MATLAB Mathematics - SERC - Index of

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5 Differential Equations Error Tolerance and Other Options (Continued) Question How do I tell the solver that I don’t care about getting an accurate answer for one of the solution components? Answer You can increase the absolute error tolerance corresponding to this solution component. If the tolerance is bigger than the component, this specifies no correct digits for the component. The solver may have to get some correct digits in this component to compute other components accurately, but it generally handles this automatically. Solving Different Kinds of Problems Question Can the solvers handle partial differential equations (PDEs) that have been discretized by the method of lines? Answer Yes, because the discretization produces a system of ODEs. Depending on the discretization, you might have a form involving mass matrices – the ODE solvers provide for this. Often the system is stiff. This is to be expected when the PDE is parabolic and when there are phenomena that happen on very different time scales such as a chemical reaction in a fluid flow. In such cases, use one of the four solvers: ode15s, ode23s, ode23t, ode23tb. If there are many equations, set the JPattern property. This might make the difference between success and failure due to the computation being too expensive. For an example that uses JPattern, see “Example: Large, Stiff, Sparse Problem” on page 5-25. When the system is not stiff, or not very stiff, ode23 or ode45 is more efficient than ode15s, ode23s, ode23t, or ode23tb. Parabolic-elliptic partial differential equations in 1-D can be solved directly with the MATLAB PDE solver, pdepe. For more information, see “Partial Differential Equations” on page 5-89. Can I solve differential-algebraic equation (DAE) systems? Yes. The solvers ode15s and ode23t can solve some DAEs of the form Mty ( , )y′ = fty ( , ) where Mty ( , ) is singular. The DAEs must be of index 1. ode15i can solve fully implicit DAEs of index 1, ft ( , y, y′ ) = 0 . For examples, see amp1dae, hb1dae, or ihb1dae. 5-46
Initial Value Problems for ODEs and DAEs Solving Different Kinds of Problems (Continued) Question Can I integrate a set of sampled data? What do I do when I have the final and not the initial value? Answer Not directly. You have to represent the data as a function by interpolation or some other scheme for fitting data. The smoothness of this function is critical. A piecewise polynomial fit like a spline can look smooth to the eye, but rough to a solver; the solver takes small steps where the derivatives of the fit have jumps. Either use a smooth function to represent the data or use one of the lower order solvers (ode23, ode23s, ode23t, ode23tb) that is less sensitive to this. All the solvers of the ODE suite allow you to solve backwards or forwards in time. The syntax for the solvers is [t,y] = ode45(odefun,[t0 tf],y0); and the syntax accepts t0 > tf. Troubleshooting Question The solution doesn’t look like what I expected. Answer If you’re right about its appearance, you need to reduce the error tolerances from their default values. A smaller relative error tolerance is needed to compute accurately the solution of problems integrated over “long” intervals, as well as solutions of problems that are moderately unstable. You should check whether there are solution components that stay smaller than their absolute error tolerance for some time. If so, you are not asking for any correct digits in these components. This may be acceptable for these components, but failing to compute them accurately may degrade the accuracy of other components that depend on them. My plots aren’t smooth enough. Increase the value of Refine from its default of 4 in ode45 and 1 in the other solvers. The bigger the value of Refine, the more output points. Execution speed is not affected much by the value of Refine. 5-47
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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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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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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 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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