Questions tagged [precision]
Issues related to the representation of numerical quantities in a finite representation in a given base differing from their exact mathematical value.
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Computing $\frac{x - y}{x - z}$ when $x,y,z$ are close to each other
What is the most stable way to compute
$$\frac{x - y}{x - z}$$
when $x$, $y$, and $z$ are all close to each other? I would like to compute expressions of this form in low precision on a GPU, but when ...
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Unexpected result when summing sorted (and unsorted) positive floating point numbers
I am exploring Higham's excellent Accuracy and Stability of Numerical Algorithms and chapter 4 is dedicated to summation.
So I decided to test the most basic thing. Summing positive random numbers ...
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Single precision vs double precision conjugate gradients
I tested my conjugate gradients implementation with float and double precision and contrary to my guess the double code was twice faster than the single precision code. The reason is that I need many ...
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Runge Kutta 4th order: unexpected result
My problem in brief: in some situations, the Runge Kutta 4th order method (RK4) doesn't seem to give 4th order improvement when using a smaller time step. I wonder how this worse-than-expected result ...
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Float equality tolerance for single and half precision
Suppose the metric is
abs(a-b) <= rtol * max(abs(a), abs(b))
i.e. math.isclose with ...
2
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Numerically stable way to implement Cramer's rule analog
Problem statement
Let $A$ be an $n\times n$ matrix and $b$ an $n$-dimensional vector. For $j\in \{1, \dots, n \}$, let $A_j$ be the matrix where we take $A$ and replace the $j^{\rm th}$ column with $b$...
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Dynamic tolerance in a conditional loop to obtain maximum precision allowed by machine floating point numbers
I have coded a simple program for a root finding problem using Halley's method. Here is the code:
...
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Summation of trigonometric functions results in error with finite precision
Consider the following expression:
$$f(t) = B+\sum_{k=1}^{N} A_k\cos(\omega_kt)$$
where $A$ and $B$ are known. the frequencies are also known but are not multiples of a fundamental frequency. However, ...
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Robust unit test for reciprocal approximation
Let $x$ and $y$ be representable floating point numbers. I'm looking for a unit test which can ensure that my user's compiler has not made the reciprocal approximation $\mathrm{fl}(x/y) \approx \...
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What are the Exact Rules for Significant Figures, Precision, and Uncertainty?
In the physical sciences (which are physics, chemistry, astronomy, materials science, etc.), we learned that the uncertainty is +/- the smallest unit (which is 1) of the last significant figure if the ...
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Accuracy loss in single-precision Euclidean norm computation
I do hydrodynamics simulations with Fortran and recently I met with this issue:
I have a single-precision array b of length ...
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High precision numerical integration of discrete data with Matlab
I have discrete data of a function plotted below:
The "Y" values of the function near "X=1.57" are very close to each other and zero, like 9.25558265263186E-11 and 5....
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Efficient way of calculating a cumulative integral with prefactor
I have a grid of points $x_i$ and corresponding function values $y_i=f(x_i)$. I'm interesting in something like the cumulant of $f$, but it has an awkward prefactor. The desired quantity we'll call
$$...
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How error scales with numerical precision in molecular dynamics?
In terms of time-step, numerical error in molecular dynamics scales with square, i.e. $error \approx dt^2$. But how it look for numerical precision ? E.g. how much bigger will be numerical error when ...
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How to include negative number in the log-sum-exp?
I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...}
If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
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Hardware supporting floats with fraction beyond 64 bit
Is there any computation accelerator (like a GPGPU) available, that natively (this means in hardware, not emulated by a library) supports arithmetics using floating point numbers with a fractional ...
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How can I detect lost of precision due to rounding in both floating point addition and multiplication?
From Computer Systems: a Programmer's Perspective:
With single-precision floating point
the expression (3.14+1e10)-1e10 evaluates to 0.0: the value 3.14 is lost ...
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How can I avoid catastrophic cancellation?
I have the following formula that I need to rewrite in order to avoid catastrophic cancellation.
$$y =\sqrt{\frac{1}{2}\left(1-\sqrt{1-x^{2}}\right)}$$
As $x$ becomes smaller, $\sqrt{1-x^{2}}$ ...
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Red flags for numerical computing?
I've learnt the hard way that you should avoid:
computing small numbers as the difference of two large numbers
evaluating chaotic functions with imprecise inputs.
Are there any other red flags a ...
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Convert decimal number in binary double precision, how to avoid the loss of the last digits after normalization?
I have the decimal number: $0.023$, and I want to convert in a binary number with $52$ bit of mantissa in Double Precision:
if I go to convert, using this utility here, in non-normalized form, with $...
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Is half precision supported by modern architecture?
I am new to computer science and I was wondering whether half precision is supported by modern architecture in the same way as single or double precision is. I thought the 2008 revision of IEEE-754 ...
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Division and Modulus for Large DIVISORS
Excel and also Excel VBA have no built-in support for arbitrary precision arithmetic. There are a few very large add-ins that can be installed to do these sorts of calculations where the operands are ...
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Comparison of integrals with a function:
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
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Find precision, or the number of digits in the mantissa, in a floating point machine number
We have that: $\epsilon$ is the smallest positive machine number that summed to $1$ resuts in $(\epsilon + 1)$: i.e. the smallest number greater than $1$:
if $p$ is precision and $\beta$ the base:
$$...
