@@ -238,7 +215,7 @@ MathJax.Hub.Config({
-
@@ -246,22 +223,30 @@ MathJax.Hub.Config({ -
+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. - +
+
-The Numerical Recipes codes have been rewritten in Fortran 90/95 and -C/C++ by us. The original source codes are taken from the widely used +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. +developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.
- +
+ +
The inverse of a matrix is defined by $$ @@ -387,7 +389,209 @@ For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all
-
+ + +
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 +
+ + +
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 +
+ + +
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 + +
+ + +
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 +
+ + +
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 +
+ + +
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 +
+ + +
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x.itemsize)
++ + +
+ + +
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 +
+ + +
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 +
+ + +
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 +
+ + +
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 +
+ + +
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) +
+ + +
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 +$$ +\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ + \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ + \sigma_{zx} & \sigma_{zy} & \sigma_{zz} + \end{bmatrix}, +$$ + +where for example +$$ +\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). +$$ + +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 \( \hat{W} \) +$$ +\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ + x_1 & y_1 & z_1 \\ + x_2 & y_2 & z_2 \\ + \dots & \dots & \dots \\ + x_{n-2} & y_{n-2} & z_{n-2} \\ + x_{n-1} & y_{n-1} & z_{n-1} + \end{bmatrix}, +$$ + +
+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. + +
+ + +
# 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)
++ + +
-
-
-
At least three possibilities in this course @@ -489,7 +693,7 @@ At least three possibilities in this course -
-
@@ -557,7 +761,7 @@ Always free space when you don't need an array anymore.
-
For people using Ubuntu, Debian, Linux Mint, simply go to the synaptic package manager and install @@ -585,7 +789,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -618,7 +822,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -649,7 +853,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -682,7 +886,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -710,7 +914,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -742,7 +946,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -772,7 +976,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
@@ -804,7 +1008,7 @@ For OS X users you may have to declare the paths to the include files and the li
-
We start with the linear set of equations @@ -839,7 +1043,7 @@ $$
-
-
The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown \( x_1 \) @@ -878,7 +1082,7 @@ what is called a backward substitution.
-
-
Our actual \( 4\times 4 \) example reads after the first operation @@ -936,7 +1140,7 @@ $$
-
The new coefficients are @@ -972,7 +1176,7 @@ We see that the system of unknowns \( x_1,\dots,x_n \) is transformed into an \(
-
This step is called forward substitution. @@ -1010,7 +1214,7 @@ adding \( 10^7+1 \). With single precision this results in \( 10^7 \).
-
The LU decomposition method means that we can rewrite @@ -1054,7 +1258,7 @@ $$
-
LU decomposition forms the backbone of other algorithms in linear algebra, such as the @@ -1083,7 +1287,7 @@ $$
-
There are at least three main advantages with LU decomposition compared with standard Gaussian elimination: @@ -1096,7 +1300,7 @@ There are at least three main advantages with LU decomposition compared with sta -
With the LU decomposition it is rather @@ -1125,7 +1329,7 @@ $$ \mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. $$
-
The previous equation can be calculated in two steps @@ -1138,7 +1342,7 @@ to rewrite our system of linear equations as $$ \mathbf{LUx}=\mathbf{w}, $$ -and since the determinat of \( \mathbf{L} \) is equal to 1 (by construction +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 @@ -1158,7 +1362,7 @@ through \( \mathbf{Ux}=\mathbf{y} \).
-
For our four-dimentional example this takes the form @@ -1191,7 +1395,7 @@ needed to solve the set of \( n \) linear equations.
-
The algorithm goes as follows @@ -1204,7 +1408,7 @@ The algorithm goes as follows -
If the inverse exists then @@ -1222,7 +1426,7 @@ $$
-
If we assume that the first column (that is column 1) of the inverse matrix @@ -1256,7 +1460,7 @@ $$
-
In a similar way we can compute the unknow entries of the second column, @@ -1280,7 +1484,7 @@ and continue till we have solved all \( n \) sets of linear equations.
