This documentation page of boost::math::tools::brent_find_minima says about its first argument:
The function to minimise: a function object (or C++ lambda) ... with no maxima occurring in that interval.
But what happens if this is not the case? (After all, this condition is rather difficult to pre-ensure, especially since the function is usually expensive to evaluate at many points.) Best would be to detect violations to this condition on the fly.
If this condition is violated, does boost throw an exception, or does it exhibit undefined behavior?
A workaround I am thinking of is to build the checking into the lambda ("function to minimize"), by capturing and maintaining a std::map<double,double> holding all the points that have been evaluated, and comparing each new evaluation with its nearest neighbor in each direction, to check whether there may be a local maximum. But I don't want to do all that if it isn't necessary.
There is no way for this to be done. If you read Corless's A Graduate Introduction to Numerical Methods, you'll read a very interesting point: All numerically defined functions are discontinuous halfway between representables, and have zero derivatives between representables. Basically they can be thought of as a sum of Heaviside functions.
So none of them are differentiable in the mathematical sense. Ok, maybe you think this is a bit unfair-the scale should be zoomed out. But how much? We know that |x-1| isn't differentiable at x=1, but how could a computer tell that? How does it know that there isn't some locally smooth mollifier that makes it differentiable between x=1-eps and x=1+eps? I don't think there's a good answer to this question.
One of the most difficult problems in this class arises in quadrature. Some of these methods work fast when the complex extension of the function has poles far from the real axis. Try to numerically determine that.
Function spaces are impossible to determine numerically. Users just have to get it right.
Related
I was told to solve this problem:
given a1, ..., an are real numbers. Need to calculate min(a1, -a1a2, a1a2a3, ...,(-1)^(n+1) a1a2,... an)
but I cannot understand the logic of the task. Could you tell me what I should do?
For example, what is (-l)^n+1? I've never seen it before.
What you should do is:
use the n real numbers of input to ...
... calculate the n numbers defined by the quoted formula (though you only need one value at a time to be more efficient)
while doing so keep track of the smallest number you encounter, that is the final result
concerning the (-1)^(n+1), it is reasonable to assume (as e.g. in the comment by Weather Vane and others) that it means powers of -1 (in a lazy and unexplained but non-C++ syntax)
note that you can easily calculate one value from the previous one by simple multiplication
probably you should do all of that by writing a program, an assumption based on the fact that you are asking on StackOverflow and tag a programming language
I am currently coding in C++, creating an all rounded calculator that, when finished, will be capable of handling all major and common mathematical procedures.
The current wall I am hitting is from the fact I am still learning about to profession we call being a programmer.
I have several ways of achieving a single result. I am curious as to whether I should pick the method that has a clear breakdown of how it got to that point in the code; or the method that is much shorter - while not sacrificing any of the redability.
Below I have posted snippets from my class showing what I mean.
This function uses if statements to determine whether or not a common denominator is even needed, but is several lines long.
Fraction Fraction::addFraction(Fraction &AddInput)
{
Fraction output;
if (m_denominator != AddInput.m_denominator)
{
getCommonDenominator(AddInput);
output.setWhole(m_whole + AddInput.m_whole);
output.setNumerator((m_numerator * firstchange) + (AddInput.m_numerator * secondchange));
output.setDenominator(commondenominator);
}
else
{
output.setWhole(m_whole + AddInput.m_whole);
output.setNumerator(m_numerator + AddInput.m_numerator);
output.setDenominator(m_denominator);
}
output.simplify();
return output;
}
This function below, gets a common denominator; repeats the steps on the numerators; then simplifies to the lowest terms.
Fraction Fraction::addFraction(Fraction &AddInput)
{
getCommonDenominator(AddInput);
Fraction output(m_whole + AddInput.m_whole, (m_numerator * firstchange) + (AddInput.m_numerator * secondchange), commondenominator);
output.simplify();
return output;
}
Both functions have been tested and always return the accurate result. When it comes to coding standards... do we pick longer and asy to follow? or shorter and easy to understand?
Your first priority with your code should be that it's correct.
Your second priority with code should be "If someone who's never seen this before is going to make a tiny change, which one is he less likely to break?
There's actually a lot that goes into this. How difficult is it to understand at a high level? How abstracted out are arcane details? Are there any surprises? What quirks do you have to know about? Are there edge cases that have to be handled?
