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Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.

A Systems Challenge among various computer algebra systems was held recently at ISSAC (International Symposium on Symbolic and Algebraic Computations) '97. The ISSAC '97 conference, held in Maui, Hawaii, on July 23, was sponsored by ACM SIGSAM and ACM SIGNUM and in federation with PASCO '97.

Below are statements of the original problems together with the Mathematica solutions.

You can also download the solutions in a Mathematica notebook. To do this you need Mathematica. It is also possible to download MathReader to view the notebook. If you would like to receive a printed version of the solutions, please send email to ISSACSolutions@wolfram.com and include your fax number or postal address.


Problem 1

What is the 4-significant-digit approximation to the condition number of the 256 by 256 Hilbert matrix?

Result
[Graphics:ISSACChallengegr1.gif]

See Mathematica solutions.


Problem 2

What is the value of [Graphics:ISSACChallengegr30.gif] to 7 significant digits?

Result
[Graphics:ISSACChallengegr31.gif]

See Mathematica solutions.


Problem 3

What is [Graphics:ISSACChallengegr43.gif]to 14 significant digits?

Result
21.19324037771154...

See Mathematica solutions.


Problem 4

What is the coefficient of [Graphics:ISSACChallengegr50.gif] in the expansion of the polynomial [Graphics:ISSACChallengegr51.gif] to 13 significant digits?

Result
[Graphics:ISSACChallengegr52.gif]

See Mathematica solutions.


Problem 5

What is the largest zero of the 1000[Graphics:ISSACChallengegr74.gif] Laguerre polynomial to 12 significant digits?

Result
3943.24739485...

See Mathematica solutions.


Problem 6

Find a lexicographic Gröbner basis for the following polynomial system.

[Graphics:ISSACChallengegr104.gif]

Result
[Graphics:ISSACChallengegr105.gif]

See Mathematica solutions.


Problem 7

What is [Graphics:ISSACChallengegr122.gif] to 9 significant digits?

[Graphics:ISSACChallengegr123.gif]
[Graphics:ISSACChallengegr124.gif]

Result
[Graphics:ISSACChallengegr125.gif]

See Mathematica solutions.


Problem 8

What is [Graphics:ISSACChallengegr142.gif]?

Result
[Graphics:ISSACChallengegr143.gif]

See Mathematica solutions.


Problem 9

Find the largest eigenvalue lambda to 13 significant digits for the following integral equation.

[Graphics:ISSACChallengegr158.gif]

Result
lambda = 37.5291455603353...

See Mathematica solutions.


Problem 10

Consider the following initial value problem.

[Graphics:ISSACChallengegr178.gif]
[Graphics:ISSACChallengegr179.gif], [Graphics:ISSACChallengegr180.gif]

Find the smallest positive number r such that the solution has a derivative singularity at x = r. Calculate r to 13 significant digits. Is y(r) infinite or finite? If y(r) is finite, then compute it to 13 significant digits.

Result
r = 1.6443766903388...
y(r) = 0.93193876511028...

See Mathematica solutions.


Timings and Memory


References