I have a lengthy symbolic expression that involves rational polynomials (basic arithmetic and integer powers). I'd like to simplify it into a single (simple) rational polynomial.
numden does it, but it seems to use some expensive optimization, which probably addresses a more general case. When tried on my example below, it crashed after a few hours--out of memory (32GB).
I believe something more efficient is possible even if I don't have a cpp access to matlab functionality (e.g. children).
Motivation: I have an objective function that involves polynomials. I manually derived it, and I'd like to verify and compare the derivatives: I subtract the two expressions, and the result should vanish.
Currently, my interest in this is academic since practically, I simply substitute some random expression, get zero, and it's enough for me.
I'll try to find the time to play with this as some point, and I'll update here about it, but I posted in case someone finds it interesting and would like to give it a try before that.
To run my function:
x = sym('x', [1 32], 'real')
e = func(x)
The function (and believe it or not, this is just the Jacobian, and I also have the Hessian) can't be pasted here since the text limit is 30K:
https://drive.google.com/open?id=1imOAa4VS87WDkOwAK0NoFCJPTK_2QIRj
Just few questions, i hope someone will find time to answer :).
What if we have COUPLED model example: system of n indepedent variables X and n nonlinear partial differential equations PDEf(X,PDEf(X)) with respect to TIME that depends of X,PDEf(X)(partial differential equation depending of variables X ). Can you give some advice? Here is one example:
Let’s say that c is output, or desired variable. Let’s say that r is independent variable.Partial differential equation looks like:
∂c/∂t=D*1/r+∂c/∂r+2(D* (∂^2 c)/(∂r^2 ))
D=constant
r=0:0.1:Rp- Matlab syntaxis, how to represent same in Modelica (I use integrator,but didn't work)?
Here is a code (does not work):
model PDEtest
/* Boundary conditions
1. delta(c)/delta(r)=0 for r=0
2. delta(c)/delta(r)=-j*d for r=Rp*/
parameter Real Rp=88*1e-3; // length
parameter Real initialConc=1000;
parameter Real Dp=1e-14;
parameter Integer np=10; // num. of points
Real cp[np](start=fill(initialConc,np));
Modelica.Blocks.Continuous.Integrator r(k=1); // independent x1
Real j;
protected
parameter Real dr=Rp/np;
parameter Real ts= 0.01; // for using when loop (sample(0,ts) )
algorithm
j:=sin(time); // this should be indepedent variable like x2
r.u:=dr;
while r.y<=Rp loop
for i in 2:np-1 loop
der(cp[i]):=2*Dp/r.y+(cp[i]-cp[i-1])/dr+2*(Dp*(cp[i+1]-2*cp[i]+cp[i-1])/dr^2);
end for;
if r.y==Rp then
cp[np]:=-j*Dp;
end if;
cp[1]:=if time >=0 then initialConc else initialConc;
end while;
annotation (uses(Modelica(version="3.2")));
end PDEtest;
Here are more questions:
This code don’t work in OpenModelica 1.8.1, also don’t work in Dymola 2013demo. How can we have continuos function of variable c, not array of functions ?
Can we place values of array cp in combiTable? And how?
If instead “algorithm” stay “equation” code can’t be succesfull checked.Why? In OpenModelica, error is :could not flattening model :S.
Is there any simplified way to use a set of equation (PDE’s) that are coupled? I know for PDEs library in Modelica, but I think they are complicated. I want to write a function for solving PDE and call these function in “main model”, so that output of function be continuos function of “c”.I don’t know what for doing with array of functions.
Can you give me advice how to understand Modelica language, if we “speak” like in Matlab? For example: Values of independent variable r,we can specife in Matlab, like r=0:TimeStep:Rp…How to do same in Modelica? And please explain me how section “equation” works, is there similarity with Matlab, and is there necessary sequancial approach?
