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## warning messages – Lagrange interpolation of 2000 equidistant points on \$frac{1}{1+x^2}\$ crashes mathematica/does not evaluate

My Code:

``````Points := Table(-5 + 10 j/2048, {j, 1, 2047})
Data := Table(1/(1 + x^2), {x, Points})
Data2 := Transpose({Points, Data})
p(x_) = InterpolatingPolynomial(Data2, x)
``````

I am not sure why this crashes Mathematica. It seems it cannot handle this. I have a HW assignment that requires for me to find the interpolating polynomial of $$frac{1}{1+x^2}$$ at the points $$-5+10j/2048$$ for $$j$$ between 1 and 2047. The code above is my best attempt at doing so. I could not figure out how to make a table of the form $${{x_1,y_1}…}$$ so I just combined the two sets by transposition. Mathematica handles The first three lines properly. However, when trying to evaluate

`p(x_) = InterpolatingPolynomial(Data2, x)`

after about 2-3 minutes crashes. I am not sure what the issue is. Is this impossible for mathematica? I tried the command parallelize but that di not help. The goal of the problem is to find the error of this interpolatoin, and compare it to the error of interpolation using chebyshev points (which I will try doing after I can do this)

## Is it wrong to consider SCAN DENSITY as a performance metric in higher versions of SQL Server apart from SQL Server 2000?

• `DBCC SHOWCONTIG` – introduced in SQL Server 2000
• `sys.dm_db_index_physical_stats` – introduced in SQL Server 2005

Question: Is it wrong to consider scan density as a performance metric in higher versions of SQL Server apart from SQL Server 2000? I am looking for an answer
whether SCAN density was or was not something we should be concerned about in higher versions of SQL Server (> SQL Server 2000), as this metric was rolled out in DMVs and functions.

Background: I have a requirement to maintain scan density of more than 90% in on of my SQL Server database (SQL Server 2016) as part of vendor’s requirement.

However, I can see in the Microsoft site, `DBCC SHOWCONTIG` as a deprecated feature which will be removed in future version of SQL Server and I don’t see the SCAN DENSITY column in function `sys.dm_db_index_physical_stats` (which is the replacement of `DBCC SHOWCONTIG`).

Example:

For one of my table with 12954223 rows, below are the stats for clustered index (page count – 830943) with different queries:

With `DBCC SHOWCONTIG`:

``````Scan density - 76.11 %(103868:106468)
Logical scan fragmentation - 4.68%
``````

With `sys.dm_db_index_physical_stats`

``````avg_page_space_used_in_precent - 92.84 %
agg_fragmentation_in_precent - 4.68%
``````

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