The reasons that this second priority is important are:
it's key to preventing you from writing bugs in the first place
it's easier to find bugs later
it's easier to fix bugs later
despite whatever you think, you won't remember the details in 6 months.
Both implementations appear about equally difficult in complexity per branch, but the first one has branches, so I'd lean toward the second for understandability. Details seem abstracted out in both, and if there's surprises or quirks, I don't immediately see them (but that's sort of the point, that they can be easily overlooked). I don't see any special handling for edge cases, so if edge cases exist in either, comments would be good.
Unrelated to picking, but while on the topic of reviewing code, It's unclear how either handles fractions that have no fractional part, but that might be part of the full class documentation, which would be fine. Both codepaths take AddArgument by mutable reference, which is bad, and require this to be mutable as well, which is also bad. Both have methods named get*() that appear to modify (getCommonDenominator), which is bad. The code appears to be using variables that are external (firstchange? secondchange?) which is a major strike against preventing bugs.
I'm reading a book about rendering 3d graphics and the author sometimes uses epsilon and sometimes doesn't.
Notice the if at the beginning using epsilon and the other ifs that don't.
What's the logic behind this? I can see he avoids any chance for division by zero but when not using epsilon in the function there's still a chance it will return a value that will make the outer code to divide by zero.
Book is Real-Time Rendering 3rd Edition, by the way.
The first statement, if(|f| > ϵ) is just checking to make sure f is significantly different from 0. It's important to do that in that specific spot in the code because the next two statements divide by f.
The other statements don't need to do that, so they don't need to use ϵ.
For example,
if(t1 > t2) swap(t1, t2);
is a self-contained statement that compares two numbers to each other and swaps them if the wrong one is greater. Since it's not comparing to see if a value is close to 0, there's no need to use ϵ.
If the value that is returned from this block of code can make the calling code divide by zero, that should be handled in the calling code.
I can't really find any answer in the Modelica specification so ill ask you guys. The specification states that
A tool is free to solve equations, reorder expressions and to not evaluate expressions if their values do not influence the result (e.g. short-circuit evaluation of Boolean expressions). If-statements and if-expressions guarantee that their clauses are only evaluated if the appropriate condition is true, but relational operators generating state or time events will during continuous integration have the value from the most recent event.
If a numeric operation overflows the result is undefined. For literals it is recommended to automatically convert the number to another type with greater precision.
Now, I wonder, can the tool choose to evaluate an expression several time in an integrator step? For example (probably not an valid example, just to give you guys an idea of what I was wondering :) )
Real x;
equation
der(x) = -t;
Modelica.Utilities.Streams.print(String(time));
This will print the same time for several times, so I figured that there is some kind of iteration going on. But I would really like to have it confirmed by some source.
That is normal.
Variable step size solvers (like dassl) can go back
and forth in time to find the direction of the curve.
Also, if you have events more values can be generated
at the same time.
If you want to print time or values just at exact time instants you need when equations:
when sample(0, 1) then
Modelica.Utilities.Streams.print(String(time));
end when;
Read more in the Modelica Spec about sample.
Is also possible to use fixed step size solvers like Euler or so.
I've recently heard that in some cases, programmers believe that you should never use literals in your code. I understand that in some cases, assigning a variable name to a given number can be helpful (especially in terms of maintenance if that number is used elsewhere). However, consider the following case studies:
Case Study 1: Use of Literals for "special" byte codes.
Say you have an if statement that checks for a specific value stored in (for the sake of argument) a uint16_t. Here are the two code samples:
Version 1:
// Descriptive comment as to why I'm using 0xBEEF goes here
if (my_var == 0xBEEF) {
//do something
}
Version 2:
const uint16_t kSuperDescriptiveVarName = 0xBEEF;
if (my_var == kSuperDescriptiveVarName) {
// do something
}
Which is the "preferred" method in terms of good coding practice? I can fully understand why you would prefer version 2 if kSuperDescriptiveVarName is used more than once. Also, does the compiler do any optimizations to make both versions effectively the same executable code? That is, are there any performance implications here?