-
@@ -1312,245 +1516,6 @@ and continue till we have solved all \( n \) sets of linear equations. return 0; }
- - -
- -
- - -
-It is a simple method for solving -$$ -\mathbf{A}\mathbf{x}=\mathbf{b}, -$$ - -where \( \mathbf{A} \) is a matrix and \( \mathbf{x} \) and \( \mathbf{b} \) are vectors. The vector \( \mathbf{x} \) is -the unknown. - -
-It is an iterative scheme where we start with a guess for the unknown, and -after \( k+1 \) iterations we have -$$ -\mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}), -$$ - -with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and -\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular -matrix. - -
-If the matrix \( \mathbf{A} \) is positive definite or diagonally dominant, one can show that this method will always converge to the exact solution. -
- - -
-We can demonstrate Jacobi's method by this \( 4\times 4 \) matrix problem. We assume a guess -for the vector elements \( x_i^{(0)} \), a guess which represents our first iteration. The new -values are obtained by substitution -$$ -\begin{align} - x_1^{(1)} =&(b_1-a_{12}x_2^{(0)} -a_{13}x_3^{(0)} - a_{14}x_4^{(0)})/a_{11} \nonumber \\ - x_2^{(1)} =&(b_2-a_{21}x_1^{(0)} - a_{23}x_3^{(0)} - a_{24}x_4^{(0)})/a_{22} \nonumber \\ - x_3^{(1)} =&(b_3- a_{31}x_1^{(0)} -a_{32}x_2^{(0)} -a_{34}x_4^{(0)})/a_{33} \nonumber \\ - x_4^{(1)}=&(b_4-a_{41}x_1^{(0)} -a_{42}x_2^{(0)} - a_{43}x_3^{(0)})/a_{44}, \nonumber -\end{align} -$$ - -which after \( k+1 \) iterations reads -$$ -\begin{align} - x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ - x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ - x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ - x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber -\end{align} -$$ -
- - -
-We can generalize the above equations to -$$ - x_i^{(k+1)}=(b_i-\sum_{j=1, j\ne i}^{n}a_{ij}x_j^{(k)})/a_{ii} -$$ - -or in an even more compact form as -$$ \mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}), -$$ - -with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and -\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular -matrix. -
- - -
-Our \( 4\times 4 \) matrix problem -$$ -\begin{align} - x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ - x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ - x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ - x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber -\end{align} -$$ - -can be rewritten as -$$ -\begin{align} - x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ - x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ - x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ - x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber -\end{align} -$$ - -which allows us to utilize the preceding solution (forward substitution). This improves normally the convergence -behavior and leads to the Gauss-Seidel method! -
- - -
-We can generalize -$$ -\begin{align} - x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ - x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ - x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ - x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber -\end{align} -$$ - -to the following form -$$ - x^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i - \sum_{j > i}a_{ij}x^{(k)}_j - \sum_{j < i}a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n. -$$ - -The procedure is generally continued until the changes made by an iteration are below some tolerance. - -
-The convergence properties of the Jacobi method and the -Gauss-Seidel method are dependent on the matrix \( \mathbf{A} \). These methods converge when -the matrix is symmetric positive-definite, or is strictly or irreducibly diagonally dominant. -Both methods sometimes converge even if these conditions are not satisfied. -
- - -
-Given a square system of n linear equations with unknown \( \mathbf x \): -$$ - \mathbf{A}\mathbf x = \mathbf b -$$ - -where -$$ - \mathbf{A}=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \qquad \mathbf{x} = \begin{bmatrix} x_{1} \\ x_2 \\ \vdots \\ x_n \end{bmatrix} , \qquad \mathbf{b} = \begin{bmatrix} b_{1} \\ b_2 \\ \vdots \\ b_n \end{bmatrix}. -$$ -
- - -
-Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U: -$$ - \mathbf{A} =\mathbf{D} + \mathbf{L} + \mathbf{U}, -$$ - -where -$$ - D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & a_{nn} \end{bmatrix}, \quad L = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ a_{21} & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & 0 \end{bmatrix}, \quad U = \begin{bmatrix} 0 & a_{12} & \cdots & a_{1n} \\ 0 & 0 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \end{bmatrix}. -$$ - -The system of linear equations may be rewritten as: -$$ - (D+\omega L) \mathbf{x} = \omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x} -$$ - -for a constant \( \omega > 1 \). -
- - -
-The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for \( x \), using previous value for \( x \) on the right hand side. Analytically, this may be written as: -$$ - \mathbf{x}^{(k+1)} = (D+\omega L)^{-1} \big(\omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}^{(k)}\big). -$$ - -However, by taking advantage of the triangular form of \( (D+\omega L) \), the elements of \( x^{(k+1)} \) can be computed sequentially using forward substitution: -$$ - x^{(k+1)}_i = (1-\omega)x^{(k)}_i + \frac{\omega}{a_{ii}} \left(b_i - \sum_{j > i} a_{ij}x^{(k)}_j - \sum_{j < i} a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n. -$$ - -The choice of relaxation factor is not necessarily easy, and depends upon the properties of the coefficient matrix. For symmetric, positive-definite matrices it can be proven that \( 0 < \omega < 2 \) will lead to convergence, but we are generally interested in faster convergence rather than just convergence. -
diff --git a/doc/pub/Linalg/html/Linalg-reveal.html b/doc/pub/Linalg/html/Linalg-reveal.html index c20710a77..1e36e4cc0 100644 --- a/doc/pub/Linalg/html/Linalg-reveal.html +++ b/doc/pub/Linalg/html/Linalg-reveal.html @@ -3,9 +3,9 @@ - + -
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
-
@@ -159,12 +159,19 @@ MathJax.Hub.Config({
+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.
