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Note that functions are not automatically vectorized. That is why we see the error above. There are a few ways to achieve that. One is to \"cast\" the input variables to objects that support vectorized operations, such as numpy.array objects.

Python has some nice features in creating functions. You can create default values for variables, have optional variables and optional keyword variables.In this function f(a,b), a and b are called positional arguments, and they are required, and must be provided in the same order as the function defines.

If we provide a default value for an argument, then the argument is called a keyword argument, and it becomes optional. You can combine positional arguments and keyword arguments, but positional arguments must come first. Here is an example.

In the first call to the function, we only define the argument a, which is a mandatory, positional argument. In the second call, we define a and n, in the order they are defined in the function. Finally, in the third call, we define a as a positional argument, and n as a keyword argument.

If all of the arguments are optional, we can even call the function with no arguments. If you give arguments as positional arguments, they are used in the order defined in the function. If you use keyword arguments, the order is arbitrary.

It is possible to have arbitrary keyword arguments. This is a common pattern when you call another function within your function that takes keyword arguments. We use **kwargs to indicate that arbitrary keyword arguments can be given to the function. Inside the function, kwargs is variable containing a dictionary of the keywords and values passed in.

In that example we wrap the matplotlib plotting commands in a function, which we can call the way we want to, with arbitrary optional arguments. In this example, you cannot pass keyword arguments that are illegal to the plot command or you will get an error.

Is that some kind of fraternity of anonymous functions What is that! There are many times where you need a callable, small function in python, and it is inconvenient to have to use def to create a named function. Lambda functions solve this problem. Let us look at some examples. First, we create a lambda function, and assign it to a variable. Then we show that variable is a function, and that we can call it with an argument.

This figure illustrates graphically what the numbers above show. The function crosses zero at approximately \\(x = 1.5\\). To get a more precise value, we must actually solve the function numerically. We use the function func:scipy.optimize.fsolve to do that. More precisely, we want to solve the equation \\(f(x) = \\cos(x) = 0\\). We create a function that defines that equation, and then use func:scipy.optimize.fsolve to solve it.

derivative!numericalderivative!forward differencederivative!backward differencederivative!centered differencenumpy has a function called numpy.diff() that is similar to the one found in matlab. It calculates the differences between the elements in your list, and returns a list that is one element shorter, which makes it unsuitable for plotting the derivative of a function.

derivative!polynomialOne way to reduce the noise inherent in derivatives of noisy data is to fit a smooth function through the data, and analytically take the derivative of the curve. Polynomials are especially convenient for this. The challenge is to figure out what an appropriate polynomial order is. This requires judgment and experience.

You can see a third order polynomial is a reasonable fit here. There are only 6 data points here, so any higher order risks overfitting. Here is the comparison of the numerical derivative and the fitted derivative. We have \"resampled\" the fitted derivative to show the actual shape. Note the derivative appears to go through a maximum near t = 0.9. In this case, that is probably unphysical as the data is related to the consumption of species A in a reaction. The derivative should increase monotonically to zero. The increase is an artefact of the fitting process. End points are especially sensitive to this kind of error.

Visually this fit is about the same as a third order polynomial. Note the difference in the derivative though. We can readily extrapolate this derivative and get reasonable predictions of the derivative. That is true in this case because we fitted a physically relevant model for concentration vs. time for an irreversible, first order reaction.

This posts introduces a novel way to numerically estimate the derivativeof a function that does not involve finite difference schemes. Finitedifference schemes are approximations to derivatives that become more andmore accurate as the step size goes to zero, except that as the step sizeapproaches the limits of machine accuracy, new errors can appear in theapproximated results. In the references above, a new way to compute thederivative is presented that does not rely on differences!

Let us use this method to verify the fundamental Theorem of Calculus, i.e.to evaluate the derivative of an integral function. Let \\(f(x) =\\int\\limits_1^{x^2} tan(t^3)dt\\), and we now want to compute df/dx.Of course, this can be doneanalytically, but it is not trivial!

You can see that away from the transition the combined function is practically equivalent to the original two functions. That is because away from the transition the sigmoid function is 0 or 1. Near Re = 3000 is a smooth transition from one curve to the other curve.

The approach demonstrated here allows one to smoothly join two discontinuous functions that describe physics in different regimes, and that must transition over some range of data. It should be emphasized that the method has no physical basis, it simply allows one to create a mathematically smooth function, which could be necessary for some optimizers or solvers to work.

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.

The default tolerance used in Matlab is max(size(A))*eps(norm(A)). Let us break that down. eps(norm(A)) is the positive distance from abs(X) to the next larger in magnitude floating point number of the same precision as X. Basically, the smallest significant number. We multiply that by the size of A, and take the largest number. We have to use some judgment in what the tolerance is, and what \"zero\" means.

The number of rows is greater than the rank, so these vectors are notindependent. Let's demonstrate that one vector can be defined as a linearcombination of the other two vectors. Mathematically we represent thisas:

the rank command roughly works in the following way: the matrix is converted to a reduced row echelon form, and then the number of rows that are not all equal to zero are counted. Matlab uses a tolerance to determine what is equal to zero. If there is uncertainty in the numbers, you may have to define what zero is, e.g. if the absolute value of a number is less than 1e-5, you may consider that close enough to be zero. The default tolerance is usually very small, of order 1e-15. If we believe that any number less than 1e-5 is practically equivalent to zero, we can use that information to compute the rank like this.

If the built in linear algebra functions in numpy and scipy do not meet your needs, it is often possible to directly call lapack functions. Here we call a function to solve a set of complex linear equations. The lapack function for this is ZGBSV. The description of this function ( ) is:

Nonlinear algebra problems are typically solved using an iterative process that terminates when the solution is found within a specified tolerance. This process is hidden from the user. The canonical standard form to solve is \\(f(X) = 0\\).

We explore a method that bypasses this problem today. The principle is to introduce a new variable, \\(\\lambda\\), which will vary from 0 to 1. at \\(\\lambda=0\\) we will have a simpler equation, preferably a linear one, which can be easily solved, or which can be analytically solved. At \\(\\lambda=1\\), we have the original equations. Then, we create a system of differential equations that start at the easy solution, and integrate from \\(\\lambda=0\\) to \\(\\lambda=1\\), to recover the final solution.

Now we have the other solution. Note if you choose the other root, \\(x=2\\), you find that 2 is a root, and learn nothing new. You could choose other values to add, e.g., if you chose to add and subtract 16, then you would find that one starting point leads to one root, and the other starting point leads to the other root. This method does not solve all problems associated with nonlinear root solving, namely, how many roots are there, and which one is \"best\" or physically reasonable But it does give a way to solve an equation where you have no idea what an initial guess should be. You can see, however, that just like you can get different answers from different initial guesses, here you can get different answers by setting up the equations differently.

Class A had 30 students who received an average test score of 78, with standard deviation of 10. Class B had 25 students an average test score of 85, with a standard deviation of 15. We want to know if the difference in these averages is statistically relevant. Note that we only have estimates of the true average and standard deviation for each class, and there is uncertainty in those estimates. As a result, we are unsure if the averages are really different. It could have just been luck that a few students in class B did better.

the true averages are the same. We need to perform a two-sample t-test of the hypothesis that \\(\\mu_1 - \\mu_2 = 0\\) (this is often called the null hypothesis). we use a two-tailed test because we do not care if the difference is positive or negative, either way means the averages are not the same. 59ce067264

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