My personal rule of thumb is that for a given level of accuracy the
most efficient method is the one where the order of the method is
equal to the number of digits of accuracy being sought.

Don't forget that Intel machines have the extended data type which is
an 80 bit floating point number allthough that data type is not
supported by evey compiler.

On Sat, 06 Jan 2007 11:46:40 GMT, Gunnar G wrote:

Do you know how can I calculate the order of the any numerical methods
using stability function.
For example for implicit/backward euler q(z)=1/1-z.

If I am not able to do it in this way please gove me another suggestion
with example.

Thx,
Magda

Gunnar G wrote:

Besides the usual CAS like Maple or Mathematica or others there are
several free software libraries extending numerics to high precision,
may be you try to google.

If you are on Windows you may consider LCC as a complete environment,
it gives you ~ 100 decimal places. It does not have many numerical
examples, but after a while one gets used to it and some of its special
handling. You will be able to compile DLLs callable from other, usual
Windows applications. It is 'free' in a non-commercial sense.

Hi,

I assume that the following inequality holds true for any real
symmetric 2x2-matrices A1,A2,B1,B2

trace( sqrtm( (A1+A2)^2 + (B1+B2)^2 ) )
<=trace( sqrtm( A1^2+B1^2 ) ) + trace( sqrtm( A2^2+B2^2 )
).

(sqrtm means the unique square root of a symm. pos. semi-definite
matrix)
Note that A1,A2,B1,B2 need not be pos. (semi-) definite.

Does anybody have an idea how to prove this inequality (or find a
counterexample)?

Remark: The matrix

sqrtm( (A1+B1)^2 )+sqrtm( (A2+B2)^2 )-sqrtm( (A1+A2)^2 + (B1+B2)^2
)

is in general not pos.semi-definite. Furthermore, I think the
inequality holds for n x n matrices, too.

Thanks,

Simon

I am trying to solve:
x^4-4x^3+8x+3=0
first i put x+y+4
and ended up with y^4-96Y^2-504Y-733
This seems weird to me.
How should I proceed?
I am followin KA Stroud 'Further Engineering Mathematics' Prog 2
Section 82.4
Any help appreciated.
I am an engineering trying to revise my degree maths after 31 years ..
not a pure mathematician!