Maple Questions and Posts

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How can I fix the error of ChangeOfVariables: Error, (in Student:-MultivariateCalculus:-ChangeOfVariables) unable to solve the change-of-variables equations for the original variables ?

Thanks!

Hi

I have trouble with solving this ODE system using dsolve command:

and 

 

This system have following solutions:

where

and

C's and A are constants of integration.

 

They're equations from this paper https://arxiv.org/abs/1710.01910 (45 and 47). 
               

However, my solution differs from correct one - in output there are hypergeometric functions everywhere.

Is there any way to fix/convert this solution? Or to get rid of these functions (my f1 solution looks very close to original one but with coupled hypergeometric function). 
 

`` ``

``

``

 

``

sysode := 2*q*(3*q-1)*f1(tau)/tau^2+2*q*(diff(f1(tau), tau))/tau+diff(f1(tau), tau, tau)+(kappa^2+f2(tau))*(1+omega)*(tau/t0)^(-(3*(3+omega))*q) = 0, (54*q^3-30*q^2+4*q)*f1(tau)/tau^3+(24*q^2-4*q)*(diff(f1(tau), tau))/tau^2+11*q*(diff(f1(tau), tau, tau))/tau+diff(f1(tau), tau, tau, tau)-3*omega*(1+omega)*(kappa^2+f2(tau))*q*(tau/t0)^(-(3*(1+omega))*q)/tau = 0;

2*q*(3*q-1)*f1(tau)/tau^2+2*q*(diff(f1(tau), tau))/tau+diff(diff(f1(tau), tau), tau)+(kappa^2+f2(tau))*(1+omega)*(tau/t0)^(-3*(3+omega)*q) = 0, (54*q^3-30*q^2+4*q)*f1(tau)/tau^3+(24*q^2-4*q)*(diff(f1(tau), tau))/tau^2+11*q*(diff(diff(f1(tau), tau), tau))/tau+diff(diff(diff(f1(tau), tau), tau), tau)-3*omega*(1+omega)*(kappa^2+f2(tau))*q*(tau/t0)^(-3*(1+omega)*q)/tau = 0

(1)

``

``

simplify(dsolve([sysode], build));

{f1(tau) = _C1*tau^(-q+1/2-(1/2)*(-20*q^2+4*q+1)^(1/2))+_C2*tau^(-q+1/2+(1/2)*(-20*q^2+4*q+1)^(1/2))+_C3*tau^(-9*q+2)*hypergeom([-(1/12)*(16*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(-16*q+(-20*q^2+4*q+1)^(1/2)+3)/q], [-(1/12)*(4*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(-4*q+(-20*q^2+4*q+1)^(1/2)+3)/q], -(1/2)*(tau/t0)^(6*q)*omega), f2(tau) = (-695520*(q^2+(11/21)*q+2/21)*(tau/t0)^(3*q*(omega+5))*_C3*(q-3/10)*omega*q*(q^2-(25/69)*q+2/69)*tau^(-9*q)*hypergeom([-(1/12)*(4*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(-4*q+(-20*q^2+4*q+1)^(1/2)+3)/q], [-(1/12)*(-8*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(8*q+(-20*q^2+4*q+1)^(1/2)+3)/q], -(1/2)*(tau/t0)^(6*q)*omega)-89424*(q^2*(tau/t0)^(3*q*(omega+7))*omega^2*tau^(-9*q)*_C3*(q^2-(25/69)*q+2/69)*hypergeom([-(1/12)*(-8*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(8*q+(-20*q^2+4*q+1)^(1/2)+3)/q], [-(1/12)*(-20*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(20*q+(-20*q^2+4*q+1)^(1/2)+3)/q], -(1/2)*(tau/t0)^(6*q)*omega)+(7/3)*(q^2+(11/21)*q+2/21)*(hypergeom([-(1/12)*(16*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(-16*q+(-20*q^2+4*q+1)^(1/2)+3)/q], [-(1/12)*(4*q+(-20*q^2+4*q+1)^(1/2)-3)/q, (1/12)*(-4*q+(-20*q^2+4*q+1)^(1/2)+3)/q], -(1/2)*(tau/t0)^(6*q)*omega)*_C3*(tau/t0)^(3*(3+omega)*q)*(q^2-(25/69)*q+2/69)*tau^(-9*q)+(1/69)*kappa^2*(1+omega)))*(q^2-(7/9)*q+2/9))/((1+omega)*(4*q-(-20*q^2+4*q+1)^(1/2)-3)*(8*q-(-20*q^2+4*q+1)^(1/2)+3)*(8*q+(-20*q^2+4*q+1)^(1/2)+3)*(4*q+(-20*q^2+4*q+1)^(1/2)-3))}

