MATLAB Mathematics - SERC - Index of

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4 Function Functions You can verify that the function value at a is close to zero by entering humps(a) ans = 8.8818e-016 Using a Starting Point Suppose you do not know two points at which the function values of humps differ in sign. In that case, you can choose a scalar x0 as the starting point for fzero. fzero first searches for an interval around this point on which the function changes sign. If fzero finds such an interval, it proceeds with the algorithm described in the previous section. If no such interval is found, fzero returns NaN. For example, if you set the starting point to -0.2, the Display option to Iter, and call fzero by a = fzero(@humps,-0.2,options) fzero returns the following output: Search for an interval around -0.2 containing a sign change: Func-count a f(a) b f(b) Procedure 1 -0.2 -1.35385 -0.2 -1.35385 initial interval 3 -0.194343 -1.26077 -0.205657 -1.44411 search 5 -0.192 -1.22137 -0.208 -1.4807 search 7 -0.188686 -1.16477 -0.211314 -1.53167 search 9 -0.184 -1.08293 -0.216 -1.60224 search 11 -0.177373 -0.963455 -0.222627 -1.69911 search 13 -0.168 -0.786636 -0.232 -1.83055 search 15 -0.154745 -0.51962 -0.245255 -2.00602 search 17 -0.136 -0.104165 -0.264 -2.23521 search 18 -0.10949 0.572246 -0.264 -2.23521 search Search for a zero in the interval [-0.10949, -0.264]: Func-count x f(x) Procedure 18 -0.10949 0.572246 initial 19 -0.140984 -0.219277 interpolation 20 -0.132259 -0.0154224 interpolation 21 -0.131617 3.40729e-005 interpolation 22 -0.131618 -6.79505e-008 interpolation 23 -0.131618 -2.98428e-013 interpolation 24 -0.131618 8.88178e-016 interpolation 25 -0.131618 8.88178e-016 interpolation Zero found in the interval [-0.10949, -0.264] 4-24
Minimizing Functions and Finding Zeros a = -0.1316 The endpoints of the current subinterval at each iteration are listed under the headings a and b, while the corresponding values of humps at the endpoints are listed under f(a) and f(b), respectively. Note The endpoints a and b are not listed in any specific order: a can be greater than b or less than b. For the first nine steps, the sign of humps is negative at both endpoints of the current subinterval, which are listed under in the output. At the tenth step, the sign of humps is positive at the endpoint, -0.10949, but negative at the endpoint, -0.264. From this point on, the algorithm continues to narrow down the interval [-0.10949 -0.264], as described in the previous section, until it reaches the value -0.1316. Tips Optimization problems may take many iterations to converge. Most optimization problems benefit from good starting guesses. Providing good starting guesses improves the execution efficiency and may help locate the global minimum instead of a local minimum. Sophisticated problems are best solved by an evolutionary approach, whereby a problem with a smaller number of independent variables is solved first. Solutions from lower order problems can generally be used as starting points for higher order problems by using an appropriate mapping. The use of simpler cost functions and less stringent termination criteria in the early stages of an optimization problem can also reduce computation time. Such an approach often produces superior results by avoiding local minima. 4-25
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6 Sparse Matrices Function Summary
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Index LU factorization 6-30 minimum
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