Mastering SciPy

Table of Contents

Mastering SciPy

Credits

About the Author

About the Reviewers

www.PacktPub.com

Support files, eBooks, discount offers, and more

Why subscribe?

Free access for Packt account holders

Preface

What this book covers

What you need for this book

Who this book is for

Conventions

Reader feedback

Customer support

Downloading the example code

Downloading the color images of this book

Errata

Piracy

Questions

1. Numerical Linear Algebra

Motivation

Creation of matrices and linear operators

Constructing matrices in the ndarray class

Constructing matrices in the matrix class

Constructing sparse matrices

Linear operators

Basic matrix manipulation

Scalar multiplication, matrix addition, and matrix multiplication

Traces and determinants

Transposes and inverses

Norms and condition numbers

Matrix functions

Matrix factorizations related to solving matrix equations

Relevant factorizations

Pivoted LU decomposition

Cholesky decomposition

QR decomposition

Singular value decomposition

Matrix equations

Back and forward substitution

Basic systems: banded matrices

Basic systems: generic square matrices

Least squares

Normal equations

QR factorization

Singular value decomposition

Regularized least squares

Other matrix equation solvers

Matrix factorizations based on eigenvalues

Spectral decomposition

Schur decomposition

Summary

2. Interpolation and Approximation

Motivation

Interpolation

Implementation details

Univariate interpolation

Nearest-neighbors interpolation

Lagrange interpolation

Hermite interpolation

Piecewise polynomial interpolation

Spline interpolation

Multivariate interpolation

Least squares approximation

Linear least squares approximation

Nonlinear least squares approximation

Summary

3. Differentiation and Integration

Motivation

Differentiation

Numerical differentiation

Symbolic differentiation

Automatic differentiation

Integration

Symbolic integration

Numerical integration

Functions without singularities on finite intervals

Functions with singularities on bounded domains

Weighted functions

General functions with singularities

Integration on unbounded domains

Numerical multivariate integration

Summary

4. Nonlinear Equations and Optimization

Motivation

Non-linear equations and systems

Iterative methods for univariate functions

Bracketing methods

Secant methods

Brent method

Systems of nonlinear equations

Simple iterative solvers

The Broyden method

Powell's hybrid solver

Large-scale solvers

Optimization

Unconstrained optimization for univariate functions

Constrained optimization for univariate functions

Unconstrained optimization for multivariate functions

The stochastic methods

Deterministic algorithms that exclusively employ function evaluations

The Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method

The conjugate gradient method

Constrained optimization for multivariate functions

Summary

5. Initial Value Problems for Ordinary Differential Equations

Symbolic solution of differential equations

Analytic approximation methods

Discrete-variable methods

One-step methods

Two-step methods

Summary

6. Computational Geometry

Plane geometry

Combinatorial computational geometry

Static problems

Convex hulls

Voronoi diagrams

Triangulations

Shortest paths

Geometric query problems

Point location

Nearest neighbors

Range searching

Dynamic problems

Numerical computational geometry

Bézier curves

Summary

7. Descriptive Statistics

Motivation

Probability

Symbolic setting

Numerical setting

Data exploration

Picturing distributions with graphs

Bar plots and pie charts

Histograms

Time plots

Describing distributions with numbers and boxplots

Relationship between quantitative variables

Scatterplots and correlation

Regression

Ordinary linear regression for moderate-sized datasets

Ordinary least-squares regression for large datasets

Linear regression beyond ordinary least-squares

Support vector machines

Ensemble methods

Analysis of the time series

Summary

8. Inference and Data Analysis

Statistical inference

Estimation of parameters

Frequentist approach

Bayesian approach

Likelihood approach

Interval estimation

Frequentist approach

Bayesian approach

Likelihood approach

Data mining and machine learning

Classification

Support vector classification

Trees

Naive Bayes

Nearest neighbors

Dimensionality reduction

Principal component analysis

Isometric mappings

Spectral embedding

Locally linear embedding

Clustering

MeanShift

Gaussian mixture models

Kmeans

Spectral clustering

Summary

9. Mathematical Imaging

Digital images

Binary

Gray-scale

Color

Alpha channels

High-level operations on digital images

Object measurements

Mathematical morphology

Smoothing filters

Multivariate calculus

Statistical filters

Fourier analysis

Wavelet decompositions

Image compression

Lossless compression

Lossy compression

Image editing

Transformations of the domain

Rescale and resize

Swirl

Geometric transformations

Intensity adjustment

Histogram equalization

Intensity clipping/resizing

Contrast enhancement

Image restoration

Noise reduction

Sharpening and blurring

Inpainting

Image analysis

Image structure

Object recognition

Edge detection

Line, circle, and ellipse detection

Blob detection

Corner detection

Beyond geometric entities

Summary

Index

Mastering SciPy

Mastering SciPy

Copyright © 2015 Packt Publishing

All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the publisher, except in the case of brief quotations embedded in critical articles or reviews.