Cheers :)
It's hard to answer your question, since you assuming that Modelica ~ Matlab, but that's not the case. So I won't comment your code, since it's really wrong. Let me give you an example model to the burger equation. Maybe you could use it as starting point.
model burgereqn
Real u[N+2](start=u0);
parameter Real h = 1/(N+1);
parameter Integer N = 10;
parameter Real v = 234;
parameter Real Pi = 3.14159265358979;
parameter Real u0[N+2]={((sin(2*Pi*x[i]))+0.5*sin(Pi*x[i])) for i in 1:N+2};
parameter Real x[N+2] = { h*i for i in 1:N+2};
equation
der(u[1]) = 0;
for i in 2:N+1 loop
der(u[i]) = - ((u[i+1]^2-u[i-1]^2)/(4*(x[i+1]-x[i-1])))
+ (v/(x[i+1]-x[i-1])^2)*(u[i+1]-2*u[i]+u[i+1]);
end for;
der(u[N+2]) = 0;
end burgereqn;
Your further questions:
cp is an continuous variable and the array is representing
every discretization point.
Why you should want to do that, as far as I understand cp is
your desired solution variable.
You should try to use almost always equation section
algorithm sections are usually used in functions. I'm pretty
sure you can represent your desire behaviour with equations.
I don't know that library, but the hard thing on a pde is the
discretization and the solving it self. You may run into issues
while solving the pde with a modelica tool, since usually
a Modelica tool has no specialized solving algorithm for pdes.
Please consider for that question further references. You could
start with Modelica.org.
Rheological models are usually build using three (or four) basics elements, which are :
The spring (existing in Modelica.Mechanics.Translational.Components for example). Its equation is f = c * (s_rel - s_rel0);
The damper (dashpot) (also existing in Modelica.Mechanics.Translational.Components). Its equation is f = d * v_rel; for a linear damper, an could be easily modified to model a non-linear damper : f = d * v_rel^(1/n);
The slider, not existing (as far as I know) in this library... It's equation is abs(f)<= flim. Unfortunately, I don't really understand how I could write the corresponding Modelica model...
I think this model should extend Modelica.Mechanics.Translational.Interfaces.PartialCompliant, but the problem is that f (the force measured between flange_b and flange_a) should be modified only when it's greater than flim...
If the slider extends PartialCompliant, it means that it already follows the equations flange_b.f = f; and flange_a.f = -f;
Adding the equation f = if abs(f)>flim then sign(f)*flim else f; gives me an error "An independent subset of the model has imbalanced number of equations and variables", which I couldn't really explain, even if I understand that if abs(f)<=flim, the equation f = f is useless...
Actually, the slider element doesn't generate a new force (just like the spring does, depending on its strain, or just like the damper does, depending on its strain rate). The force is an input for the slider element, which is sometime modified (when this force becomes greater than the limit allowed by the element). That's why I don't really understand if I should define this force as an input or an output....
If you have any suggestion, I would greatly appreciate it ! Thanks
After the first two comments, I decided to add a picture that, I hope, will help you to understand the behaviour I'm trying to model.
On the left, you can see the four elements used to develop rheological models :
a : the spring
b : the linear damper (dashpot)
c : the non-linear damper
d : the slider
On the right, you can see the behaviour I'm trying to reproduce : a and b are two associations with springs and c and d are respectively the expected stress / strain curves. I'm trying to model the same behaviour, except that I'm thinking in terms of force and not stress. As i said in the comment to Marco's answer, the curve a reminds me the behaviour of a diode :
if the force applied to the component is less than the sliding limit, there is no relative displacement between the two flanges
if the force becomes greater than the sliding limit, the force transmitted by the system equals the limit and there is relative displacement between flanges
I can't be sure, but I suspect what you are really trying to model here is Coulomb friction (i.e. a constant force that always opposes the direction of motion). If so, there is already a component in the Modelica Standard Library, called MassWithStopAndFriction, that models that (and several other flavors of friction). The wrinkle is that it is bundled with inertia.
