A method of representing numbers by a fixed number of significant digits, and the exponent of some base number. They are characterized in the form ${(significant digits)}*base^{exponent}$. Typically, numbers are represented with respect to base = 2 (binary).

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47 views

Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different ...
5
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1answer
61 views

Accurate Polynomial Evaluation in Floating Point

What are the most accurate algorithms for evaluating a polynomial using floating point arithmetic? The internet seems to suggest that Horner's method is commonly used. In particular I have a cubic ...
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3answers
156 views

How to determine the amount of FLOPs my computer is capable of

I would like to determine the theoretical number of FLOPs (Floating Point Operations) that my computer can do. Can someone please help me with this. (I would like to compare my computer to some ...
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8answers
1k views

Which algorithm is more accurate for computing the sum of a sorted array of numbers?

Given is an increasing finite sequence of positive numbers $z_{1} ,z_{2},.....z_{n}$. Which of the following two algorithms is better for computing the sum of the numbers? ...
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2answers
72 views

Re-scaling array of floats so that all items are approximately integer

I have an array of floating point values $F$. I want to input my array into an algorithm that only takes integer values. How can I efficiently determine the smallest multiplier $m$ such that all ...
5
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3answers
140 views

Machine epsilon does not limit relative rounding error for denormals. Is this a problem?

As we know, machine epsilon limits relative rounding error in the range of normalized floating point numbers. But it is easy to check that this is not true for denormalized numbers. My question is ...
2
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1answer
545 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. ...
3
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1answer
1k views

Machine epsilon (eps)

The wiki for machine epsilon says: "Machine epsilon gives an upper bound on the relative error due to rounding in floating point arithmetic" If machine epsilon is the upper bound on the relative ...
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2answers
276 views

How to handle floating point operations in HLSL?

I'm trying to write a perona malik anisotropic diffusion filter for the GPU. I'm basing my shader off a matlab implementation of the filter. I'm running into trouble because of what I suspect is ...
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3answers
2k views

Do currently available GPUs support double precision floating point arithmetic?

I have run the molecular dynamics (MD) code GROMACS on a Ubuntu Linux cluster consisting of nodes containing 24 Intel Xeon CPUs. My particular point of interest turns out to be somewhat sensitive to ...
7
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4answers
244 views

Relevance of fixed-point and arbitrary precision computations

I see very few non-floating point computing libraries/packages around. Given the various inaccuracies of floating point representation, the question arises why there aren't at least some fields where ...
5
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1answer
165 views

cancellation problem in float-point numbers

In http://en.wikipedia.org/wiki/Floating_point#Addition_and_subtraction, it gives an example about cancellation problem in float-point numbers, see I don't understand why it is written : ...
9
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1answer
171 views

Are there Improved ways of computing $p \log(p)$?

Most math libraries have a number of versions of logarithm functions. Most of the time we assume them to be perfect, but actually quite a lot of them just offer a certain number of digits of ...
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2answers
345 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 ...
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2answers
126 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 ...
11
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1answer
181 views

Catastrophic cancellation in logsum

I'm trying to implement the following function in double-precision floating point with low relative error: $$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$ This is used extensively in statistical ...
10
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3answers
284 views

Relative comparison of floating point numbers

I have a numerical function f(x, y) returning a double floating point number that implements some formula and I want to check that it is correct against analytic ...
6
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2answers
251 views

transitive floating point comparison with (absolute) tolerance

I want to compare two floating point numbers for equality relative to a known absolute tolerance. However, this is inside an algorithm I wrote quite some time ago, and I believe the logic of that ...
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4answers
301 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 ...
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5answers
296 views

What algorithm for solving a set of stiff ODEs would be easiest to port to high precision floating point arithmetic?

I want to solve a relatively small system of stiff ODEs (< 10 first-order equations) using high precision floating point arithmetic (using MPFR or alike). What would be the easiest algorithm to ...
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3answers
1k views

How to add large exponential terms reliably without overflow errors?

A very common problem in Markov Chain Monte Carlo involves computing probabilities that are sum of large exponential terms, $ e^{a_1} + e^{a_2} + ... $ where the components of $a$ can range from ...
7
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1answer
303 views

What's the right way to compare vectors in floating-point?

I know that I should use a tolerance for comparing floating point numbers. But for comparing vectors, I can think of 3 possible solutions corresponding to different distance metrics: Compare the ...
15
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5answers
502 views

Analyzing Numerical Error in C++ Function

Suppose that I have a function that takes as input several floating-point values (single or double), does some computation, and produces output floating-point values (also single or double). I am ...
10
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4answers
2k views

FLOP counting for library functions

When evaluating the number of FLOPs in a simple function, one can often just go down the expression tallying basic arithmetic operators. However, in the case of mathematical statements involving even ...
5
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2answers
3k views

How to set double precision values in Fortran

Recently, I've encountered a bizarre problem with FORTRAN95. I initialized variables X and Y as follows: X=1.0 Y=0.1 Later I add them together and print the ...
2
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2answers
286 views

Is there a Moore's law for floating-point precision, and what would it imply?

Moore's law states that the number of transistors on an integrated circuit grow exponentially, roughly doubling at a period of 20 months. This affects the amount of memory available and the speed of ...
19
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5answers
509 views

Is there software that can autogenerate numerically-accurate floating point C routines from symbolic formulae?

Given a real function of real variables, is there software available that can automatically generate numerically-accurate code to calculate the function over all inputs on a machine equipped with IEEE ...
3
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2answers
165 views

Looking for a mathematical proof of stability in floating point arithmetic of CG - any reference?

I am looking for a reference - paper, book, discussion, anything that has a mathematical proof for stability of the conjugate gradient method in floating point arithmetic. Something similar for ...
5
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3answers
306 views

Does there exist an arbitrary-precision convex optimization solver?

I have a relatively simple convex optimization problem that involves less than 100 variables but contains a terribly ill-conditioned matrix. I have tried CVX and CPLEX; even though both can typically ...