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Numerically stable and fast sum of last K elements in sequence
Suppose I have a long, possibly infinite, sequence $x := [x_1, x_2, ...]$, and I want to use it to compute another sequence $y:=[y_1, y_2, ...]$ where each element is the sum of the last K elements of ...
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Best way to check if SOR solution has converged for 2d matrix
I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10.
I can obtain the fully converged solution for ...
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log-sum-exp trick for signed/complex numbers
I need to evaluate a sum of values that are on many different orders of magnitude in scale but might be signed. I’ve had great luck with the “log-sum-exp” trick for an unsigned version of my problem, ...
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How do I globally change the precision of a piece of code in Python to debug it?
I am solving a system of non-linear equations using the Newton-Raphson method in Python. This involves using the solve(Ax,b) function (...
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Numerical stability in the product of many matrices
I have to calculate in numpy the matrix-product of many matrices (~400). Are there common practices to increase numerical stability?
If this is relevant, the matrices are $300\times 300$ orthogonal ...
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How to perform an eigendecomposition of a general complex matrix with arbitrary precision in C/C++
I need to obtain the Eigenvectors of a general complex matrix, but with quadruple precision. Is anyone aware of a means to do this?
I currently use Tux Eigen, and I see that in their unsupported ...
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Inverse of ill-conditioned symmetric matrix
I've got a matrix K, with dimensions $(n, n)$ where each element is computed using the following equation:
$$K_{i, j} = \exp(-\alpha t_i^2 -\gamma(t_i - t_j)^2 - \...
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Evaluate Nth root of a rational to a correctly rounded float
Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible.
I want to evaluate an expression in the ...
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857
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Conjugate gradient - ill-conditioning and numerical tolerance
I would like to solve system $Ax=b$, where $A$ is SPD, but very ill-conditioned ($\text{cond}(A)>10^{11}$). I am interested in using UNpreconditioned version of the conjugate gradient method.
Is ...
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Analytical convergent sequence and numerical divergent sequence
Is it possible to construct a sequence that converges in theory but when computed numerically with a computer program is diverging.
I feel that today our computer programs doesn't allow such ...
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How to compute large condition number of a matrix in Python?
I have a matrix that is extremely singular, but I am still interested in computing the exact condition number, which is the ratio between the largest and smallest singular values.
Is it possible to ...
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Compile-time error control vs. interval arithmetic?
I use interval arithmetic for reliable computing. Now, a procedure coded in a good implementation of interval arithmetic takes perhaps about eight times as much as the same procedure carried out ...
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Is it possible to proof a-b+b = a for all double floating-point numbers?
I want to know whether the equation : a-b+b = a is always true for a, b belongs to double precision floating-point number and |a|>=|b|.
If the equation is true, how can I proof it?
If not, what ...
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What good are hard-sphere event-driven molecular dynamics simulations in the face of chaos?
Simple hard-sphere dynamical systems can exhibit chaotic dynamics. Due to finite-precision arithmetic when implemented on a computer, the presence of chaos implies that for a given set of initial data,...
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Time iteration no longer smooth after using scaled units
I have a time iteration function looked on a 2D surface like this.
Since the numbers wee very small i.e. hbar=6.6260700404e-34./(2*pi), my professor told me to use our own "scaled unites" during the ...
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2
answers
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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 ...
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High precision Discrete Fourier Transform in c
I'm trying to do a high precision discrete fourier transform on a signal. To examine the precision, I use a gaussian function as the signal, because the fourier transform is also a gaussian function.
...
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Why can ill-conditioned linear systems be solved precisely?
According to the answer here, large condition number (for linear system solving) decreases the guaranteed number of correct digits in the floating point solution. Higher order differentiation matrices ...
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Numerical Precision in Matrix Inversion Routines
Let's say I have a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$, and its associated inverse, $\mathbf{A}^{-1}$. The elements of $\mathbf{A}$ are given in IEEE single precision, i.e. 23-bit mantissa, ...
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Numerically stable way of computing angles between vectors
When applying the classical formula for the angle between two vectors:
$$\alpha = \arccos \frac{\mathbf{v_1} \cdot \mathbf{v_2}}{\|\mathbf{v_1}\| \|\mathbf{v_2}\|}$$
one finds that, for very small/...
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Computing the change of function at two close points without cancellation
I want to compute the difference $\Delta f(x_1,x_2) = f(x_1)-f(x_2)$ of a smooth function $f(x)$ at two points $x_1$ and $x_2$ which are close to each other. The magnitude of the expected result, $|\...
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Half precision in Fortran
To improve the time efficiency of my code, I'd like to test a lower precision for real number, using e.g. half precision (2 bytes).
However, I'm not sure if I can do that in Fortran.
After playing ...
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Stable computation of ratio of sums of large numbers
I have two sets of large positive numbers $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$. By 'large' I mean of the order of $10^{10}$. I want to calculate the ratio $$R = \frac{a_1 - a_2 + \cdots +(-1)^{n+1}...
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How to deal with big numbers in intermediate calculations?
I have a rather long expression (https://pastebin.com/jUsxdCCs) that is an analytical solution of a set of differential equations generated symbolically from Maple. I need to solve a set of equations ...
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How to deal with very low numerical values in C?
I have to work with values such as 1e-15 in my code but I can't. Indeed, because of low precision these values are equivalent to 0. Any ideas?
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Precision loss in Matrix-Vector product when applying Finite-Difference scheme
I am applying a 6th order Finite-Difference differentiation scheme as seen in http://www.scholarpedia.org/article/Method_of_lines/example_implementation/dss006
There seems to be severe numerical/...
