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Questions tagged [computer-arithmetic]

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3
votes
2answers
195 views

Best software to do big number calculations quickly

I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (100+ digits ). I am currently using python to test these numbers. In the ...
2
votes
2answers
104 views

Evaluating $\log(\exp(x)+1)$ for negative $x$

With double precision, I get $\log(\exp(-3)+1)=0.048587351573741958$, which already has $4$ incorrect digits, and $\log(\exp(-30)+1)=9.348... \times10^{-14}$, which only has two correct digits. What ...
2
votes
1answer
82 views

Performing computations over the set of constructable numbers

The set of constructable numbers is commonly defined in one of two ways: the set of finite numbers reachable from the rational numbers through a finite sequence of addition/subtraction, ...
3
votes
1answer
964 views

Fast(er) computation of dot product of two convolutions?

Let $a,b,c,d\in\mathbb{R}^n$. Is it possible to compute $$\langle a*b,c*d\rangle$$ faster than 6 FFTs? I can do it with 6 FFTs by doing normal convolutions, 3 FFTs each. In my application I know $b, ...
-2
votes
2answers
138 views

Array initialization in C

In this bit of code, the X and Y arrays should be identical but for some reason, that I CANNOT figure out for the life of me, X[0] is always 1 rather than 0. I have tried initializing the whole array ...
11
votes
4answers
284 views

Small, unpredictable results in runs of a deterministic model

I have a sizable model (~5000 lines) written in C. It is a serial program, with no random number generation anywhere. It makes use of the FFTW library for functions using FFT - I do not know the ...
1
vote
2answers
52 views

MATLABs double arithmetic

this is a classical problem, but I need help to pinpoint what I am missing. Problem: In MATLAB (exp(1) + 10^12) - 10^12 gives you a double which equal to e, up to 5 correct digits. But I thought ...
8
votes
0answers
410 views

Implementing std::nextafter: Should denormals-are-zero mode affect it? If so, how?

This might be the wrong stackexchange site for this question. math.SE, cs.SE, programmers.SE, and of course stackoverflow are all possibilities. I'm hoping to reach an audience that might actually ...
2
votes
0answers
692 views

Divide and Conquer division algorithm explained (as used in GMP bignum)

I am trying to understand the divide and conquer division algorithm that is used in the GMP bignum arithmetic library. The code is very optimised and that makes it somewhat hard to understand. the ...
2
votes
0answers
175 views

How to construct a subring of a polynomial ring in Magma

This is a question about the computer algebra system Magma. I have been looking for a place to ask this type of question on the SE network and scicomp.SE was suggested to me; hopefully it finds a home ...
3
votes
1answer
147 views

How can I avoid roundoff error when calculating the difference $\textrm{erfc}(a) - \textrm{erfc}(b)$?

In this excellent answer, it is recommended that one make use of the $\textrm{erfcx}$ function to avoid roundoff error in calculating dealing with $x < 25$ (approximately). So, one scales their ...
5
votes
1answer
55 views

How would I figure out when a function would be at the “brink of underflow”?

User hardmath, provided an excellent overview of how to handle overflow when calculating the product of two functions, where one is likely to overflow: https://scicomp.stackexchange.com/a/20913/9466 ...
3
votes
2answers
2k views

what does -ffast-math do?

What kind of optimisations does the option -ffast-math do ? I saw that the time taken for a simple $O(n^2)$ algorithm being reduced to that of an $O(n)$ algorithm ...
1
vote
2answers
762 views

performance of icc main.cpp == g++ -ffast-math main.cpp

I have a program that has a nested loop, together with its parent running at $O(n^2)$ complexity performing floating point arithmetic. I see that the performance of the code when compiled with ...
3
votes
3answers
494 views

Exact analytical matrix inversion of sparse 100x100 matrices in C++

I need to invert a matrix. Of course, I'm not the first person in this situation, and I know that there's a wealth of powerful libraries out there, of which I only know a couple. That being said, ...
4
votes
1answer
166 views

Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and $0<x&...
0
votes
1answer
71 views

How many years it would take our laptops to be as fast as the fastest super computer in 2000 [closed]

How many years it would take our laptops to be as fast as the fastest super computer in 2000 according to Moore's law?
3
votes
1answer
135 views

Fibonacci, variation on the theme

I am trying to calculate the numbers $n$ for which the $n$-th Fibonacci number $F_n$ is a multiple of $n$; that is fib(n)%n==0. Here is the best PARI code I could come up with (for the counting ...
2
votes
1answer
2k views

range of positive mantissa in given floating-point number representation

I am a student and I came to this question while solving problems regarding the float-points. ...
6
votes
2answers
119 views

How many bits to unambiguously represent fixed-point division?

Suppose I have a function which divides an $m$-bit unsigned integer $a$ by an $n$-bit unsigned integer $b$ and returns the quotient as a fixed-point number with $t$ fractional bits, truncating towards ...
2
votes
2answers
114 views

Integer arithmetic support on future HPC systems

I writing some robust geometric algorithms using quantization + integer arithmetic for evaluating exact predicates. However, since BlueGene's integer support is so terrible, it occurred to me that ...
6
votes
2answers
2k views

Need for quad precision in scientific computing?

Even if quad precision is not directly supported by most CPUs, many Compilers (GNU, Intel) support them. Also some software packages allow to compile with quad precision, e.g. PETSc. But is there ...
6
votes
2answers
259 views

Krylov subspace iterative methods in floating point arithmetic

Is there any work that considers Krylov subspace iterative methods in floating point arithmetic? I'm especially interested in how rounding errors influence the convergence and the accuracy of the ...
13
votes
4answers
460 views

In floating point arithmetic, why does numerical imprecision result from adding a small term to a difference of large terms?

I have been reading the book Computer Simulation of Liquids by Allen and Tildesley. Starting on page 71, the authors discuss the various algorithms that are used to integrate Newton's equations of ...
10
votes
3answers
378 views

Which is computed faster, $a^b$, $\log_a c$ or $\sqrt[b]{c}$?

Which is computed faster, $a^b$ or $\log_a c$ or $\sqrt[b]{c}$? $a$, $b$ and $c$ are positive reals with $b>1$. What kinds of algorithms will you use in the comparison? What are their complexities?...
38
votes
2answers
7k views

Why is division so much more complex than other arithmetic operations?

I recently encountered a case where I needed an integer division operation on a chip that lacked one (ARM Cortex-A8). While trying to research why that must be, I found out that in general division ...