Case Study 2: Use of sizeof
I fully understand that using sizeof versus a raw literal is preferred for portability and also readability concerns. Take the two code examples into account. The scenario is that you are computing the offset into a packet buffer (an array of uint8_t) where the first part of the packet is stored as my_packet_header, which let's say is a uint32_t.
Version 1:
const int offset = sizeof(my_packet_header);
Version 2:
const int offset = 4; // good comment telling reader where 4 came from
Clearly, version 1 is preferred, but what about for cases where you have multiple data fields to skip over? What if you have the following instead:
Version 1:
const int offset = sizeof(my_packet_header) + sizeof(data_field1) + sizeof(data_field2) + ... + sizeof(data_fieldn);
Version 2:
const int offset = 47;
Which is preferred in this case? Does is still make sense to show all the steps involved with computing the offset or does the literal usage make sense here?
Thanks for the help in advance as I attempt to better my code practices.
Which is the "preferred" method in terms of good coding practice? I can fully understand why you would prefer version 2 if kSuperDescriptiveVarName is used more than once.
Sounds like you understand the main point... factoring values (and their comments) that are used in multiple places. Further, it can sometimes help to have a group of constants in one place - so their values can be inspected, verified, modified etc. without concern for where they're used in the code. Other times, there are many constants used in proximity and the comments needed to properly explain them would obfuscate the code in which they're used.
Countering that, having a const variable means all the programmers studying the code will be wondering whether it's used anywhere else, keeping it in mind as they inspect the rest of the scope in which it's declared etc. - the less unnecessary things to remember the surer the understanding of important parts of the code will be.
Like so many things in programming, it's "an art" balancing the pros and cons of each approach, and best guided by experience and knowledge of the way the code's likely to be studied, maintained, and evolved.
Also, does the compiler do any optimizations to make both versions effectively the same executable code? That is, are there any performance implications here?
There's no performance implications in optimised code.
I fully understand that using sizeof versus a raw literal is preferred for portability and also readability concerns.
And other reasons too. A big factor in good programming is reducing the points of maintenance when changes are done. If you can modify the type of a variable and know that all the places using that variable will adjust accordingly, that's great - saves time and potential errors. Using sizeof helps with that.
Which is preferred [for calculating offsets in a struct]? Does is still make sense to show all the steps involved with computing the offset or does the literal usage make sense here?
The offsetof macro (#include <cstddef>) is better for this... again reducing maintenance burden. With the this + that approach you illustrate, if the compiler decides to use any padding your offset will be wrong, and further you have to fix it every time you add or remove a field.
Ignoring the offsetof issues and just considering your this + that example as an illustration of a more complex value to assign, again it's a balancing act. You'd definitely want some explanation/comment/documentation re intent here (are you working out the binary size of earlier fields? calculating the offset of the next field?, deliberately missing some fields that might not be needed for the intended use or was that accidental?...). Still, a named constant might be enough documentation, so it's likely unimportant which way you lean....
In every example you list, I would go with the name.
In your first example, you almost certainly used that special 0xBEEF number at least twice - once to write it and once to do your comparison. If you didn't write it, that number is still part of a contract with someone else (perhaps a file format definition).
In the last example, it is especially useful to show the computation that yielded the value. That way, if you encounter trouble down the line, you can easily see either that the number is trustworthy, or what you missed and fix it.
There are some cases where I prefer literals over named constants though. These are always cases where a name is no more meaningful than the number. For example, you have a game program that plays a dice game (perhaps Yahtzee), where there are specific rules for specific die rolls. You could define constants for One = 1, Two = 2, etc. But why bother?
Generally it is better to use a name instead of a value. After all, if you need to change it later, you can find it more easily. Also it is not always clear why this particular number is used, when you read the code, so having a meaningful name assigned to it, makes this immediately clear to a programmer.
Performance-wise there is no difference, because the optimizers should take care of it. And it is rather unlikely, even if there would be an extra instruction generated, that this would cause you troubles. If your code would be that tight, you probably shouldn't rely on an optimizer effect anyway.
I can fully understand why you would prefer version 2 if kSuperDescriptiveVarName is used more than once.
I think kSuperDescriptiveVarName will definitely be used more than once. One for check and at least one for assignment, maybe in different part of your program.
There will be no difference in performance, since an optimization called Constant Propagation exists in almost all compilers. Just enable optimization for your compiler.