`Rationalize(878159.58,1*10^-12) - Rationalize(431874.32,1*10^-12) // N` returns wrong result

[tranleduy2000]

There is the input:

Rationalize(878159.58,1*10^-12) - Rationalize(431874.32,1*10^-12) // N

That gives wrong result:

𝟦𝟦𝟨𝟤𝟪𝟧.𝟥

Expected result:

446285.25999999995

[tranleduy2000]

There is too much precision loss for the following input:

m = {{-2,-2,4},
{-1,-3,7},
{2,4,6}};

s= Transpose[Eigenvectors[m]];

j = {{1+Sqrt[61],0,0},{0,1-Sqrt[61],0},{0,0,-1}};

s1 = Inverse[s];

s.j.s1 // N

The result should be

{{-2.0,-2.0,4.0},
{-1.0,-3.0,7.0},
{2.0,4.0,6.0}}

[axkr]

That gives wrong result:

𝟦𝟦𝟨𝟤𝟪𝟧.𝟥

It's only the "output format" if you get the result as "input format"

Rationalize(878159.58,1*10^-12) - Rationalize(431874.32,1*10^-12) // N //InputForm

it returns 446285.26

[axkr]

s.j.s1 // N

You can get a better result, if you increase the precision:

N[s.j.s1 ,30]

A simplification before the numeric calculation may also get a better result.

s.j.s1 // FullSimplify // N

[axkr]

The result should be

{{-2.0,-2.0,4.0},
{-1.0,-3.0,7.0},
{2.0,4.0,6.0}}

See test case testNIssue1065()

• https://github.com/axkr/symja_android_library/blob/e4ef9465816175b96116734148e6623f8f7fcc17/symja_android_library/matheclipse-core/src/test/java/org/matheclipse/core/system/LowercaseTestCase.java#L15014

with Config.USE_EXTENDED_PRECISION_IN_N = true; you can enable using apfloat for calculating in better precision.

[axkr]

with Config.USE_EXTENDED_PRECISION_IN_N = true; you can enable using apfloat for calculating in better precision.

Note: Because of using Simplify for evaluating the Dot(...) function:

• https://github.com/axkr/symja_android_library/blob/ff2a51167fa5bb9b2de8457f0af81a2a1333e310/symja_android_library/matheclipse-core/src/main/java/org/matheclipse/core/convert/Convert.java#L736

it's not necessary anymore to use Config.USE_EXTENDED_PRECISION_IN_N = true at least for this Eigenvectors example.