(2)

NULL

NULL

``

NULLNULL

NULL

NULL

NULL

NULL

``


 

Download question.mw

I know you can call python from Maple, I am thinking if there is the other way around. That is use Maple (and its toolbox) as backend engine to do calculations (e.g. Global Optimization), and say manipulate the data in Python as the front-end.

Hello,

My question is mathematical in nature, so it might be a little out of place but I though I would give it a shot. 

You have a series of chebyshev coefficients in two connecting subdomains lets say S1 = [0,0.5] and S2=[0.5,1]. So far you are still in the spectral space. If you want to compute the solution in real space you can sum the coefficients with the Chebyshev polynomials. 

Now imagine you change the interval to S1 = [0,0.6] and S2 = [0.6,1]. Is there a way to manipulate the Chebyshev coefficients from both initial subdomains to create a new set of Chebyshev coefficients that fit the solution in the new subdomains. 

The brute force method would be to create the real solution of Chebyshev polynomials and then use that to form a new set of Chebyshev coefficients. Or you can use Clenshaw to compute the solution at several points, and then use the points to create new Chebyshev coefficients.

But what if we can stay in spectral space and create the new chebyshev coefficients. Is that possible? If so, how?

Hi, 

When creating a user random variable, I would like to instanciate some of its attributes, for instance ParentName.
But it seems that it's not always possible.

​​​​​​​Is it a Maple's limitation or am I not doing the things correctly ?
​​​​​​​
Example:
 

restart:

with(Statistics):

U := RandomVariable(Uniform(0, 1)):

interface(warnlevel=0):

A := attributes(U)[3]

_ProbabilityDistribution

(1)

AllAttributes := with(A);

[CDF, Conditions, HodgesLehmann, InverseSurvivalFunction, MGF, MaximumLikelihoodEstimate, Mean, Median, Mode, PDF, Parameters, ParentName, Quantile, RandomSample, RandomSampleSetup, RandomVariate, RousseeuwCrouxSn, Support, Variance]

(2)

A:-ParentName

UniformDistribution

(3)

# Define a user random variable

v := Distribution(PDF = (z -> piecewise(0 <= t and t < 1, 1, 0))):
V := RandomVariable(v):
A := attributes(V)[3];
AllAttributes := with(A);
A:-Conditions;

_ProbabilityDistribution0

 

[Conditions, PDF]

 

[]

(4)

# its definition can be augmented by adding some recognized attributes...
# even if the result returned by Mean is strange

v := Distribution(PDF = (z -> piecewise(0 <= t and t < 1, 1, 0)), 'Mean'=1/Pi, 'Median'=exp(-1)):
V := RandomVariable(v):
A := attributes(V)[3];
AllAttributes := with(A);
[Median, Mean](V)

_ProbabilityDistribution1

 

[Conditions, Mean, Median, PDF]

 

[exp(-1), 1/Pi(_R1)]

(5)

# but not all the recognized attributes seem to be able to be instanciated:

v := Distribution(PDF = (z -> piecewise(a <= t and t < b, 1/(b-a), 0)), 'Parameters'=[a, b]);
v := Distribution(PDF = (z -> piecewise(a <= t and t < b, 1/(b-a), 0)), 'ParentNames'=MyDistribution);

Error, (in Statistics:-Distribution) invalid input: too many and/or wrong type of arguments passed to NewDistribution; first unused argument is Parameters = [a, b]

 

Error, (in Statistics:-Distribution) invalid input: too many and/or wrong type of arguments passed to NewDistribution; first unused argument is ParentNames = MyDistribution

 

 

 


 

Download Attributes.mw

I'm new to Maple.

My problem is that if I input the command sqrt(3.0), for example, I get this strange result:

1.81847767202745*10^(-58) + (7.53238114626421*10^(-59))*I

The results is the same, no matter the argument of sqrt.

Also, when using ln, I get this:

-265.745524189222 + 0.785398163397448*I

Again, no matter the argument of ln, the result is the same.