To do
+Introduction
+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.
-
-
+
-The Numerical Recipes codes have been rewritten in Fortran 90/95 and -C/C++ by us. The original source codes are taken from the widely used +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. +developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.
-Add python material on linear algebra and array handling, text on numpy etc
+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.
@@ -311,7 +333,217 @@ For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all
+
+
+
+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
+
+
+
+
+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
+
+
+
+
+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
+
+
+
+
+
+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
+
+
+
+
+or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
+
+
+
+
+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
+
+
+
+
+
+
+
+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
+
+
+
+
+We can continue this was by printing out other columns or rows. The example here prints out the second column
+
+
+
+
+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
+
+
+
+
+or initializing all elements to
+
+
+
+
+or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
+
+
+
+
+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
+
+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.
+
+
+
+
+
At least three possibilities in this course
@@ -415,7 +647,7 @@ At least three possibilities in this course
@@ -487,7 +719,7 @@ Always free space when you don't need an array anymore.
For people using Ubuntu, Debian, Linux Mint, simply go to the synaptic package manager and install
@@ -516,7 +748,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -550,7 +782,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -582,7 +814,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -616,7 +848,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -645,7 +877,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -678,7 +910,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -709,7 +941,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -742,7 +974,7 @@ For OS X users you may have to declare the paths to the include files and the li
We start with the linear set of equations
@@ -781,7 +1013,7 @@ $$
The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown \( x_1 \)
@@ -824,7 +1056,7 @@ what is called a backward substitution.
Our actual \( 4\times 4 \) example reads after the first operation
@@ -888,7 +1120,7 @@ $$
The new coefficients are
@@ -930,7 +1162,7 @@ We see that the system of unknowns \( x_1,\dots,x_n \) is transformed into an \(
This step is called forward substitution.
@@ -972,7 +1204,7 @@ adding \( 10^7+1 \). With single precision this results in \( 10^7 \).
The LU decomposition method means that we can rewrite
@@ -1025,7 +1257,7 @@ $$
LU decomposition forms the backbone of other algorithms in linear algebra, such as the
@@ -1058,7 +1290,7 @@ $$
There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:
@@ -1075,7 +1307,7 @@ There are at least three main advantages with LU decomposition compared with sta
With the LU decomposition it is rather
@@ -1110,7 +1342,7 @@ $$ \mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. $$
The previous equation can be calculated in two steps
@@ -1127,7 +1359,7 @@ to rewrite our system of linear equations as
$$ \mathbf{LUx}=\mathbf{w}, $$
For our four-dimentional example this takes the form
@@ -1188,7 +1420,7 @@ needed to solve the set of \( n \) linear equations.
The algorithm goes as follows
@@ -1205,7 +1437,7 @@ The algorithm goes as follows
If the inverse exists then
@@ -1227,7 +1459,7 @@ $$
If we assume that the first column (that is column 1) of the inverse matrix
@@ -1265,7 +1497,7 @@ $$
In a similar way we can compute the unknow entries of the second column,
@@ -1291,7 +1523,7 @@ and continue till we have solved all \( n \) sets of linear equations.
@@ -1326,260 +1558,6 @@ and continue till we have solved all \( n \) sets of linear equations.
-It is a simple method for solving
-
-It is an iterative scheme where we start with a guess for the unknown, and
-after \( k+1 \) iterations we have
-
-
-If the matrix \( \mathbf{A} \) is positive definite or diagonally dominant, one can show that this method will always converge to the exact solution.
