# Questions tagged [machine-precision]

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### Rounding in floating point operations

I'm given this set of floating point numbers: And I'm given the function: I'm then asked to find the value of f for: So I've done this: Now my guess is that I need to convert the exact value (1....
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### Given an analytic function can I derive to what precision I need to evaluate its arguments to get a result of a given precision?

Suppose I have some arbitrary analytic function over the reals e.g.: $$x \mapsto \frac{\sqrt{\sin(x)+2e^x}}{x^2 - \ln(x^x)}$$ Given some input of arbitrary precision $x$ how can I evaluate such an ...
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### Exponencial mantissa

I try to compute a function of type $e^x$. Since its natural fast-growing behaviour, the most natural behaviour is to decompose the resulting number into its mantissa and exponent components. However, ...
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### How to prove this statement concerning Machine Numbers and Primes?

So, let $F=F(b,t,e_{min},e_{max})$ be a range of machine numbers, with b being a Prime Number. Now I have to prove: "Let $x,y$ be two machine numbers ($y\neq0)$ and let $q:=x/y$ be a normalized ...
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### Why is this function not being graphed as expected?

I am interested in the Rastrigin function: https://en.wikipedia.org/wiki/Rastrigin_function. I have read that the local minimums of this function occur at integer coordinates. However, when I plug ...
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### Finding a machine number such that $fl(x)=x(1+\delta)$.

The guide book asks me for A real number $x$ in range of a machine with $\beta=2$ (binary) and $n=24$ (24 mantissa positions), such that it satisfy $fl(x)=x(1+\delta)$, with $|\delta|$ as big ...
269 views

### Machine epsilon: why is $(1 + \epsilon) + \epsilon = 1$?

My book on real analysis has the following statement: I don't understand how the first equation can possibly be true, by definition of machine epsilon. Machine epsilon is defined as the smallest ...
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### How to Calculate the Precision Required to Exactly Produce the Repeating Sequence of Digits in the Decimal Expansion of a Fraction

Say I am given a fraction $\frac{p}{q}$ and wish to express it as a decimal such that the repeating sequence of digits is accurately displayed at least once and preferably twice in the printed output. ...
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### Tough test polynomials for (finite precision) complex root finding methods, especially Aberth's method

Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing ...
112 views

### Floating-point rounding error in numerical differentiation formula

In Numerical Analysis by Timothy Sauer (Pearson, 2nd Edition) it says that $\tilde{f'}(x+h) = f(x+h) + \epsilon_{\text{mach}}$, where $\tilde{f'}(x)$ is the floating-point representation of the given ...
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### Finding the real roots of a univariate polynomial on the interval [0,1]

I have numerous, univariate polynomials with degree in excess of 100 and with very, very large coefficients (Here's an example coefficient ...
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### Bounds on real numbers

Suppose that $\{\epsilon _1 , · · · , \epsilon_m \}$are real numbers that all satisfy $|\epsilon_i | ≤ \eta$. Show that given $C > 1$, we have that $$\Pi_{j=1}^m(1+\epsilon_j) = 1 + \epsilon$$...
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### How to prove the bound for Relative Round Off error

Machine precision is defined as the smallest machine number ε. Anything smaller when added to 1 will be lost at roundoff. Prove that ε is the bound for relative round-off error. ...
299 views

### How to prove fl(x^k) definition

How would I show that fl(x^k)= (x^k)(1 + δ)^(k−1), where |δ| ≤ ,ε if x is a floating-point machine number in a computer that has machine precision . Edit: I was thinking I can use induction to ...
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### Numerical Analysis- Finding two closest machine numbers in single precision.

There is a question that is similar to this that was asked already, but the answer did not really make too much sense to me. Let's say I have a number 4096.000244 and I want to find the two closest ...
256 views

### SVD - near-zero singular value

I have difficulties to handle singular values close to zero. My SVD implementation $A = U \Sigma V {}^{T}$ performs first an Eigen decomposition of the matrix $A {}^{T} A$. That is done with the QR-...
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### Intersection of three planes and precision of computing

I have a simple 3D object made of a triangle mesh. Here is a rendering of the mesh. I would like to make a 'coat' which is another mesh over the primary one, where all new faces have equal distance ...
460 views

### Numerically find cubic polynomial roots where coefficients widely vary in magnitude

Consider the following polynomial: $$p(x) = x^3 + (C_b+K_a)x^2 - (C_aK_a + K_w)x - K_aK_w$$ Where: $x, C_a, C_b$ are concentrations, positive real numbers typically within $[10^{-7};1]$. The ...
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### 3 Circles acting as gears problem! what would be the Formula for calculating the 3rd (the driving) gears radius?

Radius of Circle A is R2 Radius of Circle B is R3 Radius of Circle C is R Distance between the center of Circle A & B is "X" Distance between the plane of the center of Circle A & B and the ...
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### Verification: Machine number immediately to right and left of $2^m$

The question I'm given is what are the machine numbers immediately to the right and left of $2^m$? How far is each from $2^m$? I'm given the machine epsilon, $\epsilon$ is $2^{-23}$. (I believe we ...
223 views

### Convergence of Newton method and machine precision

I implemented the Newton method to find the non-zero root of $f(x) = 1-bx-e^{-x}$ in Excel and I have tested it for various values of $0<b<1$. However, what I am seeing for some values of $b$ (e....
3k views

### Machine Epsilon meaning

Say we have the floating-point system $(2,3,-1,2)$ and we want to find machine epsilon. According to my textbook, this can be found as $\epsilon_m=\beta^{1-t} = 2^{1-3}=0.25$. However, my textbook ...
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### Numerically approximating limit with large numbers, bumping up against machine precision

Consider $\lim_{x\rightarrow\infty}(1+\frac1x)^{x}=e$. Using standard calculus techniques, this limit can be evaluated, however, approximating it directly with numerical code can be difficult ...
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### When will A+(B+C) != （A+B）+ C, in a finite precision system

A, B, C, have finite precisions with machine epsilon of $10^{-16}$. When will the associative law A + (B+C) = (A+B) + C fail in this finite precision system? I have difficulty to find A, B and C. ...
111 views

### Subtracting a diagonal matrix from an ill-conditioned one

I've written Python 3 + NumPy code where I perform the following calculation: $$(B - \lambda I)^{-1}$$ where $B$ is a symmetric negative semidefinite matrix where some elements can be huge (greater ...
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### Why are SDP solvers inherently inaccurate?

It's common to read in textbooks that semidefinite programming solvers are inherently inaccurate. Are the authors referring to the general machine inaccuracy (things like $10^{-16}=0$) or a special ...
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### Spreadsheet: Significant figures function [closed]

1 AU is 149597870700 meters. This is much more information than typically needed, so 150 million kilometers would be precise enough in most situations. Does spreadsheets have a function to handle ...
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### Numerically stable way to calculate (a-b)/(c-d) where a~=b and c~=d

Is there a known general numerically-stable way to calculate $\frac{a-b}{c-d}$, where a is very close to b and c is very close to d, and all variables are stored as floating-point with some precision? ...
Im having a debate between me and my study group regarding a question of numerical representation. $x = 0.002718281828459$, $\beta = 10$, $p = 6$ We are told to find $\text{fl}(x)$, $\beta$ is base ...