What is happening?

Dear maple user  any one suggest me how to solve  second order coupled differential equation using galerkin finite element method for 8 elements and 10 elements using maple codes

 

Hi, Is there a way in which i can solve the following optimal control problem numerically with Maple ??

where P(t)=N(t)+S(t)+A(t) and N(0)=0.4897, S(0)=0.4018, A(0)=0.1085.

μ=0.000833, d=0.000666, ε1=0.0020, ε2=0.000634, β1=0.002453, β2=0.25*0.02, γ1=0.0048, γ2=0.25*0.02+0.00013, k1=1, k2=0.001, k3=0.99.

 where 

p1,p2,p3 are transversality conditions 

p1(60)=0 
p2(60)=0 
p3(60)=0

Answers and advice are very appreciated. 

Thank you all for reading.

Benz.

Hi

I have an optimization problem subjects with a system of ordinary differential equations with initial conditions.

I would like to obtain u^star, x^star and y^star solution of my problem 

I prefer if possible we implement hamilton jacobi bellman if possible

 

Optimal_control_problem.mw

 

thanks

 

whats wrong with the codes while running the codes in maple 13 it will take memory and time as 41.80M, 9.29s while the same code is running in maple 18 it will take 1492.38M , 911.79s

Why the same codes take different time and memory. The codes are here

 

 

restart:
Digits:=15:
d1:=0.2:d2:=0.6:L1:=0.2:L2:=0.2:F:=0.3:Br:=0.3:
Gr:=0.2: Nb:=0.1:Nt:=0.3:B:=1:B1:=0.7:m:=1:k:=0.1:
Ro:=1:R1:=1:q:=1:alpha:=Pi/4:
h:=z->piecewise( z<=d1,    1,
                 z<=d1+L1,   1-(gamma1/(2*Ro))*(1 + cos(2*(Pi/L1)*(z - d1 - L1/2))), 
                        z<=B1-L2/2,  1 ,          
                    z<=B1,  1-(gamma2/(2*Ro))*(1 + cos(2*(Pi/L2)*(z - B1))),
                 z<=B1+L2/2,  R1-(gamma2/(2*Ro))*(1 + cos(2*(Pi/L2)*(z - B1))),
                 z<=B,    R1):
A:=(-m^2/4)-(1/4*k):
S1:=(h(z)^2)/4*A-ln(A*h(z)^2+1)*(1+h(z)^2)/4*A:
a2:=Int((1/S1),z=0..1):
b2:=Int((sin(alpha)/F),z=0..1):
c2:=(1/S1)*(-h(z)^6/(6912*A)-h(z)^4/(9216*A)+h(z)^2/(4608*A^3)+ln(1+A*h(z)^2)*(h(z)^6/(576*A)+h(z)^4/(512*A^2)-1/(4608*A^4))):
c3:=Int(c2,z=0..1):
c4:=2*Gr*(Nb-Nt)*c3:
e2:=(1/S1)*(-7*h(z)^4/(256*A)-h(z)^2/(128*A^2)+ln(1+A*h(z)^2)*(3*h(z)^4/(128*A)+h(z)^2/(32*A^2)+1/(128*A^3))):
e3:=Int(e2,z=0..1):
e4:=2*(Nt/Nb)*Br*e3:
l1:=-a2:
l2:=-b2-c4+e4:
Dp:=q*l1+l2:

igRe:=subsindets(Dp,specfunc(anything,Int),
                         u->Int(Re(op(1,u)),op(2,u),
                                   method=_d01ajc,epsilon=1e-6)):

plot([seq(eval(igRe,gamma2=j),j=[0,0.02,0.06])],gamma1=0.02..0.1,
     adaptive=false,
     legend = [gamma2 = 0.0,gamma2 = 0.02,gamma2 = 0.04],
     linestyle = [solid,dash,dot],
     color = [black,black,black],
     labels=[gamma1,'Re(Dp)'],
     gridlines=false, axes=boxed);

igIm:=subsindets(Dp,specfunc(anything,Int),
                         u->Int(Im(op(1,u)),op(2,u),
                                   method=_d01ajc,epsilon=1e-6)):

plot([seq(eval(igIm,gamma2=j),j=[0,0.02,0.06])],gamma1=0.02..0.1,
     adaptive=false,
     legend = [gamma2 = 0.0,gamma2 = 0.02,gamma2 = 0.04],
     linestyle = [solid,dash,dot],
     color = [black,black,black],
     labels=[gamma1,'Im(Dp)'],
     gridlines=false, axes=boxed);
 

 

Dears, greeting for all

I have a problem, I try to explain it by a figure

This formula does not work.

