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This is the equivalent of selecting and arranging 3 items from 5. This method is not likely to be more accurate than quad, and it does not give you an error estimate. Matlab post Python has capability to do symbolic math through the sympy package. The symbolic math in sympy is pretty good. It is not up to the capability of Maple or Mathematica, but neither is Matlab but it continues to be developed, and could be helpful in some situations.

Float numbers i. This can lead to some artifacts when you have to compare float numbers that on paper should be the same, but in silico are not. In this example, we do some simple math that should result in an answer of 1, and then see if the answer is "equal" to one. The first line shows the result is not 1.

You can see here why the equality statement fails. We will print the two numbers to sixteen decimal places. The two numbers actually are not equal to each other because of float math. They are very, very close to each other, but not the same. This leads to the idea of asking if two numbers are equal to each other within some tolerance. The question of what tolerance to use requires thought. Should it be an absolute tolerance? How large should the tolerance be? We will use the distance between 1 and the nearest floating point number this is eps in Matlab.

Below, we implement a comparison function from doi For completeness, here are the other float comparison operators from that paper. We also show a few examples.

As you can see, float comparisons can be tricky. You have to give a lot of thought to how to make the comparisons, and the functions shown above are not the only way to do it.

You need to build in testing to make sure your comparisons are doing what you want. Numpy has some gotcha features for linear algebra purists. The first is that a 1d array is neither a row, nor a column vector.

This would not be allowed in Matlab. Compare the previous behavior with this 2d array. You must transpose the second argument to make it dimensionally consistent. Try to figure this one out! Just by adding them you get a 2d array.

In the next example, we have a 3 element vector and a 4 element vector. These concepts are known in numpy as array broadcasting. These are points to keep in mind, as the operations do not strictly follow the conventions of linear algebra, and may be confusing at times. When solving linear equations, we can represent them in matrix form. The we simply use numpy. It can be useful to confirm there should be a solution, e. The matrix rank will tell us that.

Note that numpy:rank does not give you the matrix rank, but rather the number of dimensions of the array. We compute the rank by computing the number of singular values of the matrix that are greater than zero, within a prescribed tolerance. We use the numpy. In Matlab you would use the rref command to see if there are any rows that are all zero, but this command does not exist in numpy. That command does not have practical use in numerical linear algebra and has not been implemented.

Matlab comparison. Today we examine some methods of linear algebra that allow us to avoid writing explicit loops in Matlab for some kinds of mathematical operations. We can compute this with a loop, where you initialize y, and then add the product of the ith elements of a and b to y in each iteration of the loop.

This is known to be slow for large vectors. The operation defined above is actually a dot product. We an directly compute the dot product in numpy.

Note that with 1d arrays, python knows what to do and does not require any transpose operations. This operation is like a weighted sum of squares. The old-fashioned way to do this is with a loop. We can also express this in matrix algebra form. Consider the sum of the product of three vectors. This is like a weighted sum of products. We showed examples of the following equalities between traditional sum notations and linear algebra. These relationships enable one to write the sums as a single line of python code, which utilizes fast linear algebra subroutines, avoids the construction of slow loops, and reduces the opportunity for errors in the code.

Admittedly, it introduces the opportunity for new types of errors, like using the wrong relationship, or linear algebra errors due to matrix size mismatches. Matlab post Occasionally we have a set of vectors and we need to determine whether the vectors are linearly independent of each other.

This may be necessary to determine if the vectors form a basis, or to determine how many independent equations there are, or to determine how many independent reactions there are. Matlab provides a rank command which gives you the number of singular values greater than some tolerance.

It returns the number of dimensions in the array. We will just compute the rank from singular value decomposition. Let us break that down. 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.

Let us show that one row can be expressed as a linear combination of the other rows. The number of rows is greater than the rank, so these vectors are not independent. Let's demonstrate that one vector can be defined as a linear combination of the other two vectors.

Mathematically we represent this as:. To get there, we transpose each side of the equation to get:. 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. The default tolerance is usually very small, of order 1e 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.

