of lectures focuses both on using central Python packages like tensorflow and scikit-learn as well as writing your own codes for some central ML algorithms. The latter can be written in a language of your choice, be it Python, Julia, R, Rust, C++, Fortran etc. In order to avoid confusion however, in these lectures we will limit our attention to Python, C++ and Fortran.
There are several central software packages for linear algebra and eigenvalue problems. Several of the more popular ones have been wrapped into ofter software packages like those from the widely used text Numerical Recipes. The original source codes in many of the available packages are often taken from the widely used software package LAPACK, which follows two other popular packages developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.
LINPACK: package for linear equations and least square problems.
LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.
BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.
When dealing with matrices and vectors a central issue is memory handling and allocation. If our code is written in Python the way we declare these objects and the way they are handled, interpreted and used by say a linear algebra library, requires codes that interface our Python program with such libraries. For Python programmers, Numpy is by now the standard Python package for numerical arrays in Python as well as the source of functions which act on these arrays. These functions span from eigenvalue solvers to functions that compute the mean value, variance or the covariance matrix. If you are not familiar with how arrays are handled in say Python or compiled languages like C++ and Fortran, the sections in this chapter may be useful. For C++ programmer, Armadillo is widely used library for linear algebra and eigenvalue problems. In addition it offers a convenient way to handle and organize arrays. We discuss this library as well. Before we proceed we believe it may be convenient to repeat some basic features of matrices and vectors.
Matrix properties reminder.
Matrix Properties Reminder.
| Relations | Name | matrix elements |
|---|---|---|
| $A = A^{T}$ | symmetric | $a_{ij} = a_{ji}$ |
| $A = \left (A^{T} \right )^{-1}$ | real orthogonal | $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$ |
| $A = A^{ * }$ | real matrix | $a_{ij} = a_{ij}^{ * }$ |
| $A = A^{\dagger}$ | hermitian | $a_{ij} = a_{ji}^{ * }$ |
| $A = \left (A^{\dagger} \right )^{-1}$ | unitary | $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$ |
Diagonal if $a_{ij}=0$ for $i\ne j$
Upper triangular if $a_{ij}=0$ for $i > j$
Lower triangular if $a_{ij}=0$ for $i < j$
Upper Hessenberg if $a_{ij}=0$ for $i > j+1$
Lower Hessenberg if $a_{ij}=0$ for $i < j+1$
Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$
Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$
Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$
Banded, block upper triangular, block lower triangular....
Some Equivalent Statements.
For an $N\times N$ matrix $\mathbf{A}$ the following properties are all equivalent
If the inverse of $\mathbf{A}$ exists, $\mathbf{A}$ is nonsingular.
The equation $\mathbf{Ax}=0$ implies $\mathbf{x}=0$.
The rows of $\mathbf{A}$ form a basis of $R^N$.
The columns of $\mathbf{A}$ form a basis of $R^N$.
$\mathbf{A}$ is a product of elementary matrices.
$0$ is not eigenvalue of $\mathbf{A}$.
Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as
In [1]:
import numpy as np
n = 10
x = np.random.normal(size=n)
print(x)
Here we have defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$. Another alternative is to declare a vector as follows
In [2]:
import numpy as np
x = np.array([1, 2, 3])
print(x)
Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++ start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as
In [3]:
import numpy as np
x = np.log(np.array([4, 7, 8]))
print(x)
Here we have used Numpy's unary function $np.log$. This function is highly tuned to compute array elements since the code is vectorized and does not require looping. We normaly recommend that you use the Numpy intrinsic functions instead of the corresponding log function from Python's math module. The looping is done explicitely by the np.log function. The alternative, and slower way to compute the logarithms of a vector would be to write
In [3]:
import numpy as np
from math import log
x = np.array([4, 7, 8])
for i in range(0, len(x)):
x[i] = log(x[i])
print(x)
We note that our code is much longer already and we need to import the log function from the math module. The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automacally our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as
In [4]:
import numpy as np
x = np.log(np.array([4, 7, 8], dtype = np.float64))
print(x)
or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
In [6]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x)
To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the itemsize functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as
In [7]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x.itemsize)
In [8]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
print(A)
If we use the shape function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as
In [9]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[:,0])
We can continue this was by printing out other columns or rows. The example here prints out the second column
In [10]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[1,:])
Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero
In [11]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to zero
A = np.zeros( (n, n) )
print(A)
or initializing all elements to
In [12]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to one
A = np.ones( (n, n) )
print(A)
or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
In [13]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
A = np.random.rand(n, n)
print(A)
As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors $\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as
where for example
The Numpy function np.cov calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. For a more in-depth discussion of the covariance and covariance matrix and its meaning, we refer you to the lectures on statistics. The following simple function uses the np.vstack function which takes each vector of dimension $1\times n$ and produces a $ 3\times n$ matrix $\hat{W}$
which in turn is converted into into the $3 times 3$ covariance matrix $\hat{\Sigma}$ via the Numpy function np.cov(). In our review of statistical functions and quantities we will discuss more about the meaning of the covariance matrix. Here we note that we can calculate the mean value of each set of samples $\hat{x}$ etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function.