Every effort has been made in the preparation of this book to ensure the accuracy of the information presented. However, the information contained in this book is sold without warranty, either express or implied. Neither the author, nor Packt Publishing, and its dealers and distributors will be held liable for any damages caused or alleged to be caused directly or indirectly by this book.

Packt Publishing has endeavored to provide trademark information about all of the companies and products mentioned in this book by the appropriate use of capitals. However, Packt Publishing cannot guarantee the accuracy of this information.

First published: November 2015

Production reference: 1301015

Published by Packt Publishing Ltd.

Livery Place

35 Livery Street

Birmingham B3 2PB, UK.

ISBN 978-1-78398-474-9

www.packtpub.com

Credits

Author

Francisco J. Blanco-Silva

Reviewers

Raiyan Kamal

Kristen Thyng

Patrick Varilly

Jonathan Whitmore

Commissioning Editor

Kartikey Pandey

Acquisition Editor

Shaon Basu

Content Development Editor

Nikhil Potdukhe

Technical Editor

Bharat Patil

Copy Editors

Tani Kothari

Merilyn Pereira

Project Coordinator

Judie Jose

Proofreader

Safis Editing

Indexer

Tejal Soni

Production Coordinator

Aparna Bhagat

Cover Work

Aparna Bhagat

About the Author

Francisco J. Blanco-Silva is the owner of a scientific consulting company called Tizona Scientific Solutions, a faculty member of the Department of Mathematics, and an associate member of the Interdisciplinary Mathematics Institute at the University of South Carolina. He obtained his formal training as an applied mathematician from Purdue University. He enjoys problem solving, learning, and teaching alike. Being an avid programmer and blogger, when it comes to writing, he relishes finding the common denominator among his passions and skills and making it available to everyone.

He wrote the prequel to this book, Learning SciPy for Numerical and Scientific Computing, Packt Publishing, and coauthored Chapter 5 of the book, Modeling Nanoscale Imaging in Electron Microscopy, Springer.

I will always be indebted to Bradley J. Lucier and Rodrigo Bañuelos for being constant sources of inspiration and for their guidance and teachings. Special thanks to my editors, Sriram Neelakantam, Bharat Patil, Nikhil Potdukhe, Mohammad Rizvi, and the many colleagues who have contributed by giving me encouragement and participating in helpful discussions. In particular, I would like to mention Parsa Bakhtary, Aaron Dutle, Edsel Peña, Pablo Sprechmann, Adam Taylor, and Holly Watson.

The most special thanks, without a doubt, goes to my wife and daughter. Grace's love and smiles alone provided all the motivation, enthusiasm, and skills to overcome the difficulties encountered during the writing of this book and everything that life threw at me ever since she was born.

About the Reviewers

Raiyan Kamal is a strong proponent of the open source movement and everything related to Python. He holds a bachelor's degree in computer science from BUET, Dhaka, Bangladesh, and a master's degree from the University of Windsor, Ontario, Canada. He has been working in the software industry for several years, developing software for mobile, web, and desktop platforms. Although he is in his early thirties, Raiyan feels that his boyhood has not ended yet. He often looks for hidden treasures in science, engineering, programming, art, and nature. He is currently working at IOU Concepts, exploring different ways of saying thank you. When he isn't on a computer, he plants trees and composts kitchen scraps.

Kristen Thyng has worked on scientific computing for most of her career. She has a bachelor's degree in physics from Whitman College, master's degree in applied mathematics from the University of Washington, and PhD in mechanical engineering from the University of Washington. She uses Python on a daily basis for analysis and visualization in physical oceanography at Texas A&M University, where she works as an assistant research scientist.

