Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Construct a sequence of interpolating values $Y_n, to,f(1 + \sqrt{10})$, where $f(x) = (1 + X^2)^{-1}$ for $-5 \leq X \leq 5$, as follows: For each n = 1,2, ... ,10, let $h = \frac{10} {n}$

$x_j^n= -5 + jh$, for each j = 0, 1,2, ... ,n. What would my sequence be exactly. I have come up with something but it seems incorrect.

The first term would be : $-5+0*\frac{10} {1}$

The second ther would be : $-5+1*\frac{10} {2}$

And so forth. But I have a feeling that this is incorrect.

share|cite|improve this question
what kind of interpolation are you using? – robjohn Sep 19 '12 at 14:56
Lagrange interpolation / Neville's Method but that is the easy part. Once I figure out what my x-values are then the rest is quite simple. – math101 Sep 19 '12 at 15:03
up vote 3 down vote accepted

It seems as if you are being asked to use $n$ subintervals (therefore, $n+1$ points) to interpolate the function $f$ on $[-5,5]$. Since the width of the interval is $10$, the width of each subinterval is $h=\frac{10}n$. The mesh for $n$ intervals consists of the points $x_j^n=-5+jh=-5+j\frac{10}n$ for $j=0,1,2,\dots,n$.

For a particular $n$, compute $f(x_j^n)$ for $j=0,1,2,\dots,n$ and interpolate to get $f(1+\sqrt{10})$.

For example, if $n=4$, use the $x$ values $\{-5,-2.5,0,2.5,5\}$ and the $y$ values $\{f(-5),f(-2.5),f(0),f(2.5),f(5)\}$.

The interpolating polynomial is $1-\frac{129}{754}x^2+\frac{2}{377}x^4$ and when that is evaluated at $1+\sqrt{10}$, we get $-\frac1{754}(21+82\sqrt{10})\approx-.3717596394$, which is the fourth value your book gives.

share|cite|improve this answer
Ok here is the answer that is in the back of my book: The first 10 terms of the sequences – math101 Sep 19 '12 at 21:36
The first 10 terms of the sequence are $.038462, .333671, .116605, -.371760, -.0548919, .605935, .190249, -.513353, -.0668173, .448335 \text{ and } f(1+ \sqrt{10})=.0545716$ – math101 Sep 19 '12 at 21:47
As you can see these 10 terms are in reality the y-values or f(x). They have been plugged into the equation $(1+x^{2})^{-1}$ – math101 Sep 19 '12 at 21:52
@math101: how are you getting negative values for $(1+x^2)^{-1}$ ? – robjohn Sep 19 '12 at 22:03
@math101: The answers in the back of the book are for the interpolated values of $f(1+\sqrt{10})$ using $n=1,2,3,\dots,10$. For $n=1$ you are only using $f(-5)=\frac1{26}$ and $f(5)=\frac1{26}$, so the value of $f(1+\sqrt{10})=\frac1{26}$. – robjohn Sep 19 '12 at 22:20

You want $11$ equally spaced points from $-5$ to $5,$ which are all the integers: $-5, -4, -3, \ldots ,5$. $h$ should be one value, here $1$, which is the spacing of the points. It should be the length of the interval $(10)$ divided by the number of spaces $(10)$. I don't know where you got $h=\frac{10}n.$ So it should be $-5+0*1, -5+1*1, \ldots -5+10*1$

share|cite|improve this answer
Your answer seems correct but then it doesn't match the answer I have found unless the answer is incorrect. I wonder – math101 Sep 19 '12 at 5:19
$h=\frac{10} {n}$ was given in the question – math101 Sep 19 '12 at 14:18
@math101: that is fine, but it should not be "For each $n=1,2,\ldots ,10$ because $n$ doesn't vary. Maybe this fragment should come before $x_j^n=-5+jh$ – Ross Millikan Sep 19 '12 at 14:40
For some reason it doesnt match the answers given in my book. For some reason the question was worded unclearly and its hard to put the pieces together – math101 Sep 19 '12 at 14:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.