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cosolver/basis_transformation/FMlib/SampleFM.chk
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Sample 1. Real root of f(x) = x**5 - 3x**4 + x**3 - 4x**2 + x - 6 = 0.
Iteration Newton approximation
0 3.120000000000000000000000000000000000000000000000000000000000
1 3.120656718532108533919391265947916793506741449899073468862023
2 3.120656215327022122238354686569835883519704471397219749798884
3 3.120656215326726500470956115551705969611230193197937042123082
4 3.120656215326726500470956013523797484654623935599078168006617
5 3.120656215326726500470956013523797484654623935599066014988828
6 3.120656215326726500470956013523797484654623935599066014988828
Sample 2. Find the root above to 500 decimal places.
3.12065621532672650047095601352379748465462393559906601498882843581902649995179546
89783257450017151095811923431332682839420040840535954560118152245371792881305271951
01711893889821240366205830730398354737691328200011005827350420283867070989561927541
34845215492825918911569452007894158183875295120109996021551313210767970990266642369
92803703462570149559724389392331827597552460610612200485579529156910428115547013787
71442370857816102564155509748117996917502839010590478683168012838433114325930915557
71717
Sample 3. 100 terms were added in the zeta sum.
zeta(3) = 1.202056903159594285399738161511449990764986292340498881792272
Sample 4. 25 values were checked before finding a prime p.
p = 9552131129056058313103536357738804884840825498503088946760768419490591
Sample 5. Log integral. Estimate the number of primes less than 10**30.
It should be accurate to about 15 significant digits.
int(li(1.0e+30)) = 14692398897720447639079087669
Sample 6. Check that gamma(1/2) = sqrt(pi).
gamma(1/2) = 1.772453850905516027298167483341145182797549456122387128213808
Sample 7. Psi and polygamma functions.
Sum (n=1 to infinity) 1/(n**2 * (8n+1)**2) =
.013499486145413024755107829105035147950644978635837270816327
Sample 8. Incomplete gamma and gamma functions.
Probability = .193733130114871446327510259182505999534723186071213869730663
Sample 9. Error function. Probability that a value drawn from a normal
distribution is within k standard deviations from the mean.
k = 1, probability = .68268949213708589717046509126407584495582593345321
k = 2, probability = .95449973610364158559943472566693312505644755259664
k = 3, probability = .99730020393673981094669637046481004524434126368324
Sample 10. Eigenvalue from matrix powers.
Iteration eigenvalue approximation
1 24.238372093023255813953488372093023255813953488372093023256
2 23.913064457596406344892377530895927499788142161414894216112
3 23.912767173080067549422508320051584489821761621077098788515
4 23.912767172321328589362041859914215096468342615030281071820
5 23.912767172321328589357039228003304505549129411762921508588
6 23.912767172321328589357039228003304505549129195999272982168
7 23.912767172321328589357039228003304505549129195999272982168
The corresponding eigenvector is
.261767370446674646792962130450573876653229364533233926774
.427657441555113861658087609816426954331976576625316869561
.702622032727792741311136111781916313570256073497278071852
.331326981028071028200774826381622268381700049086256824155
.380961503593547602959695624571404019706840413977365377249
Sample 11. Function and subroutine example.
Integral = .520499877813046537682746653892
Sample 12. Complex root of f(x) = x**5 - 3x**4 + x**3 - 4x**2 + x - 6 = 0.
Iteration Newton approximation
0 .560000000000000000000000000000 + 1.060000000000000000000000000000 i
1 .561964780980333719745880263787 + 1.061135231152741154895778904059 i
2 .561958308372772219534516409947 + 1.061134679566247415769456345141 i
3 .561958308335403235495113920123 + 1.061134679604332556981397796290 i
4 .561958308335403235498111195347 + 1.061134679604332556983391239059 i
5 .561958308335403235498111195347 + 1.061134679604332556983391239059 i
Sample 13. 44 terms were added to get exp(1.23-2.34i).
Result= -2.379681796854777515745457977697 - 2.458032970832342652397461908326 i
Sample 14. Exception handling.
Iterate exp(x) starting at 1.0 until overflow occurs.
Iteration 1 2.7182818284590452353602874713526624977572M+0
Iteration 2 1.5154262241479264189760430272629911905529M+1
Iteration 3 3.8142791047602205922092195940982035710239M+6
Iteration 4 2.3315043990071954622896899110121376663320M+1656520
Iteration 5 + OVERFLOW
Overflow was correctly detected.
All results were ok -- no errors were found.
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