Jonathan Whitmore is a data scientist at Silicon Valley Data Science. He has a diverse range of interests and is excited by the challenges in data science and data engineering. Before moving into the tech industry, he worked as an astrophysicist in Melbourne, Australia, researching whether the fundamental physical constants have changed over the lifespan of the universe. He has a long-standing commitment to the public's understanding of science and technology, and has contributed to FOSS projects. He co-starred in the 3D IMAX film Hidden Universe, which was playing in theaters around the world at the time of writing this book. Jonathan is a sought-after conference speaker on science and technical topics. He received his PhD in physics from the University of California, San Diego, and graduated magna cum laude from Vanderbilt University with a bachelor's degree in science. He is also a triple major in physics (with honors), philosophy, and mathematics.

www.PacktPub.com

Support files, eBooks, discount offers, and more

For support files and downloads related to your book, please visit www.PacktPub.com.

Did you know that Packt offers eBook versions of every book published, with PDF and ePub files available? You can upgrade to the eBook version at www.PacktPub.com and as a print book customer, you are entitled to a discount on the eBook copy. Get in touch with us at for more details.

At www.PacktPub.com, you can also read a collection of free technical articles, sign up for a range of free newsletters and receive exclusive discounts and offers on Packt books and eBooks.

https://www2.packtpub.com/books/subscription/packtlib

Do you need instant solutions to your IT questions? PacktLib is Packt's online digital book library. Here, you can search, access, and read Packt's entire library of books.

Why subscribe?

Fully searchable across every book published by Packt

Copy and paste, print, and bookmark content

On demand and accessible via a web browser

Free access for Packt account holders

If you have an account with Packt at www.PacktPub.com, you can use this to access PacktLib today and view 9 entirely free books. Simply use your login credentials for immediate access.

Preface

The idea of writing Mastering SciPy arose but 2 months after publishing Learning SciPy for Numerical and Scientific Computing. During a presentation of that book at the University of South Carolina, I had the privilege of speaking about its contents to a heterogeneous audience of engineers, scientists, and students, each of them with very different research problems and their own set of preferred computational resources. In the weeks following that presentation, I helped a few professionals transition to a SciPy-based environment. During those sessions, we discussed how SciPy is, under the hood, the same set of algorithms (and often the same code) that they were already using. We experimented with some of their examples and systematically obtained comparable performance. We immediately saw the obvious benefit of a common environment based upon a robust scripting language. Through the SciPy stack, we discovered an easier way to communicate and share our results with colleagues, students, or employers. In all cases, the switch to the SciPy stack provided a faster setup for our groups, where newcomers could get up to speed quickly.

Everybody involved in the process went from novice to advanced user, and finally mastered the SciPy stack in no time. In most cases, the scientific background of the individuals with whom I worked made the transition seamless. The process toward mastering materialized when they were able to contrast the theory behind their research with the solutions offered. The aha moment always happened while replicating some of their experiments with a careful guidance and explanation of the process.

That is precisely the philosophy behind this book. I invite you to participate in similar sessions. Each chapter has been envisioned as a conversation with an individual with certain scientific needs expressed as numerical computations. Together, we discover relevant examples—the different possible ways to solve those problems, the theory behind them, and the pros and cons of each route.

The process of writing followed a similar path to obtain an engaging collection of examples. I entered into conversations with colleagues in several different fields. Each section clearly reflects these exchanges. This was crucial while engaged in the production of the most challenging chapters—the last four. To ensure the same quality throughout the book, always trying to commit to a rigorous set of standards, these chapters took much longer to be completed to satisfaction. Special mentions go to Aaron Dutle at NASA Langley Research Center, who helped shape parts of the chapter on computational geometry, and Parsa Bakhtary, a data analyst at Facebook, who inspired many of the techniques in the chapter on applications of statistical computing to data analysis.

It was an amazing journey that helped deepen my understanding of numerical methods, broadened my perspective in problem solving, and strengthened my scientific maturity. It is my wish that it has the same impact on you.

What this book covers

Chapter 1, Numerical Linear Algebra, presents an overview of the role of matrices to solve problems in scientific computing. It is a crucial chapter for understanding most of the processes and ideas of subsequent chapters. You will learn how to construct and store large matrices effectively in Python. We then proceed to reviewing basic manipulation and operations on them, followed by factorizations, solutions of matrix equations, and the computation of eigenvalues/eigenvectors.

Chapter 2, Interpolation and Approximation, develops advanced techniques to approximate functions, and their applications to scientific computing. This acts as a segway for the next two chapters.

