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4 Function Functions 100 80 60 40 20 0 −20 −1 −0.5 0 0.5 1 1.5 2 Using a Starting Interval The graph of humps indicates that the function is negative at x = -1 and positive at x = 1. You can confirm this by calculating humps at these two points. humps(1) ans = 16 humps(-1) ans = -5.1378 Consequently, you can use [-1 1] as a starting interval for fzero. 4-22
Minimizing Functions and Finding Zeros The iterative algorithm for fzero finds smaller and smaller subintervals of [-1 1]. For each subinterval, the sign of humps differs at the two endpoints. As the endpoints of the subintervals get closer and closer, they converge to a zero for humps. To show the progress of fzero at each iteration, set the Display option to iter using the function optimset. options = optimset('Display','iter'); Then call fzero as follows: a = fzero(@humps,[-1 1],options) This returns the following iterative output: a = fzero(@humps,[-1 1],options) Func-count x f(x) Procedure 2 -1 -5.13779 initial 3 -0.513876 -4.02235 interpolation 4 -0.513876 -4.02235 bisection 5 -0.473635 -3.83767 interpolation 6 -0.115287 0.414441 bisection 7 -0.115287 0.414441 interpolation 8 -0.132562 -0.0226907 interpolation 9 -0.131666 -0.0011492 interpolation 10 -0.131618 1.88371e-007 interpolation 11 -0.131618 -2.7935e-011 interpolation 12 -0.131618 8.88178e-016 interpolation 13 -0.131618 8.88178e-016 interpolation Zero found in the interval [-1, 1] a = -0.1316 Each value x represents the best endpoint so far. The Procedure column tells you whether each step of the algorithm uses bisection or interpolation. 4-23
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MATLAB® The Language of Technical
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2 Polynomials and Interpolation Pol
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Introduction to Initial Value ODE P
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1 Matrices and Linear Algebra Funct
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2 Polynomials and Interpolation Cal
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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 Function d
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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 Importing Sparse
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6 Sparse Matrices The find Function
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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 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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