-
-We can demonstrate Jacobi's method by this \( 4\times 4 \) matrix problem. We assume a guess
-for the vector elements \( x_i^{(0)} \), a guess which represents our first iteration. The new
-values are obtained by substitution
-
-We can generalize the above equations to
-
-Our \( 4\times 4 \) matrix problem
-
-We can generalize
-
-The convergence properties of the Jacobi method and the
-Gauss-Seidel method are dependent on the matrix \( \mathbf{A} \). These methods converge when
-the matrix is symmetric positive-definite, or is strictly or irreducibly diagonally dominant.
-Both methods sometimes converge even if these conditions are not satisfied.
-
-Given a square system of n linear equations with unknown \( \mathbf x \):
-
-Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U:
-
-The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for \( x \), using previous value for \( x \) on the right hand side. Analytically, this may be written as:
-
@@ -179,27 +163,35 @@ MathJax.Hub.Config({
-
+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.
-
+
-The Numerical Recipes codes have been rewritten in Fortran 90/95 and
-C/C++ by us. The original source codes are taken from the widely used
+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.
+developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.
-
+
The inverse of a matrix is defined by
@@ -316,7 +325,209 @@ For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all
+
+
+
+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
+
+
+
+
+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
+
+
+
+
+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
+
+
+
+
+
+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
+
+
+
+
+or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
+
+
+
+
+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
+
+
+
+
+
+
+
+
+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
+
+
+
+
+We can continue this was by printing out other columns or rows. The example here prints out the second column
+
+
+
+
+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
+
+
+
+
+or initializing all elements to
+
+
+
+
+or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
+
+
+
+
+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
+$$
+\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
+ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
+ \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
+ \end{bmatrix},
+$$
+
+where for example
+$$
+\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
+$$
+
+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 \( \hat{W} \)
+$$
+\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
+ x_1 & y_1 & z_1 \\
+ x_2 & y_2 & z_2 \\
+ \dots & \dots & \dots \\
+ x_{n-2} & y_{n-2} & z_{n-2} \\
+ x_{n-1} & y_{n-1} & z_{n-1}
+ \end{bmatrix},
+$$
+
+
+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.
+
+
+
+
+
+
At least three possibilities in this course
@@ -415,7 +626,7 @@ At least three possibilities in this course
@@ -482,7 +693,7 @@ Always free space when you don't need an array anymore.
For people using Ubuntu, Debian, Linux Mint, simply go to the synaptic package manager and install
@@ -510,7 +721,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -543,7 +754,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -574,7 +785,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -607,7 +818,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -635,7 +846,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -667,7 +878,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -697,7 +908,7 @@ For OS X users you may have to declare the paths to the include files and the li
@@ -729,7 +940,7 @@ For OS X users you may have to declare the paths to the include files and the li
We start with the linear set of equations
@@ -764,7 +975,7 @@ $$
The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown \( x_1 \)
@@ -803,7 +1014,7 @@ what is called a backward substitution.
Our actual \( 4\times 4 \) example reads after the first operation
@@ -861,7 +1072,7 @@ $$
The new coefficients are
@@ -897,7 +1108,7 @@ We see that the system of unknowns \( x_1,\dots,x_n \) is transformed into an \(
This step is called forward substitution.
@@ -935,7 +1146,7 @@ adding \( 10^7+1 \). With single precision this results in \( 10^7 \).
The LU decomposition method means that we can rewrite
@@ -979,7 +1190,7 @@ $$
LU decomposition forms the backbone of other algorithms in linear algebra, such as the
@@ -1008,7 +1219,7 @@ $$
There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:
@@ -1021,7 +1232,7 @@ There are at least three main advantages with LU decomposition compared with sta
With the LU decomposition it is rather
@@ -1050,7 +1261,7 @@ $$ \mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. $$
The previous equation can be calculated in two steps
@@ -1063,7 +1274,7 @@ to rewrite our system of linear equations as
$$ \mathbf{LUx}=\mathbf{w}, $$
-and since the determinat of \( \mathbf{L} \) is equal to 1 (by construction
+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
@@ -1083,7 +1294,7 @@ through \( \mathbf{Ux}=\mathbf{y} \).
For our four-dimentional example this takes the form
@@ -1116,7 +1327,7 @@ needed to solve the set of \( n \) linear equations.
The algorithm goes as follows
@@ -1129,7 +1340,7 @@ The algorithm goes as follows
If the inverse exists then
@@ -1147,7 +1358,7 @@ $$
If we assume that the first column (that is column 1) of the inverse matrix
@@ -1181,7 +1392,7 @@ $$
In a similar way we can compute the unknow entries of the second column,
@@ -1205,7 +1416,7 @@ and continue till we have solved all \( n \) sets of linear equations.