Chapter 3, Differentiation and Integration, explores the different techniques to produce derivatives of functions and, more importantly, how to compute areas and volumes effectively by integration processes. This is the first of two chapters devoted to the core of numerical methods in scientific computing. This second part is also an introduction to Chapter 5, Initial Value Problems for Ordinary Differential Equations that mentions ordinary differential equations.

Chapter 4, Nonlinear Equations and Optimization, is a very technical chapter in which we discuss the best methods of obtaining the roots and extrema of systems of functions depending on the kinds of functions involved.

Chapter 5, Initial Value Problems for Ordinary Differential Equations, is the first of five chapters on applications to real-world problems. We show you, by example, the most popular techniques to solve systems of differential equations, as well as some applications.

Chapter 6, Computational Geometry, takes a tour of the most significant algorithms in this branch of computer science.

Chapter 7, Descriptive Statistics, is the first of two chapters on statistical computing and its applications to Data Analysis. In this chapter, we focus on probability and data exploration.

Chapter 8, Inference and Data Analysis, is the second chapter on Data Analysis. We focus on statistical inference, machine learning, and data mining.

Chapter 9, Mathematical Imaging, is the last chapter of this book. In it, we explore techniques for image compression, edition, restoration, and analysis.

What you need for this book

To work with the examples and try out the code of this book, all you need is a recent version of Python (2.7 or higher) with the SciPy stack: NumPy, the SciPy library, matplotlib, IPython, pandas, and SymPy. Although recipes to install all these independently are provided throughout the book, we recommend that you perform a global installation through a scientific Python distribution such as Anaconda.

Who this book is for

Although this book and technology are ultimately intended for applied mathematicians, engineers, and computer scientists, the material presented here is targeted at a broader audience. All that is needed is proficiency in Python, familiarity with iPython, some knowledge of the numerical methods in scientific computing, and a keen interest in developing serious applications in science, engineering, or data analysis.

Conventions

In this book, you will find a number of text styles that distinguish between different kinds of information. Here are some examples of these styles and an explanation of their meaning.

Code words in text, database table names, folder names, filenames, file extensions, pathnames, dummy URLs, user input, and Twitter handles are shown as follows: We can include other contexts through the use of the include directive.

Any command-line input or output is written as follows:

In [7]: %time eigvals, v = spspla.eigsh(A, 5, which='SM') CPU times: user 19.3 s, sys: 532 ms, total: 19.8 s Wall time: 16.7 s In [8]: print eigvals [ 10.565523  10.663114  10.725135  10.752737  10.774503]

Note

Warnings or important notes appear in a box like this.

Tip

Tips and tricks appear like this.

Reader feedback

Feedback from our readers is always welcome. Let us know what you think about this book—what you liked or disliked. Reader feedback is important for us as it helps us develop titles that you will really get the most out of.

To send us general feedback, simply e-mail <feedback@packtpub.com>, and mention the book's title in the subject of your message.

If there is a topic that you have expertise in and you are interested in either writing or contributing to a book, see our author guide at www.packtpub.com/authors.

Customer support

Now that you are the proud owner of a Packt book, we have a number of things to help you to get the most from your purchase.

Downloading the example code

You can download the example code files from your account at http://www.packtpub.com for all the Packt Publishing books you have purchased. If you purchased this book elsewhere, you can visit http://www.packtpub.com/support and register to have the files e-mailed directly to you. You can also download the code files from GitHub repository at https://github.com/blancosilva/Mastering-Scipy.

Downloading the color images of this book

We also provide you with a PDF file that has color images of the screenshots/diagrams used in this book. The color images will help you better understand the changes in the output. You can download this file from https://www.packtpub.com/sites/default/files/downloads/4749OS_ColorImages.pdf.

Errata

Although we have taken every care to ensure the accuracy of our content, mistakes do happen. If you find a mistake in one of our books—maybe a mistake in the text or the code—we would be grateful if you could report this to us. By doing so, you can save other readers from frustration and help us improve subsequent versions of this book. If you find any errata, please report them by visiting http://www.packtpub.com/submit-errata, selecting your book, clicking on the Errata Submission Form link, and entering the details of your errata. Once your errata are verified, your submission will be accepted and the errata will be uploaded to our website or added to any li st of existing errata under the Errata section of that title.

