For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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2
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3answers
120 views

Approximation of a sum $\sum ^{\sqrt{n} }_{k=5}\frac{\log\log(k)}{k\log(k)} $

What method could I use to obtain an approximation of this sum $$\sum^{\sqrt{n}}_{k=5}\frac{\log\log(k)}{k\log(k)}$$ Should I proceed by an integral? How can I calculate its lower and upper bound?
0
votes
1answer
15 views

approximation using floating point arithmetic

Let $x=2.14366$ and $y=2.14363$ and $d=x-y.$ If $d*$ is the value of d computed using $5-$digit decimal floating point arithmetic, find the relative error. For this question I know how to calculate ...
0
votes
3answers
42 views

Approximation of an expression (no calculator please!)

Today I had my college admission exam, It was good, but there was a question which I found a bit interesting (but unable to solve at the moment). It says, Question: Find the positive integer which ...
0
votes
2answers
30 views

Can step functions approximate trigonometric functions?

I have read a notion from a number of different sources simply stating that a step function can approximate any trigonometric function. I am not convinced by simply reading this notion, for example ...
0
votes
0answers
51 views

Equivalence of two linear least squares problem

Here I want to build a subspace representation $Uq$ to approximate $x$, where $q$ is the reduced coordinates. We know that the best approximation to $x$ is the linear least squares solution $q_1 = ...
2
votes
1answer
31 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
3
votes
3answers
248 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
1
vote
1answer
47 views

Need help with a Crank Nicholson Method example problem.

I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have ...
0
votes
1answer
24 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
2
votes
1answer
34 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
0
votes
0answers
10 views

Approximating an Appell series with a second-order polynomial

Taking this Appell F1 hypergeometric series $$ f(t)=F_1\left(2;\frac{3}{2} (1-m),\frac{3}{2} (\lceil r\rceil +m-1);3;\frac{\frac{r}{2} t^2 \left(1-\frac{r}{\lfloor r\rfloor +m}\right)+\lfloor r\rfloor ...
1
vote
1answer
27 views

Any function in $L^p$ space is a linear combination of simple functions? True OR not?

Any function in $L^p$ space is a linear combination of simple functions for $1<p<\infty$. Is this true? So any function in $L^p$ is measurable. So any measurable function can be represented ...
0
votes
1answer
24 views

How to determine if an equation represents a cubic spline?

Given the equation $$ f(x) = \left\{ \begin{array}{lr} 2x^3+x^2+4x+5 & : 0 \le x \le 1\\ (x-1)^3 + 7(x-1)^2 + 12(x-1)+12 & : 1 \le x \le 2 \end{array} \right. ...
0
votes
1answer
115 views

Why are these sums approximately equal?

Let $T$ be a finite set. Let $\rho:T\rightarrow (0,1)$ be such that $\sum_{t\in T}\rho(t)=1$. Let $F:\mathbb N\cup\{0\}\rightarrow(0,1)$ be such that $\sum_{i=0}^\infty F(i)=1$. Let ...
7
votes
4answers
108 views

How to evaluate $\log x$ to high precision “by hand”

I want to prove $$\log 2<\frac{253}{365}.$$ This evaluates to $0.693147\ldots<0.693151\ldots$, so it checks out. (The source of this otherwise obscure numerical problem is in the verification ...
0
votes
1answer
19 views

How to construct a continuous function that is (mean) convergent to a given square integrable function

(In the Riemann Sense, this is a lemma before the Fouriers-Mean-Convergence Theorem) Suppose we have a square integrable function f:$[0,2\pi]\rightarrow \mathbf{C}$. We know that $\int_{a}^{b} f^2 ...
1
vote
1answer
29 views

An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
2
votes
2answers
81 views

Terms needed to approximate with given error?

How many terms of this series would one need to add to get an approximation of $\pi$ with error less than $10^{-4}$? $$ 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots $$ So far, I wrote the ...
0
votes
1answer
13 views

Approximation of monthly payment using Taylor expansion

I am trying to understand what does APR(annual percentage rate) and how it is calculated. Thanks to Wikipedia, I got the formula of monthly payment for a fixed rate multi-year mortgage in the ...
0
votes
0answers
19 views

Numerical integration to find mean arterial pressure, as used in cardiovascular physiology

In physiology the mean arterial pressure (MAP) is calculated as $MAP = DP + \frac1{3}PP = DP + \frac1{3}(SP-DP)$ (where PP is pulse pressure, DP is diastolic pressure and SP is systolic pressure), and ...
15
votes
3answers
2k views

How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks
3
votes
1answer
48 views

Bounds on Maclaurin series of $e^{-x^2}$

This is a problem from a textbook: By taking the 4th degree Maclaurin polynomial for $e^{-x^2}$ find an approximation to $\int^1_0 e^{-x^2} \text{dx}$. Place bounds on the error in this ...
0
votes
0answers
16 views

Maclaurin polynomial error term

this is a problem from a textbook, What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}$ The answer from textbook is $n=17$, but I ...
0
votes
0answers
25 views

Computing integrals in order to find an approximation function

For a project in scientific computing I am trying to find an approximation of an unknown function $f(x)$. Given: data points $(x, f(x))$ A basis with which we can approximate $f(x)$ consists ...
0
votes
0answers
32 views

Uniqueness of best approximation. (the sketch of the proof)

Let $X$ be a compact Hausdorff space. Let $A = C(X)$, the space of real-valued continuous functions with supremum norm. Prove : if $X$ has at least 2 points, then there is a one-dimensional subspace ...
-2
votes
2answers
54 views

Integral approximation - [closed]

Whole day I can not figure out how can be proved the equality: $$\int_0^1 x^2 dx = \frac{1}{n} \sum_{i=1}^n \left(\frac{2i-1}{2n}\right)^2 + \frac{1}{12n^2}$$ Can someone help me, what should I use ...
0
votes
0answers
19 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
0
votes
0answers
20 views

the set of Best Approximation is a norm-closed convex set.