@@ -1237,236 +1448,6 @@ and continue till we have solved all \( n \) sets of linear equations.
return 0;
}
-
-
-
-
-It is a simple method for solving
-$$
-\mathbf{A}\mathbf{x}=\mathbf{b},
-$$
-
-where \( \mathbf{A} \) is a matrix and \( \mathbf{x} \) and \( \mathbf{b} \) are vectors. The vector \( \mathbf{x} \) is
-the unknown.
-
-
-It is an iterative scheme where we start with a guess for the unknown, and
-after \( k+1 \) iterations we have
-$$
-\mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}),
-$$
-
-with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and
-\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular
-matrix.
-
-
-If the matrix \( \mathbf{A} \) is positive definite or diagonally dominant, one can show that this method will always converge to the exact solution.
-
-
-We can demonstrate Jacobi's method by this \( 4\times 4 \) matrix problem. We assume a guess
-for the vector elements \( x_i^{(0)} \), a guess which represents our first iteration. The new
-values are obtained by substitution
-$$
-\begin{align}
- x_1^{(1)} =&(b_1-a_{12}x_2^{(0)} -a_{13}x_3^{(0)} - a_{14}x_4^{(0)})/a_{11} \nonumber \\
- x_2^{(1)} =&(b_2-a_{21}x_1^{(0)} - a_{23}x_3^{(0)} - a_{24}x_4^{(0)})/a_{22} \nonumber \\
- x_3^{(1)} =&(b_3- a_{31}x_1^{(0)} -a_{32}x_2^{(0)} -a_{34}x_4^{(0)})/a_{33} \nonumber \\
- x_4^{(1)}=&(b_4-a_{41}x_1^{(0)} -a_{42}x_2^{(0)} - a_{43}x_3^{(0)})/a_{44}, \nonumber
-\end{align}
-$$
-
-which after \( k+1 \) iterations reads
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber
-\end{align}
-$$
-
-
-We can generalize the above equations to
-$$
- x_i^{(k+1)}=(b_i-\sum_{j=1, j\ne i}^{n}a_{ij}x_j^{(k)})/a_{ii}
-$$
-
-or in an even more compact form as
-$$ \mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}),
-$$
-
-with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and
-\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular
-matrix.
-
-
-Our \( 4\times 4 \) matrix problem
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber
-\end{align}
-$$
-
-can be rewritten as
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber
-\end{align}
-$$
-
-which allows us to utilize the preceding solution (forward substitution). This improves normally the convergence
-behavior and leads to the Gauss-Seidel method!
-
-
-We can generalize
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber
-\end{align}
-$$
-
-to the following form
-$$
- x^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i - \sum_{j > i}a_{ij}x^{(k)}_j - \sum_{j < i}a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n.
-$$
-
-The procedure is generally continued until the changes made by an iteration are below some tolerance.
-
-
-The convergence properties of the Jacobi method and the
-Gauss-Seidel method are dependent on the matrix \( \mathbf{A} \). These methods converge when
-the matrix is symmetric positive-definite, or is strictly or irreducibly diagonally dominant.
-Both methods sometimes converge even if these conditions are not satisfied.
-
-
-Given a square system of n linear equations with unknown \( \mathbf x \):
-$$
- \mathbf{A}\mathbf x = \mathbf b
-$$
-
-where
-$$
- \mathbf{A}=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \qquad \mathbf{x} = \begin{bmatrix} x_{1} \\ x_2 \\ \vdots \\ x_n \end{bmatrix} , \qquad \mathbf{b} = \begin{bmatrix} b_{1} \\ b_2 \\ \vdots \\ b_n \end{bmatrix}.
-$$
-
-
-Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U:
-$$
- \mathbf{A} =\mathbf{D} + \mathbf{L} + \mathbf{U},
-$$
-
-where
-$$
- D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & a_{nn} \end{bmatrix}, \quad L = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ a_{21} & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & 0 \end{bmatrix}, \quad U = \begin{bmatrix} 0 & a_{12} & \cdots & a_{1n} \\ 0 & 0 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \end{bmatrix}.
-$$
-
-The system of linear equations may be rewritten as:
-$$
- (D+\omega L) \mathbf{x} = \omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}
-$$
-
-for a constant \( \omega > 1 \).