I need to substitute n=0 to give G_n+1 as a function of the parameter s, then find the limit. 

.where G_n is a function in s.

this is the result

 

how to express eigenvector or eigenvalues in terms of fibonacci or lucas or golden ratio?

fibonacci ratio has many 

f(n)/f(n-1) , all eigenvector can not divided by any one of them

 

Hello!

I want to calculate Eigenvalues. Depending on values for digits and which datatype I choose Maple sometimes returns zero as Eigenvalues. Maybe there is a problem with the used routines: CLAPACK sw_dgeevx_, CLAPACK sw_zgeevx_.

Thank you for your suggestions!
 

``

 

Problems LinearAlgebra:-Eigenvalues, Digits, ':-datatype' = ':-sfloat', ':-datatype' = ':-complex'( ':-sfloat' )

 

restart;

interface( ':-displayprecision' = 5 ):
 

infolevel['LinearAlgebra'] := 5;
myPlatform := kernelopts( ':-platform' );
myVersion := kernelopts( ':-version' );

5

 

"windows"

 

`Maple 2018.2, X86 64 WINDOWS, Nov 16 2018, Build ID 1362973`

(1.1)

Example 1

 

A1 := Matrix( 5, 5, [[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [-10201/1000, 30199/10000, -5049/250, 97/50, -48/5]] );

Matrix(5, 5, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 1, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 1, (5, 1) = -10201/1000, (5, 2) = 30199/10000, (5, 3) = -5049/250, (5, 4) = 97/50, (5, 5) = -48/5})

(1.1.1)

LinearAlgebra:-Eigenvalues( A1 );

CharacteristicPolynomial: working on determinant of minor 2
CharacteristicPolynomial: working on determinant of minor 3
CharacteristicPolynomial: working on determinant of minor 4
CharacteristicPolynomial: working on determinant of minor 5

 

Vector(5, {(1) = -10, (2) = 1/10+I, (3) = 1/10-I, (4) = 1/10+I, (5) = 1/10-I})

(1.1.2)

A11 := Matrix( op( 1, A1 ),( i,j ) -> evalf( A1[i,j] ), ':-datatype' = ':-sfloat' );

Matrix(5, 5, {(1, 1) = 0., (1, 2) = 1.00000, (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 1.00000, (2, 4) = 0., (2, 5) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 0., (3, 4) = 1.00000, (3, 5) = 0., (4, 1) = 0., (4, 2) = 0., (4, 3) = 0., (4, 4) = 0., (4, 5) = 1.00000, (5, 1) = -10.20100, (5, 2) = 3.01990, (5, 3) = -20.19600, (5, 4) = 1.94000, (5, 5) = -9.60000})

(1.1.3)

Digits := 89;
LinearAlgebra:-Eigenvalues( A11 );

Digits := 89

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881249354686)

(1.1.4)

Digits := 90;
LinearAlgebra:-Eigenvalues( A11 );

Digits := 90

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881249352150)

(1.1.5)

A12 := Matrix( op( 1, A1 ),( i,j ) -> evalf( A1[i,j] ), ':-datatype' = ':-complex'( ':-sfloat' ) );

Matrix(5, 5, {(1, 1) = 0.+0.*I, (1, 2) = 1.00000+0.*I, (1, 3) = 0.+0.*I, (1, 4) = 0.+0.*I, (1, 5) = 0.+0.*I, (2, 1) = 0.+0.*I, (2, 2) = 0.+0.*I, (2, 3) = 1.00000+0.*I, (2, 4) = 0.+0.*I, (2, 5) = 0.+0.*I, (3, 1) = 0.+0.*I, (3, 2) = 0.+0.*I, (3, 3) = 0.+0.*I, (3, 4) = 1.00000+0.*I, (3, 5) = 0.+0.*I, (4, 1) = 0.+0.*I, (4, 2) = 0.+0.*I, (4, 3) = 0.+0.*I, (4, 4) = 0.+0.*I, (4, 5) = 1.00000+0.*I, (5, 1) = -10.20100+0.*I, (5, 2) = 3.01990+0.*I, (5, 3) = -20.19600+0.*I, (5, 4) = 1.94000+0.*I, (5, 5) = -9.60000+0.*I})

