MATLAB Mathematics - SERC - Index of

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1 Matrices and Linear Algebra Vector and Matrix Norms The p-norm of a vector x x p = ⎛Σ x p⎞ 1 ⁄ p ⎝ i ⎠ is computed by norm(x,p). This is defined by any value of p > 1, but the most common values of p are 1, 2, and ∞ . The default value is p = 2, which corresponds to Euclidean length: v = [2 0 -1]; [norm(v,1) norm(v) norm(v,inf)] ans = 3.0000 2.2361 2.0000 The p-norm of a matrix A, A p = max x Ax -------------- p x p can be computed for p = 1, 2, and ∞ by norm(A,p). Again, the default value is p = 2. C = fix(10*rand(3,2)); [norm(C,1) norm(C) norm(C,inf)] ans = 19.0000 14.8015 13.0000 1-12
Solving Linear Systems of Equations Solving Linear Systems of Equations This section describes • Computational considerations • The general solution to a system It also discusses particular solutions to • Square systems • Overdetermined systems • Underdetermined systems Computational Considerations One of the most important problems in technical computing is the solution of simultaneous linear equations. In matrix notation, this problem can be stated as follows. Given two matrices A and B, does there exist a unique matrix X so that AX = B or XA = B? It is instructive to consider a 1-by-1 example. Does the equation 7x = 21 have a unique solution ? The answer, of course, is yes. The equation has the unique solution x = 3. The solution is easily obtained by division: x = 21 ⁄ 7 = 3 The solution is not ordinarily obtained by computing the inverse of 7, that is 7 -1 = 0.142857…, and then multiplying 7 -1 by 21. This would be more work and, if 7 -1 is represented to a finite number of digits, less accurate. Similar considerations apply to sets of linear equations with more than one unknown; MATLAB solves such equations without computing the inverse of the matrix. Although it is not standard mathematical notation, MATLAB uses the division terminology familiar in the scalar case to describe the solution of a general system of simultaneous equations. The two division symbols, slash, /, and 1-13
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2 Polynomials and Interpolation T =
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2 Polynomials and Interpolation Int
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2 Polynomials and Interpolation 2-3
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2 Polynomials and Interpolation 2-3
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3 Fast Fourier Transform (FFT) Intr
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3 Fast Fourier Transform (FFT) 200
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3 Fast Fourier Transform (FFT) 2 x
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3 Fast Fourier Transform (FFT) 500
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3 Fast Fourier Transform (FFT) Func
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4 Function Functions Function Summa
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4 Function Functions You can also c
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4 Function Functions 25 20 15 10 5
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4 Function Functions Minimizing Fun
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4 Function Functions To try fminsea
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4 Function Functions Plotting the R
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4 Function Functions The number of
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4 Function Functions and displays t
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4 Function Functions end function s
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4 Function Functions States of the
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4 Function Functions 100 80 60 40 2
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4 Function Functions You can verify
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4 Function Functions Troubleshootin
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4 Function Functions A three-dimens
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4 Function Functions Parameterizing
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4 Function Functions “Anonymous F
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5 Differential Equations Initial Va
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5 Differential Equations odephas3 o
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5 Differential Equations This secti
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5 Differential Equations The basic
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5 Differential Equations 2 Code the
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5 Differential Equations function.
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5 Differential Equations One way to
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5 Differential Equations y0, yp0 Ve
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5 Differential Equations For exampl
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5 Differential Equations 1 0.8 0.6
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5 Differential Equations 2.5 Soluti
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5 Differential Equations To run thi
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5 Differential Equations 2 u′ i =
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5 Differential Equations dydt(i+1,:
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5 Differential Equations refine = 4
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5 Differential Equations Example: A
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5 Differential Equations function [
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5 Differential Equations Note The R
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5 Differential Equations 1 Robertso
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5 Differential Equations The MATLAB
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5 Differential Equations Summary of
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5 Differential Equations Problem Si
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5 Differential Equations Error Tole
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5 Differential Equations Troublesho
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5 Differential Equations Function d
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5 Differential Equations dde23 prod
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5 Differential Equations The exampl
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5 Differential Equations solution y
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5 Differential Equations Example: C
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5 Differential Equations Changing D
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5 Differential Equations BVP Functi
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5 Differential Equations two-point
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5 Differential Equations The input
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5 Differential Equations 2 Pass the
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5 Differential Equations Finding Un
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5 Differential Equations vectorized
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5 Differential Equations There is a
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5 Differential Equations 3 Solve on
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5 Differential Equations hold off T
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5 Differential Equations Note The d
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5 Differential Equations legend('An
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5 Differential Equations Here, v(1-
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5 Differential Equations solution v
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5 Differential Equations After disc
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5 Differential Equations The output
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5 Differential Equations Note See t
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5 Differential Equations and the ri
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5 Differential Equations u1(x,t) 1
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5 Differential Equations Selected B
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6 Sparse Matrices Function Summary
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6 Sparse Matrices Function Summary
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6 Sparse Matrices This matrix requi
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6 Sparse Matrices S = (3,1) 1 (2,2)
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6 Sparse Matrices Now F = full(S) d
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6 Sparse Matrices Importing Sparse
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6 Sparse Matrices west0479 west0479
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6 Sparse Matrices The find Function
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6 Sparse Matrices of the rows and c
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6 Sparse Matrices The vertices of o
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6 Sparse Matrices 0 10 20 30 40 50
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6 Sparse Matrices 0 500 1000 1500 2
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6 Sparse Matrices simply sparse(m,n
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6 Sparse Matrices Similarly, S(:,p)
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6 Sparse Matrices The following MAT
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6 Sparse Matrices 0 Original 0 Reve
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6 Sparse Matrices QR Factorization
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6 Sparse Matrices shows that A has
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6 Sparse Matrices Functions for Ite
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6 Sparse Matrices set up the five-p
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6 Sparse Matrices Manipulating Spar
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6 Sparse Matrices Selected Bibliogr
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Index comparing sparse and full mat
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Index H hb1dae demo 5-35 hb1ode dem
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Index nonstiff ODE examples rigid b
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Index LU factorization 6-30 minimum
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Index twobvp demo 5-63 two-dimensio
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