-
-
-The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for \( x \), using previous value for \( x \) on the right hand side. Analytically, this may be written as:
-$$
- \mathbf{x}^{(k+1)} = (D+\omega L)^{-1} \big(\omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}^{(k)}\big).
-$$
-
-However, by taking advantage of the triangular form of \( (D+\omega L) \), the elements of \( x^{(k+1)} \) can be computed sequentially using forward substitution:
-$$
- x^{(k+1)}_i = (1-\omega)x^{(k)}_i + \frac{\omega}{a_{ii}} \left(b_i - \sum_{j > i} a_{ij}x^{(k)}_j - \sum_{j < i} a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n.
-$$
-
-The choice of relaxation factor is not necessarily easy, and depends upon the properties of the coefficient matrix. For symmetric, positive-definite matrices it can be proven that \( 0 < \omega < 2 \) will lead to convergence, but we are generally interested in faster convergence rather than just convergence.
-
diff --git a/doc/pub/Linalg/html/Linalg.html b/doc/pub/Linalg/html/Linalg.html
index c28ebdded..80ec79dd2 100644
--- a/doc/pub/Linalg/html/Linalg.html
+++ b/doc/pub/Linalg/html/Linalg.html
@@ -6,9 +6,9 @@ Automatically generated HTML file from DocOnce source
Matrix Handling in C/C++, Static and Dynamical allocation
+Numpy and arrays
+Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as
+import numpy as np
+n = 10
+x = np.random.normal(size=n)
+print(x)
+
import numpy as np
+x = np.array([1, 2, 3])
+print(x)
+
import numpy as np
+x = np.log(np.array([4, 7, 8]))
+print(x)
+
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)
+
import numpy as np
+x = np.log(np.array([4, 7, 8], dtype = np.float64))
+print(x)
+
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x)
+
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x.itemsize)
+
Matrices in Python
+Having defined vectors, we are now ready to try out matrices. We can define a \( 3 \times 3 \) real matrix \( \hat{A} \)
+as (recall that we user lowercase letters for vectors and uppercase letters for matrices)
+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)
+
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])
+
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,:])
+
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)
+
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)
+
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)
+
+$$
+\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
+ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
+ \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
+ \end{bmatrix},
+$$
+
+
+where for example
+
+$$
+\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
+$$
+
+
+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 \( \hat{W} \)
+
+$$
+\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
+ x_1 & y_1 & z_1 \\
+ x_2 & y_2 & z_2 \\
+ \dots & \dots & \dots \\
+ x_{n-2} & y_{n-2} & z_{n-2} \\
+ x_{n-1} & y_{n-1} & z_{n-1}
+ \end{bmatrix},
+$$
+
+
+# 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)
+
Matrix Handling in C/C++, Static and Dynamical allocation
Matrix Handling in C/C++
+Matrix Handling in C/C++
Matrix Handling in C/C++
+Matrix Handling in C/C++
Dynamic memory allocation in C/C++
+Dynamic memory allocation in C/C++
Matrix Handling in C/C++, Dynamic Allocation
+Matrix Handling in C/C++, Dynamic Allocation
Armadillo, recommended!!
+Armadillo, recommended!!
@@ -463,7 +695,7 @@ Always free space when you don't need an array anymore.
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, how to compile and install
+Armadillo, how to compile and install
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Armadillo, simple examples
+Armadillo, simple examples
Gaussian Elimination
+Gaussian Elimination
Gaussian Elimination
+Gaussian Elimination
or
@@ -798,7 +1030,7 @@ $$
Gaussian Elimination
+Gaussian Elimination
Gaussian Elimination
+Gaussian Elimination
This process can be expressed mathematically as
@@ -844,7 +1076,7 @@ the result from the $j$th equation. We assume obviously that \( a_{11}\ne 0 \) a
Gaussian Elimination
+Gaussian Elimination
Gaussian Elimination
+Gaussian Elimination
Gaussian Elimination
+Gaussian Elimination
Linear Algebra Methods
+Linear Algebra Methods
@@ -992,7 +1224,7 @@ adding \( 10^7+1 \). With single precision this results in \( 10^7 \).
LU Decomposition
+LU Decomposition
LU Decomposition
+LU Decomposition
LU Decomposition, why?
+LU Decomposition, why?
LU Decomposition, linear equations
+LU Decomposition, linear equations
LU Decomposition, linear equations
+LU Decomposition, linear equations
-and since the determinat of \( \mathbf{L} \) is equal to 1 (by construction
+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
@@ -1151,7 +1383,7 @@ through \( \mathbf{Ux}=\mathbf{y} \).