In [14]:
# Importing various packages
import numpy as np
n = 100
x = np.random.normal(size=n)
print(np.mean(x))
y = 4+3*x+np.random.normal(size=n)
print(np.mean(y))
z = x**3+np.random.normal(size=n)
print(np.mean(z))
W = np.vstack((x, y, z))
Sigma = np.cov(W)
print(Sigma)
Eigvals, Eigvecs = np.linalg.eig(Sigma)
print(Eigvals)
int N = 100;
double A[100][100];
// initialize all elements to zero
for(i=0 ; i < N ; i++) {
for(j=0 ; j < N ; j++) {
A[i][j] = 0.0;In C/C++ this would be coded like
for(i=0 ; i < N ; i++) {
for(j=0 ; j < N ; j++) {
a[i][j] = b[i][j]+c[i][j]In C/C++ this would be coded like
for(i=0 ; i < N ; i++) {
for(j=0 ; j < N ; j++) {
for(k=0 ; k < N ; k++) {
a[i][j]+=b[i][k]*c[k][j];At least three possibilities in this course
Do it yourself
Use the functions provided in the library package lib.cpp
Use Armadillo http://arma.sourceforgenet (a C++ linear algebra library, discussion both here and at lab).
Do it yourself.
int N;
double ** A;
A = new double*[N]
for ( i = 0; i < N; i++)
A[i] = new double[N];Always free space when you don't need an array anymore.
for ( i = 0; i < N; i++)
delete[] A[i];
delete[] A;Armadillo is a C++ linear algebra library (matrix maths) aiming towards a good balance between speed and ease of use. The syntax is deliberately similar to Matlab.
Integer, floating point and complex numbers are supported, as well as a subset of trigonometric and statistics functions. Various matrix decompositions are provided through optional integration with LAPACK, or one of its high performance drop-in replacements (such as the multi-threaded MKL or ACML libraries).
A delayed evaluation approach is employed (at compile-time) to combine several operations into one and reduce (or eliminate) the need for temporaries. This is accomplished through recursive templates and template meta-programming.
Useful for conversion of research code into production environments, or if C++ has been decided as the language of choice, due to speed and/or integration capabilities.
The library is open-source software, and is distributed under a license that is useful in both open-source and commercial/proprietary contexts.
#include <iostream>
#include <armadillo>
using namespace std;
using namespace arma;
int main(int argc, char** argv)
{
mat A = randu<mat>(5,5);
mat B = randu<mat>(5,5);
cout << A*B << endl;
return 0;For people using Ubuntu, Debian, Linux Mint, simply go to the synaptic package manager and install armadillo from there. You may have to install Lapack as well. For Mac and Windows users, follow the instructions from the webpage http://arma.sourceforge.net. To compile, use for example (linux/ubuntu)
c++ -O2 -o program.x program.cpp -larmadillo -llapack -lblaswhere the -l option indicates the library you wish to link to.