If you don't want the inertia effect it might be possible to set the inertia to zero. I suspect that could cause a singularity. One way you might be able to avoid the singularity is to "evaluate" the parameter (at least that is what it is called in Dymola when you set the Evaluate flat to be true in the command line). No promises whether that will work since it is model and tool dependent whether such a simplification can be properly handled.
If Coulomb friction is what you want and you really don't want inertia and the approach above doesn't work, let me know and I think I can create a simple model that will work (so long as you don't have inertia).
A few considerations:
- The force is not an input and neither an output, but it is just a relation that you add into the component in order to define how the force will be propagated between the two translational flanges of the component. When you deal with acausal connectors I think it is better to think about the degrees of freedom of your component instead of inputs and outputs. In this case you have two connectors and independently at which one of the two frames you will recieve informations about the force, the equation you implement will define how that information will be propagated to the other frame.
- I tested this:
model slider
extends
Modelica.Mechanics.Translational.Interfaces.PartialCompliantWithRelativeStates;
parameter Real flim = 1;
equation
f = if abs(f)>flim then sign(f)*flim else f;
end slider;
on Dymola and it works. It is correct modelica code so it should be work also in OpenModelica, I can't think of a reason why it should be seen as an unbalance mathematical model.
I hope this helps,
Marco
There is one thing I do not like on Matlab: It tries sometimes to be too smart. For instance, if I have a negative square root like
a = -1; sqrt(a)
Matlab does not throw an error but switches silently to complex numbers. The same happens for negative logarithms. This can lead to hard to find errors in a more complicated algorithm.
A similar problem is that Matlab "solves" silently non quadratic linear systems like in the following example:
A=eye(3,2); b=ones(3,1); x = A \ b
Obviously x does not satisfy A*x==b (It solves a least square problem instead).
Is there any possibility to turn that "features" off, or at least let Matlab print a warning message in this cases? That would really helps a lot in many situations.
I don't think there is anything like "being smart" in your examples. The square root of a negative number is complex. Similarly, the left-division operator is defined in Matlab as calculating the pseudoinverse for non-square inputs.
If you have an application that should not return complex numbers (beware of floating point errors!), then you can use isreal to test for that. If you do not want the left division operator to calculate the pseudoinverse, test for whether A is square.
Alternatively, if for some reason you are really unable to do input validation, you can overload both sqrt and \ to only work on positive numbers, and to not calculate the pseudoinverse.
You need to understand all of the implications of what you're writing and make sure that you use the right functions if you're going to guarantee good code. For example:
For the first case, use realsqrt instead
For the second case, use inv(A) * b instead
Or alternatively, include the appropriate checks before/after you call the built-in functions. If you need to do this every time, then you can always write your own functions.
I programmed in MATLAB for many years, but switched to using R exclusively in the past few years so I'm a little out of practice. I'm interviewing a candidate today who describes himself as a MATLAB expert.
What MATLAB interview questions should I ask?
Some other sites with resources for this:
"Matlab interview questions" on Wilmott
"MATLAB Questions and Answers" on GlobaleGuildLine
"Matlab Interview Questions" on CoolInterview
This is a bit subjective, but I'll bite... ;)
For someone who is a self-professed MATLAB expert, here are some of the things that I would personally expect them to be able to illustrate in an interview:
How to use the arithmetic operators for matrix or element-wise operations.
A familiarity with all the basic data types and how to convert effortlessly between them.
A complete understanding of matrix indexing and assignment, be it logical, linear, or subscripted indexing (basically, everything on this page of the documentation).
An ability to manipulate multi-dimensional arrays.
The understanding and regular usage of optimizations like preallocation and vectorization.
An understanding of how to handle file I/O for a number of different situations.
A familiarity with handle graphics and all of the basic plotting capabilities.
