Matrix Computations     
          
1.   Systems of Equations
       1.1   Applications to Differential Equations
               1.1.1   Boundary Value Problems   pdf
               1.1.2   Systems of equations    pdf
               1.1.3   Vectors and Matrices in Mathematica
               1.1.4   Vectors and Matrices in MATLAB
       1.2   Matrix Multiplication    pdf
       1.3   Implementing Matrix Operations in Programming Languages
               1.3.1   General Principles    pdf
               1.3.2   An Example with C
       1.4   LU Decompositions Without Pivoting
               1.4.1   The Algorithm    pdf
               1.4.2a  Gaussian elimination without pivoting in Mathematica
               1.4.2b  LU decomposition without pivoting in Mathematica
               1.4.3   LU decomposition without pivoting in MATLAB
               1.4.4   LU decomposition without pivoting in C
       1.5   LU Decompositions With Partial Pivoting
               1.5.1   The Algorithm    pdf
               1.5.2   Gaussian elimination with partial pivoting in Mathematica
               1.5.4   LU decomposition with partial pivoting in C
       1.6   How Long Does it Take to Solve Linear Equations
               1.6.1   Basic Priciples    pdf
               1.6.2   The Time to Solve Linear Equations in Mathematicaa
       1.7   Electrical Networks
               1.7.1   Nodal Analysis    pdf
               1.7.2   Loop Analysis    pdf
               1.7.3   Network Calculations in Mathematica    
       1.8   Positive definite systems and Cholesky decomposition    pdf

2.   Errors   
       2.1   Describing Errors    pdf
       2.2   Propagation of Errors - Componentwise Analysis    pdf
               2.2.2   Errors in the boundary value problem
               2.2.3   Finding the quadratic through three points
       2.3   Propagation of Errors - Estimation using norms    pdf
               2.3.2   Errors in the boundary value problem
               2.3.3   Finding the quadratic through three points
       2.4   General Vector Norms    pdf
       2.5   General Matrix Norms    pdf
       2.6   Geometric description of Matrix Norms    pdf
       2.7   Propagation of Coefficient Errors    pdf
       2.8   Floating Point Numbers and Round-off Errors    pdf



3.   Least Squares Problems and QR Factorizations
       3.1   Projections & Reflections
       3.2   General Least Squares Problems
       3.3   Householder Triangularization
               3.3.1   The algorithm
    PDF               3.3.2   The calculations illustrated in Mathematica
       3.4   Least Squrares and Householder Triangularization
               3.4.1   Solving least squares problems with Householder triangularization
               3.4.2   Solving least squares problems with Householder triangularization in Mathematica
               3.4.3   Fitting a parabola revisted
       3.5   The Gram Schmidt Process

4.   Eigenvalues and Eigenvectors
       4.1   The Characteristic Equation for the Eigenvalues
       4.2   Matrix Powers
       4.3   Difference Equations
       4.4   Power Iteration
               4.4.1   The Power Iteration Algorithm
               4.4.2   Power Iteration in Mathematica
               4.4.3   Deflation
               4.4.4   Power Iteration with Deflation in Mathematica
       4.5   Inverse Power Iteration
               4.5.1   The Inverse Power Iteration Algorithm
               4.5.2   Inverse Power Iteration in Mathematica 
       4.6   Shifted Inverse Power Iteration
               4.6.1   The Shifted Inverse Power Iteration Algorithm
               4.6.2   Shifted Inverse Power Iteration in Mathematica 
       4.7   Different Coordinate Systems
               4.7.1   Different Coordinate Systems 
       4.8   Simultaneous Power Iteration
               4.8.1   Simultaneous Power Iteration
               4.8.2   Simultaneous Power Iteration 
        4.9  The QR Algorithm
               4.9.1   The Alogrithm
               4.9.2   The QR Algorithm in Mathematica  
        4.10 Reduction to Upper Hessenberg Form
               4.10.2   Reduction to upper Hessenberg form in Mathematica  
        4.11 Reduction to Upper Hessenberg Form
               4.11.2   The QR algorithm applied to upper Hessenberg in Mathematica  
        4.12 The QR algorithm with shifts
               4.12.2   The QR algorithm with shifts in Mathematica  
        4.13 The QR algorithm with reduction in size
               4.13.2   The QR algorithm with reduction of size in Mathematica  
        4.14 Complex Eigenvalues
        4.15 The Eigenvalues of a 2x2 Matrix
        4.16 The Gram Schmidt Process with Complex Vectors
 
 An Introduction to Mathematica

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