For OS X users you may have to declare the paths to the include files and the libraries as
c++ -O2 -o program.x program.cpp -L/usr/local/lib -I/usr/local/include -larmadillo -llapack -lblas #include <iostream>
#include "armadillo"
using namespace arma;
using namespace std;
int main(int argc, char** argv)
{
// directly specify the matrix size (elements are uninitialised)
mat A(2,3);
// .n_rows = number of rows (read only)
// .n_cols = number of columns (read only)
cout << "A.n_rows = " << A.n_rows << endl;
cout << "A.n_cols = " << A.n_cols << endl;
// directly access an element (indexing starts at 0)
A(1,2) = 456.0;
A.print("A:");
// scalars are treated as a 1x1 matrix,
// hence the code below will set A to have a size of 1x1
A = 5.0;
A.print("A:");
// if you want a matrix with all elements set to a particular value
// the .fill() member function can be used
A.set_size(3,3);
A.fill(5.0); A.print("A:"); mat B;
// endr indicates "end of row"
B << 0.555950 << 0.274690 << 0.540605 << 0.798938 << endr
<< 0.108929 << 0.830123 << 0.891726 << 0.895283 << endr
<< 0.948014 << 0.973234 << 0.216504 << 0.883152 << endr
<< 0.023787 << 0.675382 << 0.231751 << 0.450332 << endr;
// print to the cout stream
// with an optional string before the contents of the matrix
B.print("B:");
// the << operator can also be used to print the matrix
// to an arbitrary stream (cout in this case)
cout << "B:" << endl << B << endl;
// save to disk
B.save("B.txt", raw_ascii);
// load from disk
mat C;
C.load("B.txt");
C += 2.0 * B;
C.print("C:"); // submatrix types:
//
// .submat(first_row, first_column, last_row, last_column)
// .row(row_number)
// .col(column_number)
// .cols(first_column, last_column)
// .rows(first_row, last_row)
cout << "C.submat(0,0,3,1) =" << endl;
cout << C.submat(0,0,3,1) << endl;
// generate the identity matrix
mat D = eye<mat>(4,4);
D.submat(0,0,3,1) = C.cols(1,2);
D.print("D:");
// transpose
cout << "trans(B) =" << endl;
cout << trans(B) << endl;
// maximum from each column (traverse along rows)
cout << "max(B) =" << endl;
cout << max(B) << endl; // maximum from each row (traverse along columns)
cout << "max(B,1) =" << endl;
cout << max(B,1) << endl;
// maximum value in B
cout << "max(max(B)) = " << max(max(B)) << endl;
// sum of each column (traverse along rows)
cout << "sum(B) =" << endl;
cout << sum(B) << endl;
// sum of each row (traverse along columns)
cout << "sum(B,1) =" << endl;
cout << sum(B,1) << endl;
// sum of all elements
cout << "sum(sum(B)) = " << sum(sum(B)) << endl;
cout << "accu(B) = " << accu(B) << endl;
// trace = sum along diagonal
cout << "trace(B) = " << trace(B) << endl;
// random matrix -- values are uniformly distributed in the [0,1] interval
mat E = randu<mat>(4,4);
E.print("E:"); // row vectors are treated like a matrix with one row
rowvec r;
r << 0.59499 << 0.88807 << 0.88532 << 0.19968;
r.print("r:");
// column vectors are treated like a matrix with one column
colvec q;
q << 0.81114 << 0.06256 << 0.95989 << 0.73628;
q.print("q:");
// dot or inner product
cout << "as_scalar(r*q) = " << as_scalar(r*q) << endl;
// outer product
cout << "q*r =" << endl;
cout << q*r << endl;
// sum of three matrices (no temporary matrices are created)
mat F = B + C + D;
F.print("F:");
return 0; #include <iostream>
#include "armadillo"
using namespace arma;
using namespace std;
int main(int argc, char** argv)
{
cout << "Armadillo version: " << arma_version::as_string() << endl;
mat A;
A << 0.165300 << 0.454037 << 0.995795 << 0.124098 << 0.047084 << endr
<< 0.688782 << 0.036549 << 0.552848 << 0.937664 << 0.866401 << endr
<< 0.348740 << 0.479388 << 0.506228 << 0.145673 << 0.491547 << endr
<< 0.148678 << 0.682258 << 0.571154 << 0.874724 << 0.444632 << endr
<< 0.245726 << 0.595218 << 0.409327 << 0.367827 << 0.385736 << endr;
A.print("A =");
// determinant
cout << "det(A) = " << det(A) << endl; // inverse
cout << "inv(A) = " << endl << inv(A) << endl;
double k = 1.23;
mat B = randu<mat>(5,5);
mat C = randu<mat>(5,5);
rowvec r = randu<rowvec>(5);
colvec q = randu<colvec>(5);
// examples of some expressions
// for which optimised implementations exist
// optimised implementation of a trinary expression
// that results in a scalar
cout << "as_scalar( r*inv(diagmat(B))*q ) = ";
cout << as_scalar( r*inv(diagmat(B))*q ) << endl;
// example of an expression which is optimised
// as a call to the dgemm() function in BLAS:
cout << "k*trans(B)*C = " << endl << k*trans(B)*C;
return 0;We assume also that the matrix $\mathbf{A}$ is non-singular and that the matrix elements along the diagonal satisfy $a_{ii} \ne 0$. Simple $4\times 4 $ example
The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown $x_1$ from the remaining $n-1$ equations. Then we use the new second equation to eliminate the second unknown $x_2$ from the remaining $n-2$ equations. With $n-1$ such eliminations we obtain a so-called upper triangular set of equations of the form
To arrive at such an upper triangular system of equations, we start by eliminating the unknown $x_1$ for $j=2,n$. We achieve this by multiplying the first equation by $a_{j1}/a_{11}$ and then subtract the result from the $j$th equation. We assume obviously that $a_{11}\ne 0$ and that $\mathbf{A}$ is not singular.