An intimate knowledge of the types of functions in MATLAB, in particular nested functions. Specifically, given the following function:
function fcnHandle = counter
value = 0;
function currentValue = increment
value = value+1;
currentValue = value;
end
fcnHandle = #increment;
end
They should be able to tell you what the contents of the variable output will be in the following code, without running it in MATLAB:
>> f1 = counter();
>> f2 = counter();
>> output = [f1() f1() f2() f1() f2()]; %# WHAT IS IT?!
We get several new people in the technical support department here at MathWorks. This is all post-hiring (I am not involved in the hiring), but I like to get to know people, so I give them the "Impossible and adaptive MATLAB programming challenge"
I start out with them at MATLAB and give them some .MAT file with data in it. I ask them to analyze it, without further instruction. I can very quickly get a feel for their actual experience.
http://blogs.mathworks.com/videos/2008/07/02/puzzler-data-exploration/
The actual challenge does not mean much of anything, I learn more from watching them attempt it.
Are they making scripts, functions, command line or GUI based? Do they seem to have a clear idea where they are going with it? What level of confidence do they have with what they are doing?
Are they computer scientists or an engineer that learned to program. CS majors tend to do things like close their parenthesis immediately, and other small optimizations like that. People that have been using MATLAB a while tend to capture the handles from plotting commands for later use.
How quickly do they navigate the documentation? Once I see they are going down the 'right' path then I will just change the challenge to see how quickly they can do plots, pull out submatrices etc...
I will throw out some old stuff from Project Euler. Mostly just ramp up the questions until one of us is stumped.
Floating Point Questions
Given that Matlab's main (only?) data type is the double precision floating point matrix, and that most people use floating point arithmetic -- whether they know it or not -- I'm astonished that nobody has suggested asking basic floating point questions. Here are some floating point questions of variable difficulty:
What is the range of |x|, an IEEE dp fpn?
Approximately how many IEEE dp fpns are there?
What is machine epsilon?
x = 10^22 is exactly representable as a dp fpn. What are the fpns xp
and xs just below and just above x ?
How many dp fpns are in [1,2)? How many atoms are on an edge of a
1-inch sugar cube?
Explain why sin(pi) ~= 0, but cos(pi) = -1.
Why is if abs(x1-x2) < 1e-10 then a bad convergence test?
Why is if f(a)*f(b) < 0 then a bad sign check test?
The midpoint c of the interval [a,b] may be calculated as:
c1 = (a+b)/2, or
c2 = a + (b-a)/2, or
c3 = a/2 + b/2.
Which do you prefer? Explain.
Calculate in Matlab: a = 4/3; b = a-1; c = b+b+b; e = 1-c;
Mathematically, e should be zero but Matlab gives e = 2.220446049250313e-016 = 2^(-52), machine epsilon (eps). Explain.
Given that realmin = 2.225073858507201e-308, and Matlab's u = rand gives a dp fpn uniformly distributed over the open interval (0,1):
Are the floating point numbers [2^(-400), 2^(-100), 2^(-1)]
= 3.872591914849318e-121, 7.888609052210118e-031, 5.000000000000000e-001
equally likely to be output by rand ?
Matlab's rand uses the Mersenne Twister rng which has a period of
(2^19937-1)/2, yet there are only about 2^64 dp fpns. Explain.
Find the smallest IEEE double precision fpn x, 1 < x < 2, such that x*(1/x) ~= 1.
Write a short Matlab function to search for such a number.
Answer: Alan Edelman, MIT
Would you fly in a plane whose software was written by you?
Colin K would not hire me (and probably fire me) for saying "that
Matlab's main (only?) data type is the double precision floating
point matrix".
When Matlab started that was all the user saw, but over the years
they have added what they coyly call 'storage classes': single,
(u)int8,16,32,64, and others. But these are not really types
because you cannot do USEFUL arithmetic on them. Arithmetic on
these storage classes is so slow that they are useless as types.
Yes, they do save storage but what is the point if you can't do
anything worthwhile with them?