Let $V$ be a normed vector space. Let $W$ be a proper closed subspace of $V$. Let $M$ be the set of best approximations in $W$. Prove that $M$ is a norm-closed convex set. I've shown that the ...
0
votes
0answers
14 views

how to understand 'expected max approximation error'

The background is that: E() denotes the expectation and $y$ satisfies a certain probability distribution $g(y)$, then we independently sample $y_1,y_2$ from $g(y)$. It is assumed that $E(y_1-E(y))=0, ...
0
votes
0answers
17 views

Approximation of a trisecting an angle

I learned a proof that it is impossible to trisect an angle. Is there some research that if we have been given an angle, a ruler and a compass and we are allowed to draw $m$ circles and $n$ lines/line ...
1
vote
2answers
44 views

Why does this approximation of square roots using derivatives work this way?

I came up with this way to estimate square roots by hand, but part of it doesn't seem to make sense. Consider how $f(n) = \sqrt{n^2+\varepsilon} \approx n$ when $\varepsilon$ is small. Therefore, ...
0
votes
0answers
80 views

set cover problem: d-cover

I've the following problem: Let the universe U and a list of subsets (S_1,..., S_k), be the usual input for the (unweighted) SET COVER PROBLEM, Consider the following doubling scheme for producing a ...
0
votes
1answer
41 views

Error of lagrange interpolation

If the original function I want to approximate using Lagrange interpolation is a polynomial the error function $(x-x_{0})...(x-x_{n})\frac{f^{(n+1)}(\xi)}{(n+1)!}$ is not working because the $n+1$ ...
1
vote
1answer
20 views

Can $e^\frac{1+z}{z-1}$ be uniformly approximated in the disc by harmonic continuous functions

Ie: For $f(z)=e^\frac{1+z}{z-1}$ is it true: $\inf\{||f-u||: \mbox{u is harmonic in the unit disc and continuous on the closed unit disc}\}=0$. Note the infinum is taken over the interior (ie: ...
2
votes
1answer
23 views

Boundary of the limit when $a<<1$

Let's say I have the expression $-b+\frac{3}{2}b\cdot a^2$. Can I say that by taking the limit when $a<<1$, that expression is $\approx -b$ ? can a constant number, like $\frac{3}{2}$ can "ruin" ...
1
vote
1answer
35 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
0
votes
3answers
31 views

Use a linear approximation to estimate the given number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
1
vote
2answers
48 views

Use a linear approximation to estimate the number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
1
vote
0answers
15 views

How to analytically find these rounding issues

Let's say we have a fixed yearly amount that we have to divide equally among an amount of days. For instance for $1,600 we may have: ...
1
vote
1answer
21 views

Approximate using differentials when partial derivatives are given?

I have ran into this problem on my online math assignment, this week we are covering partial derivatives and higher order partial derivatives, but I don't think I have learnt anything that can help me ...
2
votes
1answer
50 views

least squres vs. lagrange interpolation

can some one tell me the differences between these two approximation techniques, what are the strengths and weaknesses of each, and which better one to use. Thanks
0
votes
1answer
26 views

How does $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximate $\mathbf{H}$?

Page 3 of a guide to Levenberg-Marquardt optimization says that $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximates the Hessian matrix of $f$. I do ...
3
votes
0answers
21 views

Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
0
votes
0answers
32 views

Approximation to the negative-binomial and negative-hypergeometric distributions

It is known that a binomial distribution can be approximated by a normal variable with the same mean and variance, for sufficiently large $n$. What approximations are known for the negative binomial ...
1
vote
1answer
42 views

Approximate an integral using Monte Carlo method

I have a question on an assignment Calculate the value of the integral I = $\int_0^\pi sin^2(x)dx$ using the Monte Carlo Method (by generating $ 10^4 $ uniform random numbers within domain [0, ...
2
votes
3answers
30 views

Errors and Taylor Polynomials

For $g(x)=x^{1/3}$, $a=1$, degree $3$ I found the Taylor polynomial: $$p_3(x) = 1 + (x-1)/3 - ((x-1)^2)/9 + (5(x-1)^3)/81$$ How do I use the error formula for the Taylor polynomial of degree 3 to ...
0
votes
1answer
43 views

if $x\ll 1$ is it safe to assume that $x\ll \frac{1}{2}$

I know that: if $x\ll 1$ then we can write $\frac{x}{x+1}\rightarrow x$ but is it safe to write $\frac{2x+1}{x+1}\rightarrow 1$?
0
votes
0answers
53 views

Historical calculations of $tan^{-1}x $ and $e^x$

SineBhaskara_I One reads that $tan^{-1}(x) $ series expansion existed in early (Indian) history. But like the Sine trigonometric function, did any similar approximation exist as well? The query ...
2
votes
4answers
128 views

Parabolic sine approximation

Problem Find a parabola ($f(x)=ax^2+bx+c$) that approximate the function sine the best on interval [0,$\pi$]. The distance between two solutions is calculated this way (in relation to scalar ...
0
votes
1answer
33 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...