Our actual $4\times 4$ example reads after the first operation
or
where each $a_{1k}^{(1)}$ is equal to the original $a_{1k}$ element. The other coefficients are
with a new right-hand side given by
We have also set $w_1^{(1)}=w_1$, the original vector element. We see that the system of unknowns $x_1,\dots,x_n$ is transformed into an $(n-1)\times (n-1)$ problem.
This step is called forward substitution. Proceeding with these substitutions, we obtain the general expressions for the new coefficients
with $m=1,\dots,n-1$ and a right-hand side given by
This set of $n-1$ elimations leads us to an equations which is solved by back substitution. If the arithmetics is exact and the matrix $\mathbf{A}$ is not singular, then the computed answer will be exact.
Even though the matrix elements along the diagonal are not zero, numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added to a small number may yield losses of precision. Suppose for example that our first division in $(a_{22}-a_{21}a_{12}/a_{11})$ results in $-10^{-7}$ and that $a_{22}$ is one. one. We are then adding $10^7+1$. With single precision this results in $10^7$.
Gaussian elimination, $O(2/3n^3)$ flops, general matrix
LU decomposition, upper triangular and lower tridiagonal matrices, $O(2/3n^3)$ flops, general matrix. Get easily the inverse, determinant and can solve linear equations with back-substitution only, $O(n^2)$ flops
Cholesky decomposition. Real symmetric or hermitian positive definite matrix, $O(1/3n^3)$ flops.
Tridiagonal linear systems, important for differential equations. Normally positive definite and non-singular. $O(8n)$ flops for symmetric. Special case of banded matrices.
Singular value decomposition
the QR method will be discussed in chapter 7 in connection with eigenvalue systems. $O(4/3n^3)$ flops.
The LU decomposition method means that we can rewrite this matrix as the product of two matrices $\mathbf{L}$ and $\mathbf{U}$ where
The above set of equations is conveniently solved by using LU decomposition as an intermediate step.
The matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$ has an LU factorization if the determinant is different from zero. If the LU factorization exists and $\mathbf{A}$ is non-singular, then the LU factorization is unique and the determinant is given by
There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:
It is straightforward to compute the determinant of a matrix
If we have to solve sets of linear equations with the same matrix but with different vectors $\mathbf{y}$, the number of FLOPS is of the order $n^3$.
The inverse is such an operation
With the LU decomposition it is rather simple to solve a system of linear equations
This can be written in matrix form as
where $\mathbf{A}$ and $\mathbf{w}$ are known and we have to solve for $\mathbf{x}$. Using the LU dcomposition we write
To show that this is correct we use to the LU decomposition to rewrite our system of linear equations as
and since the determinant of $\mathbf{L}$ is equal to 1 (by construction since the diagonals of $\mathbf{L}$ equal 1) we can use the inverse of $\mathbf{L}$ to obtain
which yields the intermediate step
and
This example shows the basis for the algorithm needed to solve the set of $n$ linear equations.
The algorithm goes as follows
Set up the matrix $\bf A$ and the vector $\bf w$ with their correct dimensions. This determines the dimensionality of the unknown vector $\bf x$.
Then LU decompose the matrix $\bf A$ through a call to the function ludcmp(double a, int n, int indx, double &d). This functions returns the LU decomposed matrix $\bf A$, its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned.
Thereafter you call the function lubksb(double a, int n, int indx, double w) which uses the LU decomposed matrix $\bf A$ and the vector $\bf w$ and returns $\bf x$ in the same place as $\bf w$. Upon exit the original content in $\bf w$ is destroyed. If you wish to keep this information, you should make a backup of it in your calling function.
If the inverse exists then
the identity matrix. With an LU decomposed matrix we can rewrite the last equation as
then we have a linear set of equations
and continue till we have solved all $n$ sets of linear equations.
#include <iostream>
#include "armadillo"
using namespace arma;
using namespace std;
int main()
{
mat A = randu<mat>(5,5);
vec b = randu<vec>(5);
A.print("A =");
b.print("b=");
// solve Ax = b
vec x = solve(A,b);
// print x
x.print("x=");
// find LU decomp of A, if needed, P is the permutation matrix
mat L, U;
lu(L,U,A);
// print l
L.print(" L= ");
// print U
U.print(" U= ");
//Check that A = LU
(A-L*U).print("Test of LU decomposition");
return 0;
}