Mastering SciPy电子书

Mastering SciPy

Table of Contents

Mastering SciPy

Credits

About the Author

About the Reviewers

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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