A stoichiometric coefficient of 0 is used for species not participating in the reaction. You can see that reaction 6 is just the opposite of reaction 2, so it is clearly not independent. Also, reactions 3 and 5 are just the reverse of each other, so one of them can also be eliminated. There are many possible independent reactions.

In the code above, we use sympy to put the matrix into reduced row echelon form, which enables us to identify three independent reactions, and shows that three rows are all zero, i.

The choice of independent reactions is not unique. There is a nice discussion here on why there is not a rref command in numpy, primarily because one rarely actually needs it in linear algebra. Still, it is so often taught, and it helps visually see what the rank of a matrix is that I wanted to examine ways to get it. This rref form is a bit different than you might get from doing it by hand. The rows are also normalized. Let us check it out. There are "solutions", but there are a couple of red flags that should catch your eye.

First, the determinant is within machine precision of zero. Second the elements of the inverse are all "large". Third, the solutions are all "large".

All of these are indications of or artifacts of numerical imprecision. LU decomposition,determinant There are a few properties of a matrix that can make it easy to compute determinants.

So we simply subtract the sum of the diagonal from the length of the diagonal and then subtract 1 to get the number of swaps. According to the numpy documentation, a method similar to this is used to compute the determinant. 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.

But, one day it might be helpful to know this can be done, e. 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. In Matlab, the default tolerance was not sufficient to get a good solution.

Here it is. Original post in Matlab. What is the exit molar flow rate? We need to solve the following equation:. We start by creating a function handle that describes the integrand. We can use this function in the quad command to evaluate the integral. This example seemed a little easier in Matlab, where the quad function seemed to get automatically vectorized. Here we had to do it by hand. In principle this is easy, we simply need some initial guesses and a nonlinear solver.

The challenge here is what would you guess? There could be many solutions. The equations are implicit, so it is not easy to graph them, but let us give it a shot, starting on the x range -5 to 5. The idea is set a value for x, and then solve for y in each equation. We can even use that guess with fsolve.

It is disappointly easy! But, keep in mind that in 3 or more dimensions, you cannot perform this visualization, and another method could be required.

We explore a method that bypasses this problem today. Why do we do that? From calculus, you can show that:. You can use Cramer's rule to solve for these to yield:. The approximation could be improved by lowering the tolerance on the ODE solver. The functions evaluate to a small number, close to zero. You have to apply some judgment to determine if that is sufficiently accurate.

For instance if the units on that answer are kilometers, but you need an answer accurate to a millimeter, this may not be accurate enough. This is a fair amount of work to get a solution! The idea is to solve a simple problem, and then gradually turn on the hard part by the lambda parameter. What happens if there are multiple solutions? For problems with lots of variables, this would be a good approach if you can identify the easy problem.

Matlab post Yesterday in Post we looked at a way to solve nonlinear equations that takes away some of the burden of initial guess generation. Today we look at a simpler example and explain a little more about what is going on. We will use the method of continuity to solve this equation to illustrate a few ideas. The total derivative is:. What about the other solution? Now we have the other solution. You could choose other values to add, e. 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.

Matlab post The goal here is to determine how many roots there are in a nonlinear function we are interested in solving. For this example, we use a cubic polynomial because we know there are three roots. Now we consider several approaches to counting the number of roots in this interval. Visually it is pretty easy, you just look for where the function crosses zero. Computationally, it is tricker. Count the number of times the sign changes in the interval. What we have to do is multiply neighboring elements together, and look for negative values.

That indicates a sign change. For example the product of two positive or negative numbers is a positive number. You only get a negative number from the product of a positive and negative number, which means the sign changed. Using events in an ODE solver python can identify events in the solution to an ODE, for example, when a function has a certain value, e.

We can take advantage of this to find the roots and number of roots in this case. We take the derivative of our function, and integrate it from an initial starting point, and define an event function that counts zeros.

We examine an approach to finding these roots. This function is pretty well behaved, so if you make a good guess about the solution you will get an answer, but if you make a bad guess, you may get the wrong root.

We examine next a way to do it without guessing the solution. All we have to do now is set up the problem and run it. You can work this out once, and then you have all the roots in the interval and you can select the one you want. To solve this we need to setup a function that is equal to zero at the solution. We have two equations, so our function must return two values.