LU Decomposition, why?
+LU Decomposition, why?
LU Decomposition, linear equations
+LU Decomposition, linear equations
LU Decomposition, the inverse of a matrix
+LU Decomposition, the inverse of a matrix
LU Decomposition, the inverse of a matrix
+LU Decomposition, the inverse of a matrix
LU Decomposition, the inverse
+LU Decomposition, the inverse
Using Armadillo to perform an LU decomposition
+Using Armadillo to perform an LU decomposition
Iterative methods, Chapter 6
-
-
-Iterative methods, Jacobi's method
-
-$$
-\mathbf{A}\mathbf{x}=\mathbf{b},
-$$
-
-
-where \( \mathbf{A} \) is a matrix and \( \mathbf{x} \) and \( \mathbf{b} \) are vectors. The vector \( \mathbf{x} \) is
-the unknown.
-
-
-$$
-\mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}),
-$$
-
-
-with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and
-\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular
-matrix.
-
-Iterative methods, Jacobi's method
-
-$$
-\begin{align}
- x_1^{(1)} =&(b_1-a_{12}x_2^{(0)} -a_{13}x_3^{(0)} - a_{14}x_4^{(0)})/a_{11} \nonumber \\
- x_2^{(1)} =&(b_2-a_{21}x_1^{(0)} - a_{23}x_3^{(0)} - a_{24}x_4^{(0)})/a_{22} \nonumber \\
- x_3^{(1)} =&(b_3- a_{31}x_1^{(0)} -a_{32}x_2^{(0)} -a_{34}x_4^{(0)})/a_{33} \nonumber \\
- x_4^{(1)}=&(b_4-a_{41}x_1^{(0)} -a_{42}x_2^{(0)} - a_{43}x_3^{(0)})/a_{44}, \nonumber
-\end{align}
-$$
-
-
-which after \( k+1 \) iterations reads
-
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber
-\end{align}
-$$
-
-Iterative methods, Jacobi's method
-
-$$
- x_i^{(k+1)}=(b_i-\sum_{j=1, j\ne i}^{n}a_{ij}x_j^{(k)})/a_{ii}
-$$
-
-
-or in an even more compact form as
-
-$$ \mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}),
-$$
-
-
-with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and
-\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular
-matrix.
-Iterative methods, Gauss-Seidel's method
-
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber
-\end{align}
-$$
-
-
-can be rewritten as
-
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber
-\end{align}
-$$
-
-
-which allows us to utilize the preceding solution (forward substitution). This improves normally the convergence
-behavior and leads to the Gauss-Seidel method!
-Iterative methods, Gauss-Seidel's method
-
-$$
-\begin{align}
- x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\
- x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\
- x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\
- x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber
-\end{align}
-$$
-
-
-to the following form
-
-$$
- x^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i - \sum_{j > i}a_{ij}x^{(k)}_j - \sum_{j < i}a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n.
-$$
-
-
-The procedure is generally continued until the changes made by an iteration are below some tolerance.
-
-Iterative methods, Successive over-relaxation
-
-$$
- \mathbf{A}\mathbf x = \mathbf b
-$$
-
-
-where
-
-$$
- \mathbf{A}=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \qquad \mathbf{x} = \begin{bmatrix} x_{1} \\ x_2 \\ \vdots \\ x_n \end{bmatrix} , \qquad \mathbf{b} = \begin{bmatrix} b_{1} \\ b_2 \\ \vdots \\ b_n \end{bmatrix}.
-$$
-
-Iterative methods, Successive over-relaxation
-
-$$
- \mathbf{A} =\mathbf{D} + \mathbf{L} + \mathbf{U},
-$$
-
-
-where
-
-$$
- D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & a_{nn} \end{bmatrix}, \quad L = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ a_{21} & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & 0 \end{bmatrix}, \quad U = \begin{bmatrix} 0 & a_{12} & \cdots & a_{1n} \\ 0 & 0 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \end{bmatrix}.
-$$
-
-
-The system of linear equations may be rewritten as:
-
-$$
- (D+\omega L) \mathbf{x} = \omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}
-$$
-
-
-for a constant \( \omega > 1 \).
-Iterative methods, Successive over-relaxation
-
-$$
- \mathbf{x}^{(k+1)} = (D+\omega L)^{-1} \big(\omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}^{(k)}\big).
