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Mastering SciPy电子书

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作       者:Francisco J. Blanco-Silva

出  版  社:Packt Publishing

出版时间:2015-11-01

字       数:412.6万

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Implement state-of-the-art techniques to visualize solutions to challenging problems in scientific computing, with the use of the SciPy stackAbout This BookMaster the theory and algorithms behind numerical recipes and how they can be applied to real-world problemsLearn to combine the most appropriate built-in functions from the SciPy stack by understanding the connection between the sources of your problem, volume of data, or computer architectureA comprehensive coverage of all the mathematical techniques needed to solve the presented topics, with a discussion of the relevant algorithms built in the SciPy stackWho This Book Is ForIf you are a mathematician, engineer, or computer scientist with a proficiency in Python and familiarity with IPython, this is the book for you. Some basic knowledge of numerical methods in scientific computing would be helpful.What You Will LearnMaster relevant algorithms used in symbolic or numerical mathematics to address approximation, interpolation, differentiation, integration, root-finding, and optimization of scalar or multi-variate functionsDevelop different algorithms and strategies to efficiently store and manipulate large matrices of data, in particular to solve systems of linear equations, or compute their eigenvalues/eigenvectorsUnderstand how to model physical problems with systems of differential equations and distinguish the factors that dictate the strategies to solve themPerform statistical analysis, hypothesis test design and resolution, or data mining at a higher level, and apply them to real-life problems in the field of data analysisGain insights on the power of distances, Delaunay triangulations and Voronoi diagrams for Computational Geometry, and apply them to various engineering problemsFamiliarize yourself with different techniques in signal/image processing, including filtering audio, images, or video to extract information, features, or remove componentsIn DetailThe SciPy stack is a collection of open source libraries of the powerful *ing language Python, together with its interactive shells. This environment offers a cutting-edge platform for numerical computation, programming, visualization and publishing, and is used by some of the world’s leading mathematicians, scientists, and engineers. It works on any operating system that supports Python and is very easy to install, and completely free of charge! It can effectively transform into a data-processing and system-prototyping environment, directly rivalling MATLAB and Octave.This book goes beyond a mere de*ion of the different built-in functions coded in the libraries from the SciPy stack. It presents you with a solid mathematical and computational background to help you identify the right tools for each problem in scientific computing and visualization. You will gain an insight into the best practices with numerical methods depending on the amount or type of data, properties of the mathematical tools employed, or computer architecture, among other factors.The book kicks off with a concise exploration of the basics of numerical linear algebra and graph theory for the treatment of problems that handle large data sets or matrices. In the subsequent chapters, you will delve into the depths of algorithms in symbolic algebra and numerical analysis to address modeling/simulation of various real-world problems with functions (through interpolation, approximation, or creation of systems of differential equations), and extract their representing features (zeros, extrema, integration or differentiation).Lastly, you will move on to advanced concepts of data analysis, image/signal processing, and computational geometry.Style and approachPacked with real-world examples, this book explores the mathematical techniques needed to solve the presented topics, and focuses on the algorithms built in the SciPy stack.
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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

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