To view the previously submitted errata, go to https://www.packtpub.com/books/content/support and enter the name of the book in the search field. The required information will appear under the Errata section.

Piracy

Piracy of copyrighted material on the Internet is an ongoing problem across all media. At Packt, we take the protection of our copyright and licenses very seriously. If you come across any illegal copies of our works in any form on the Internet, please provide us with the location address or website name immediately so that we can pursue a remedy.

Please contact us at <copyright@packtpub.com> with a link to the suspected pirated material.

We appreciate your help in protecting our authors and our ability to bring you valuable content.

Questions

If you have a problem with any aspect of this book, you can contact us at <questions@packtpub.com>, and we will do our best to address the problem.

Chapter 1. Numerical Linear Algebra

The term Numerical Linear Algebra refers to the use of matrices to solve computational science problems. In this chapter, we start by learning how to construct these objects effectively in Python. We make an emphasis on importing large sparse matrices from repositories online. We then proceed to reviewing basic manipulation and operations on them. The next step is a study of the different matrix functions implemented in SciPy. We continue on to exploring different factorizations for the solution of matrix equations, and for the computation of eigenvalues and their corresponding eigenvectors.

Motivation

The following image shows a graph that represents a series of web pages (numbered from 1 to 8):

An arrow from a node to another indicates the existence of a link from the web page, represented by the sending node, to the page represented by the receiving node. For example, the arrow from node 2 to node 1 indicates that there is a link in web page 2 pointing to web page 1. Notice how web page 4 has two outer links (to pages 2 and 8), and there are three pages that link to web page 4 (pages 2, 6, and 7). The pages represented by nodes 2, 4, and 8 seem to be the most popular at first sight.

Is there a mathematical way to actually express the popularity of a web page within a network? Researchers at Google came up with the idea of a PageRank to roughly estimate this concept by counting the number and quality of links to a page. It goes like this:

We construct a transition matrix of this graph, T={a[i,j]}, in the following fashion: the entry a[i,j] is 1/k if there is a link from web page i to web page j, and the total number of outer links in web page i amounts to k. Otherwise, the entry is just zero. The size of a transition matrix of N web pages is always N × N. In our case, the matrix has size 8 × 8:

0  1/2  0  0    0  0  0  0 1  0  1/2 1/2  0  0  0  0 0  0  0  0    0  0  1/3  0 0  1/2  0  0    0  1  1/3  0 0  0  1/2  0    0  0  0  0 0  0  0  0    0  0  0  1/2 0  0  0  0  1/2  0  0  1/2 0  0  0  1/2  1/2  0  1/3  0

Let us open an iPython session and load this particular matrix to memory.

Note

Remember that in Python, indices start from zero, not one.

In [1]: import numpy as np, matplotlib.pyplot as plt, \   ...: scipy.linalg as spla, scipy.sparse as spsp, \   ...: scipy.sparse.linalg as spspla In [2]: np.set_printoptions(suppress=True, precision=3) In [3]: cols = np.array([0,1,1,2,2,3,3,4,4,5,6,6,6,7,7]); \   ...: rows = np.array([1,0,3,1,4,1,7,6,7,3,2,3,7,5,6]); \   ...: data = np.array([1., 0.5, 0.5, 0.5, 0.5, \   ...:                  0.5, 0.5, 0.5, 0.5, 1., \   ...:                  1./3, 1./3, 1./3, 0.5, 0.5]) In [4]: T = np.zeros((8,8)); \   ...: T[rows,cols] = data

From the transition matrix, we create a PageRank matrix G by fixing a positive constant p between 0 and 1, and following the formula G = (1-p)*T + p*B for a suitable damping factor p. Here, B is a matrix with the same size as T, with all its entries equal to 1/N. For example, if we choose p = 0.15, we obtain the following PageRank matrix:

In [5]: G = (1-0.15) * T + 0.15/8; \   ...: print G [[ 0.019  0.444  0.019  0.019  0.019  0.019  0.019  0.019] [ 0.869  0.019  0.444  0.444  0.019  0.019  0.019  0.019] [ 0.019  0.019  0.019  0.019  0.019  0.019  0.302  0.019] [ 0.019  0.444  0.019  0.019  0.019  0.869  0.302  0.019] [ 0.019  0.019  0.444  0.019  0.019  0.019  0.019  0.019] [ 0.019  0.019  0.019  0.019  0.019  0.019  0.019  0.444] [ 0.019  0.019  0.019  0.019  0.444  0.019  0.019  0.444] [ 0.019  0.019  0.019  0.444  0.444  0.019  0.302  0.019]]

PageRank matrices have some interesting properties:

1 is an eigenvalue of multiplicity one.

