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4 votes
Accepted

Is it possible to apply the matrix inverse $(H(\textrm{Id}+F^T F)^{-1}H^T)^{-1}$ to a vector without explicitly calculating the inverse?

Yes. You wish to compute $$x = (H(I + F^TF)^{-1}H^T)^{-1}y$$ for any $y$. You do this by solving the linear system $$\begin{bmatrix} I + F^TF & H^T \\ H & 0 \end{bmatrix} \begin{bmatrix} ...
Carl Christian's user avatar
3 votes
Accepted

If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?

$$\begin{align*} \log_{225}70&=\log_{225}(2\times 5\times 7)=\log_{225}2+\log_{225}5+\log_{225}7\\ &=\dfrac{1}{\log_2 225}+\dfrac{1}{\log_5 225}+\dfrac{1}{\log_7 225}\\ &=\dfrac{1}{2\log_2 ...
rushusuixing's user avatar
3 votes
Accepted

Find all polynomials in $\mathbb{P_n}$ such that the Fixed point iteration converges for all initial guesses

If $\deg(f) \ge 2$ then $p(x)/x \to \infty$ for $n \to \infty$, so that $|p(x)| \ge 2 |x|$ for all sufficiently large $x$. It follows that the fixed point iteration does not converge for all initial ...
Martin R's user avatar
  • 114k
3 votes

Evaluate or estimate $\int_{8}^{27}\frac{\mathrm{d}x}{\sqrt{x}-\sqrt[3]{x}}$ without a calculator in a fast way

Evaluating $2\sqrt{x}+3\sqrt[3]{x}+6\sqrt[6]{x}+6\log(\sqrt[6]{x}-1)$ at $x=27$ we get $$2\sqrt{27}+3\sqrt[3]{27}+6\sqrt[6]{27}+6\ln(\sqrt[6]{27}-1)$$ As $27=3^3$ , we have $\sqrt{27}=\sqrt{3^3}=3\...
Julio Puerta's user avatar
  • 7,707
2 votes

If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?

So we could rewrite and create new logarithm terms with respect to a,b and c. In the question we need $$\log_{225} 70 .$$ Here 225 can be written as 3x3x5x5. Amd 70 can be written as 2×5×7. Now we ...
Mahit Chopra's user avatar
1 vote

Evaluate or estimate $\int_{8}^{27}\frac{\mathrm{d}x}{\sqrt{x}-\sqrt[3]{x}}$ without a calculator in a fast way

For a multiple choice question, an exact answer isn't required. A very rough estimate using numerical integration may be performed using the midpoint rule with one rectangle. This is the area of a ...
Marc Shelikoff's user avatar
1 vote

Evaluate or estimate $\int_{8}^{27}\frac{\mathrm{d}x}{\sqrt{x}-\sqrt[3]{x}}$ without a calculator in a fast way

In the $u$-integral, substituting $u\to\dfrac1u$ produces an integrand with a simple partial fraction expansion, namely $$\int_{\sqrt2}^{\sqrt3} \frac{6u^3}{u-1} \, du \stackrel{u\to\tfrac1u}= \int_\...
user170231's user avatar
  • 19.8k
1 vote

finite diffrence scheme in two dimension

You are considering an explicit scheme, why do you feel the need to have a matrix representation? You can just implement your recursive formula exactly as it is. If you still want a matrix ...
PierreCarre's user avatar
  • 21.1k
1 vote
Accepted

Best uniform approximation of $x^{n+2}$ in $\mathbb{P_n}$

The same argument still works because the Chebyshev polynomials $T_k$ only consist of odd or even-degree terms. The $(n+2)$th Chebyshev $T_{n+2}(x) = 2^{n+2}( x^{n+2} + a_n x^n + a_{n-2} x^{n-2} + \...
Yimin's user avatar
  • 3,406
1 vote

If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?

As an alternative way, using exponential, we have that $\log_7 5 = a \iff 7^a=5$ $\log_5 3 = b \iff 5^b=3$ $\log_3 2 = c\iff 3^c=2$ then $$\log_{225}70 =x \iff 225^x=70 \iff (3^2\cdot 5^2)^x=(2 \...
user's user avatar
  • 156k

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