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## CRAN Task View: Optimization and Mathematical Programming

Open Live Script. Estimate Nonlinear Grey-Box Model. Configure Estimation Using Process Model. Input Arguments collapse all data — Estimation data iddata idfrd. The command used to create the option set depends on the initial model type: Model Type Use idss ssestOptions idtf tfestOptions idproc procestOptions idpoly polyestOptions idgrey greyestOptions idnlarx nlarxOptions idnlhw nlhwOptions idnlgrey nlgreyestOptions.

Output Arguments collapse all sys — Identified model linear model nonlinear model. This is a nice trick to get an analytical function with continuous derivatives for a transition between two constants. What is the difference between quad and trapz?

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The short answer is that quad integrates functions via a function handle using numerical quadrature, and trapz performs integration of arrays of data using the trapezoid method. This will be our benchmark for comparison to the numerical methods. The trapezoid method is overestimating the area significantly. With more points, we get much closer to the analytical value. You might want to combine numerical data with the quad function if you want to perform integrals easily.

Let us say you are given this data:. We do not have data in those regions, so some interpolation is going to be needed. Here is one approach. These approaches are very similar, and both rely on linear interpolation. The second approach is simpler, and uses fewer lines of code.

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Both can be used with numerical data if interpolation is used. The syntax for the quad and trapz function is different in scipy than in Matlab. Finally, see this post for an example of solving an integral equation using quad and fsolve. Polynomials can be represented as a list of coefficients. Here are some ways to create a polynomial object, and evaluate it.

There are applications of polynomials in thermodynamics. The roots of this equation tell you the volume of the gas at those conditions. Note that only one root is real and even then, we have to interpret 0. Also, in a cubic polynomial, there can only be two imaginary roots. In this case that means there is only one phase present. Polynomials in numpy are even better than in Matlab, because you get a polynomial object that acts just like a function. Otherwise, they are functionally equivalent. This innocent looking function has 20 roots, which are 1,2,3,…,19, Here is a plot of the function.

Let us consider the expanded version of the polynomial. We will use sympy to expand the polynomial. The coefficients are orders of magnitude apart in size. This should make you nervous, because the roots of this equation are between , but there are numbers here that are O This is likely to make any rounding errors in the number representations very significant, and may lead to issues with accuracy of the solution. Let us explore that.

The roots are not exact. Even more to the point, the polynomial does not evaluate to zero at the calculated roots! Something is clearly wrong here. The polynomial function is fine, and it does evaluate to zero at the known roots which are integers. It is subtle, but up to that point, we are using only integers, which can be represented exactly. The roots function is evidently using some float math, and the floats are not the same as the integers.

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If we simply change the roots to floats, and reevaluate our polynomial, we get dramatically different results. This also happens if we make the polynomial coefficients floats. That happens because in Python whenever one element is a float the results of math operations with that element are floats. Let us try to understand what is happening here. It turns out that the integer and float representations of the numbers are different! It is known that you cannot exactly represent numbers as floats.

Now you can see the issue. Many of these numbers are identical in integer and float form, but some of them are not.

### 4. Log-Cosh Loss

The integer cannot be exactly represented as a float, and there is a difference in the representations. It is a small difference compared to the magnitude, but these kinds of differences get raised to high powers, and become larger. That is because pj in that loop is an object from sympy, which prints as a string.

This is a famous, and well known problem that is especially bad for this case. This illustrates that you cannot simply rely on what a computer tells you the answer is, without doing some critical thinking about the problem and the solution. Especially in problems where there are coefficients that vary by many orders of magnitude you should be cautious.

There are a few interesting webpages on this topic, which inspired me to work this out in python. These webpages go into more detail on this problem, and provide additional insight into the sensitivity of the solutions to the polynomial coefficients. The analytical answer is 2. We will use this example to illustrate the difference in performance between loops and vectorized operations in python. In the last example, there may be loop buried in the sum command. Let us do one final method, using linear algebra, in a single line.

The key to understanding this is to recognize the sum is just the result of a dot product of the x differences and y sums. The loop method is straightforward to code, and looks alot like the formula that defines the trapezoid method. However, the vectorized methods are much faster than the loop, so the loss of readability could be worth it for very large problems. The times here are considerably slower than in Matlab.

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I am not sure if that is a totally fair comparison. Here I am running python through emacs, which may result in slower performance. I also used a very crude way of timing the performance which lumps some system performance in too. Simpson's rule A more accurate numerical integration than the trapezoid method is Simpson's rule. The syntax is similar to trapz, but the method is in scipy. The syntax in dblquad is a bit more complicated than in Matlab. We have to provide callable functions for the range of the y-variable. Here they are constants, so we create lambda functions that return the constants.

Also, note that the order of arguments in the integrand is different than in Matlab.

https://propserfalksirrump.tk The syntax differs significantly for these simple examples, but the use of functions for the limits enables freedom to integrate over non-constant limits. A common need in engineering calculations is to integrate an equation over some range to determine the total change. An alternative to the scipy.