See my post (No. 13) here, where I show that arithmetic on int32s is 12 times slower than
double arithmetic and where MathWorkser Loren Shure says "By
default, MATLAB variables are double precision arrays. In the olden
days, these were the ONLY kind of arrays in MATLAB. Back then even
character arrays were stored as double values."
For me the biggest flaw in Matlab is its lack of proper types,
such as those available in C and Fortran.
By the way Colin, what was your answer to Question 14?
Ask questions about his expertise and experience in applying MATLAB in your domain.
Ask questions about how he would approach designing an application for implementation in MATLAB. If he refers to recent features of MATLAB, ask him to explain them, and how they are different from the older features they replace or supplement, and why they are preferable (or not).
Ask questions about his expertise with MATLAB data structures. Many of the MATLAB 'experts' I've come across are very good at writing code, but very poor at determining what are the best data structures for the job in hand. This is often a direct consequence of their being domain experts who've picked up MATLAB rather than having been trained in computerism. The result is often good code which has to compensate for the wrong data structures.
Ask questions about his experience, if any, with other languages/systems and invite him to expand upon his observations about the relative strengths and weaknesses of MATLAB.
Ask for top tips on optimising MATLAB programs. Expect the answers: vectorisation, pre-allocation, clearing unused variables, etc.
Ask about his familiarity with the MATLAB profiler, debugger and lint tools. I've recently discovered that the MATLAB 'expert' over in the corner here had never, in 10 years using the tool, found the profiler.
That should get you started.
I. I think this recent SO question
on indexing is a very good question
for an "expert".
I have a 2D array, call it 'A'. I have
two other 2D arrays, call them 'ix'
and 'iy'. I would like to create an
output array whose elements are the
elements of A at the index pairs
provided by x_idx and y_idx. I can do
this with a loop as follows:
for i=1:nx
for j=1:ny
output(i,j) = A(ix(i,j),iy(i,j));
end
end
How can I do this without the loop? If
I do output = A(ix,iy), I get the
value of A over the whole range of
(ix)X(iy).
II. Basic knowledge of operators like element-wise multiplication between two matrices (.*).
III. Logical indexing - generate a random symmetric matrix with values from 0-1 and set all values above T to 0.
IV. Read a file with some properly formatted data into a matrix (importdata)
V. Here's another sweet SO question
I have three 1-d arrays where elements
are some values and I want to compare
every element in one array to all
elements in other two.
For example:
a=[2,4,6,8,12]
b=[1,3,5,9,10]
c=[3,5,8,11,15]
I want to know if there are same
values in different arrays (in this
case there are 3,5,8)
Btw, there's an excellent chance your interviewee will Google "MATLAB interview questions" and see this post :)
Possible question:
I have an array A of n R,G,B triplets. It is a 3xn matrix. I have another array B in the form 1xn which stores an index value (association to a cluster) for each triplet.
How do I plot the triplets of A in 3D space (using plot3 function), coloring each triplet according to its index in B? (The goal is to qualitatively evaluate my clustering)
Really, really good programmers who are MATLAB novices won't be able to give you an efficient (== MATLAB style) solution. However, it is a very simple problem if you do know your MATLAB.
Depends a bit what you want to test.
To test MATLAB fluency, there are several nice Stack Overflow questions that you could use to test e.g. array manipulations (example 1, example 2), or you could use fix-this problems like this question (I admit, I'm rather fond of that one), or look into this list for some highly MATLAB-specific stuff. If you want to be a bit mean, throw in a question like this one, where the best solution is a loop, and the typical MATLAB-way-of-thinking solution would just fill up the memory.
However, it may be more useful to ask more general programming questions that are related to your area of work and see whether they get the problem solved with MATLAB.
For example, since I do image analysis, I may ask them to design a class for loading images of different formats (a MATLAB expert should know how to do OOP, after all, it has been out for two years now), and then ask follow-ups as to how to deal with large images (I want to see a check on how much memory would be used - or maybe they know memory.m - and to hear about how MATLAB usually works with doubles), etc.