1 is actually the largest eigenvalue; all the other eigenvalues are in modulus smaller than 1.

The eigenvector corresponding to eigenvalue 1 has all positive entries. In particular, for the eigenvalue 1, there exists a unique eigenvector with the sum of its entries equal to 1. This is what we call the PageRank vector.

A quick computation with scipy.linalg.eig finds that eigenvector for us:

In [6]: eigenvalues, eigenvectors = spla.eig(G); \   ...: print eigenvalues [ 1.000+0.j    -0.655+0.j    -0.333+0.313j -0.333-0.313j –0.171+0.372j -0.171-0.372j  0.544+0.j    0.268+0.j  ] In [7]: PageRank = eigenvectors[:,0]; \   ...: PageRank /= sum(PageRank); \   ...: print PageRank.real [ 0.117  0.232  0.048  0.219  0.039  0.086  0.102  0.157]

Those values correspond to the PageRank of each of the eight web pages depicted on the graph. As expected, the maximum value of those is associated to the second web page (0.232), closely followed by the fourth (0.219) and then the eighth web page (0.157). These values provide us with the information that we were seeking: the second web page is the most popular, followed by the fourth, and then, the eight.

Note

Note how this problem of networks of web pages has been translated into mathematical objects, to an equivalent problem involving matrices, eigenvalues, and eigenvectors, and has been solved with techniques of Linear Algebra.

The transition matrix is sparse: most of its entries are zeros. Sparse matrices with an extremely large size are of special importance in Numerical Linear Algebra, not only because they encode challenging scientific problems but also because it is extremely hard to manipulate them with basic algorithms.

Rather than storing to memory all values in the matrix, it makes sense to collect only the non-zero values instead, and use algorithms which exploit these smart storage schemes. The gain in memory management is obvious. These methods are usually faster for this kind of matrices and give less roundoff errors, since there are usually far less operations involved. This is another advantage of SciPy, since it contains numerous procedures to attack different problems where data is stored in this fashion. Let us observe its power with another example:

The University of Florida Sparse Matrix Collection is the largest database of matrices accessible online. As of January 2014, it contains 157 groups of matrices arising from all sorts of scientific disciplines. The sizes of the matrices range from very small (1 × 2) to insanely large (28 million × 28 million). More matrices are expected to be added constantly, as they arise in different engineering problems.

Tip

More information about this database can be found in ACM Transactions on Mathematical Software, vol. 38, Issue 1, 2011, pp 1:1-1:25, by T.A. Davis and Y.Hu, or online at http://www.cise.ufl.edu/research/sparse/matrices/.

For example, the group with the most matrices in the database is the original Harwell-Boeing Collection, with 292 different sparse matrices. This group can also be accessed online at the Matrix Market: http://math.nist.gov/MatrixMarket/.

Each matrix in the database comes in three formats:

Matrix Market Exchange format [Boisvert et al. 1997]

Rutherford-Boeing Exchangeformat [Duff et al. 1997]

Proprietary Matlab.matformat.

Let us import to our iPython session two matrices in the Matrix Market Exchange format from the collection, meant to be used in a solution of a least squares problem. These matrices are located at www.cise.ufl.edu/research/sparse/matrices/Bydder/mri2.html.The numerical values correspond to phantom data acquired on a Sonata 1.5-T scanner (Siemens, Erlangen, Germany) using a magnetic resonance imaging (MRI) device. The object measured is a simulation of a human head made with several metallic objects. We download the corresponding tar bundle and untar it to get two ASCII files:

mri2.mtx (the main matrix in the least squares problem)

mri2_b.mtx (the right-hand side of the equation)

The first twenty lines of the file mri2.mtx read as follows:

The first sixteen lines are comments, and give us some information about the generation of the matrix.

The computer vision problem where it arose: An MRI reconstruction

Author information: Mark Bydder, UCSD

Procedures to apply to the data: Solve a least squares problem A * x - b, and posterior visualization of the result

The seventeenth line indicates the size of the matrix, 63240 rows × 147456 columns, as well as the number of non-zero entries in the data, 569160.

The rest of the file includes precisely 569160 lines, each