There are two variables, so the argument to our function will be an array of values. Interesting, we have to specify the divisor in numpy. The default for this in Matlab is 1, the default for this function is 0. You subtract 1 because one degree of freedom is lost from calculating the average.

This is useful for computing confidence intervals using the student-t tables. Class A had 30 students who received an average test score of 78, with standard deviation of Class B had 25 students an average test score of 85, with a standard deviation of 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. Here we simply subtract one from each sample size to account for the estimation of the average of each sample.

The difference between two averages determined from small sample numbers follows the t-distribution. A way to approach determining if the difference is significant or not is to ask, does our computed average fall within a confidence range of the hypothesized value zero? If it does, then we can attribute the difference to statistical variations at that confidence level.

If it does not, we can say that statistical variations do not account for the difference at that confidence level, and hence the averages must be different. Let us consider a smaller confidence interval. An alternative way to get the confidence that the averages are different is to directly compute it from the cumulative t-distribution function. We compute the difference between all the t-values less than tscore and the t-values less than -tscore, which is the fraction of measurements that are between them.

In this example, we show some ways to choose which of several models fit data the best. We have data for the total pressure and temperature of a fixed amount of a gas in a tank that was measured over the course of several days. We want to select a model that relates the pressure to the gas temperature. We need to read the data in, and perform a regression analysis on P vs.

In python we start counting at 0, so we actually want columns 3 and 4. We will use linear algebra to compute the line coefficients. Hence, a value close to one means nearly all the variations are described by the model, except for random variations. There are a few ways to examine this. We want to make sure that there are no systematic trends in the errors between the fit and the data, and we want to make sure there are not hidden correlations with other variables.

The residuals are the error between the fit and the data. The residuals should not show any patterns when plotted against any variables, and they do not in this case.

There may be some correlations in the residuals with the run order. That could indicate an experimental source of error. We assume all the errors are uncorrelated with each other. We can use a lag plot to assess this, where we plot residual[i] vs residual[i-1], i.

This plot should look random, with no correlations if the model is good. That is a good indication this additional parameter is not significant. This is an example of overfitting the data.

Since the constant in this model is apparently not significant, let us consider the simplest model with a fixed intercept of zero.

Let us examine the residuals again. You can see a slight trend of decreasing value of the residuals as the Temperature increases. This may indicate a deficiency in the model with no intercept.

Since the molar density of a gas is pretty small, the intercept may be close to, but not equal to zero. Computer Mathematical Topics. Course will also include the application of software to the solution of a variety of geometric and algebraic problems. Generally offered: Spring, Summer. History of Mathematics. Selected subjects in mathematics developed through historical perspectives and biographies. Real Analysis I.

Continuous functions, uniform continuity; theory of differentiation; applications of the derivative to properties of functions; antiderivatives; Riemann integral; connection between differentiation and integration. Real Analysis II. Lebesgue integral on the real line; n-dimensional spaces; vectors; calculus of functions of several variables; multidimensional integration.

Modern Abstract Algebra. Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, The Sylow Theorems, rings and fields and their basic properties, ideals, polynomial rings.

A study of non-Euclidean geometries, including spherical geometry, hyperbolic geometry and others. Set theory, including cardinal and ordinal numbers. Topological properties of the real-line and metric spaces.

Generally offered: Fall. Capstone Course for Mathematics. This course is for any interested mathematics major, particularly for those students who intend to pursue secondary certification in Mathematics. The goals of the course are to enable students to build connections among the mathematical areas they have studied and between undergraduate mathematics and high school mathematics, to develop their understanding of mathematics as an integrated discipline, and to strengthen their oral and written communication skills in mathematics.

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Topics include control charts and acceptance sampling plans. Same as STA Independent Study. Special Studies in Mathematics. Prerequisite: Consent of instructor. Resend confirmation email. Not a free member yet? Here's what you're missing out on! Sign Up. A text message with your code has been sent to:. Didn't receive the code? Don't have your phone? Please contact support.

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