-$$
-
-
-However, by taking advantage of the triangular form of \( (D+\omega L) \), the elements of \( x^{(k+1)} \) can be computed sequentially using forward substitution:
-
-$$
- x^{(k+1)}_i = (1-\omega)x^{(k)}_i + \frac{\omega}{a_{ii}} \left(b_i - \sum_{j > i} a_{ij}x^{(k)}_j - \sum_{j < i} a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n.
-$$
-
-
-The choice of relaxation factor is not necessarily easy, and depends upon the properties of the coefficient matrix. For symmetric, positive-definite matrices it can be proven that \( 0 < \omega < 2 \) will lead to convergence, but we are generally interested in faster convergence rather than just convergence.
-Data analysis and Machine Learning Lectures: Linear Algebra methods
Data analysis and Machine Learning Lectures: Linear Algebra and Handling of Arrays
May 22, 2018
May 24, 2018
-To do
+Introduction
+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.
-
-
+
+Important Matrix and vector handling packages
-Add python material on linear algebra and array handling, text on numpy etc
+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.
+
Basic Matrix Features
@@ -239,6 +246,8 @@ $$
Basic Matrix Features
-Matrix Handling in C/C++, Static and Dynamical allocation
+Numpy and arrays
+Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as
+import numpy as np
+n = 10
+x = np.random.normal(size=n)
+print(x)
+
import numpy as np
+x = np.array([1, 2, 3])
+print(x)
+
import numpy as np
+x = np.log(np.array([4, 7, 8]))
+print(x)
+
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)
+
import numpy as np
+x = np.log(np.array([4, 7, 8], dtype = np.float64))
+print(x)
+
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x)
+
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x.itemsize)
+
+
+Matrices in Python
+Having defined vectors, we are now ready to try out matrices. We can define a \( 3 \times 3 \) real matrix \( \hat{A} \)
+as (recall that we user lowercase letters for vectors and uppercase letters for matrices)
+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)
+
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])
+
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,:])
+
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)
+
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)
+
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)
+
# 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)
+
+
+Matrix Handling in C/C++, Static and Dynamical allocation
-Matrix Handling in C/C++
+Matrix Handling in C/C++
-Matrix Handling in C/C++
+Matrix Handling in C/C++
-Dynamic memory allocation in C/C++
+Dynamic memory allocation in C/C++
-Matrix Handling in C/C++, Dynamic Allocation
+Matrix Handling in C/C++, Dynamic Allocation
-Armadillo, recommended!!
+Armadillo, recommended!!
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, how to compile and install
+Armadillo, how to compile and install
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Armadillo, simple examples
+Armadillo, simple examples
-Gaussian Elimination
+Gaussian Elimination
-Gaussian Elimination
+Gaussian Elimination
or
$$
@@ -779,7 +990,7 @@ $$
-Gaussian Elimination
+Gaussian Elimination
-Gaussian Elimination
+Gaussian Elimination
This process can be expressed mathematically as
$$
@@ -821,7 +1032,7 @@ the result from the $j$th equation. We assume obviously that \( a_{11}\ne 0 \) a
-Gaussian Elimination
+Gaussian Elimination
-Gaussian Elimination
+Gaussian Elimination
-Gaussian Elimination
+Gaussian Elimination
-Linear Algebra Methods
+Linear Algebra Methods
-LU Decomposition
+LU Decomposition
-LU Decomposition
+LU Decomposition
-LU Decomposition, why?
+LU Decomposition, why?
-LU Decomposition, linear equations
+LU Decomposition, linear equations
-LU Decomposition, linear equations
+LU Decomposition, linear equations
-LU Decomposition, why?
+LU Decomposition, why?
-LU Decomposition, linear equations
+LU Decomposition, linear equations
-LU Decomposition, the inverse of a matrix
+LU Decomposition, the inverse of a matrix
-LU Decomposition, the inverse of a matrix
+LU Decomposition, the inverse of a matrix
-LU Decomposition, the inverse
+LU Decomposition, the inverse
-Using Armadillo to perform an LU decomposition
+Using Armadillo to perform an LU decomposition
-
-Iterative methods, Chapter 6
-
-
-
-
-Iterative methods, Jacobi's method
-
-
-Iterative methods, Jacobi's method
-
-
-Iterative methods, Jacobi's method
-
-
-Iterative methods, Gauss-Seidel's method
-
-
-Iterative methods, Gauss-Seidel's method
-
-
-Iterative methods, Successive over-relaxation
-
-
-Iterative methods, Successive over-relaxation
-
-
-Iterative methods, Successive over-relaxation
-