7
votes
1answer
290 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
28
votes
1answer
527 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
8
votes
3answers
251 views

Continued fractions for $\sqrt{x} $ and beyond, valid formula?

For $x > 0$, is this trick valid? I use $$ ( \sqrt{x}-1)(\sqrt{x}+1)=x-1 $$ then $$ \sqrt{x}+1 = \frac{x-1}{\sqrt{x}+1-2} $$ so I can use iterations to get the rational approximant $$ \sqrt{x} ...
1
vote
1answer
161 views

How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?

I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products: $A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
1
vote
1answer
240 views

Continued fractions with rational functions

Express the following rational function in continued-fraction form: $${4x^2+3x-7\over 2x^3+x^2-x+5}$$ The answer is : $${4 \over 2x- \frac{1}{2}} + { \frac{23}{8} \over ...
1
vote
2answers
163 views

Continued Fractions Approximation

I have come across continued fractions approximation but I am unsure what the steps are. For example How would you express the following rational function in continued-fraction form: $${x^2+3x+2 ...
4
votes
0answers
259 views

Evaluating matrix-continued fractions?

I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...