(1.1.6)

Digits := 100;
LinearAlgebra:-Eigenvalues( A12 );

Digits := 100

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881249345038)

(1.1.7)

Digits := 250;
LinearAlgebra:-Eigenvalues( A12 );

Digits := 250

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881342643606)

(1.1.8)

 

 

Example 2

 

A2 := Matrix(3, 3, [[0, 1, 0], [0, 0, 1], [3375, -675, 45]]);

Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 3375, (3, 2) = -675, (3, 3) = 45})

(1.2.1)

LinearAlgebra:-Eigenvalues( A2 );

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 3 x 3 matrix

IntegerCharacteristicPolynomial: Using prime 33554393
IntegerCharacteristicPolynomial: Using prime 33554383
IntegerCharacteristicPolynomial: Used total of  2  prime(s)

 

Vector(3, {(1) = 15, (2) = 15, (3) = 15})

(1.2.2)

A21 := Matrix( op( 1, A2 ),( i,j ) -> evalf( A2[i,j] ), ':-datatype' = ':-sfloat' );

Matrix(3, 3, {(1, 1) = 0., (1, 2) = 1.00000, (1, 3) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 1.00000, (3, 1) = 3375.00000, (3, 2) = -675.00000, (3, 3) = 45.00000})

(1.2.3)

Digits := 77;
LinearAlgebra:-Eigenvalues( A21 );

Digits := 77

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881342621686)

(1.2.4)

Digits := 78;
LinearAlgebra:-Eigenvalues( A21 );

Digits := 78

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881342617230)

(1.2.5)

A22 := Matrix( op( 1, A2 ),( i,j ) -> evalf( A2[i,j] ), ':-datatype' = ':-complex'( ':-sfloat' ) );

Matrix(3, 3, {(1, 1) = 0.+0.*I, (1, 2) = 1.00000+0.*I, (1, 3) = 0.+0.*I, (2, 1) = 0.+0.*I, (2, 2) = 0.+0.*I, (2, 3) = 1.00000+0.*I, (3, 1) = 3375.00000+0.*I, (3, 2) = -675.00000+0.*I, (3, 3) = 45.00000+0.*I})

(1.2.6)

Digits := 58;
LinearAlgebra:-Eigenvalues( A22 );

Digits := 58

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881342614934)

(1.2.7)

Digits := 59;
LinearAlgebra:-Eigenvalues( A22 );

Digits := 59

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881325525942)

(1.2.8)

 

 

Example 3

 

A3 := Matrix(4, 4, [[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [-48841, 8840, -842, 40]]);

Matrix(4, 4, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 1, (4, 1) = -48841, (4, 2) = 8840, (4, 3) = -842, (4, 4) = 40})

(1.3.1)

LinearAlgebra:-Eigenvalues( A3 );

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
IntegerCharacteristicPolynomial: Using prime 33554393

IntegerCharacteristicPolynomial: Using prime 33554383
IntegerCharacteristicPolynomial: Used total of  2  prime(s)

 

Vector(4, {(1) = 10+11*I, (2) = 10-11*I, (3) = 10+11*I, (4) = 10-11*I})

(1.3.2)

A31 := Matrix( op( 1, A3 ),( i,j ) -> evalf( A3[i,j] ), ':-datatype' = ':-sfloat' );

Matrix(4, 4, {(1, 1) = 0., (1, 2) = 1.00000, (1, 3) = 0., (1, 4) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 1.00000, (2, 4) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 0., (3, 4) = 1.00000, (4, 1) = -48841.00000, (4, 2) = 8840.00000, (4, 3) = -842.00000, (4, 4) = 40.00000})

(1.3.3)

Digits := 75;
LinearAlgebra:-Eigenvalues( A31 );

Digits := 75

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881324662046)

(1.3.4)

Digits := 76;
LinearAlgebra:-Eigenvalues( A31 );

Digits := 76

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881324657710)

(1.3.5)

A32 := Matrix( op( 1, A3 ),( i,j ) -> evalf( A3[i,j] ), ':-datatype' = ':-complex'( ':-sfloat' ) );

Matrix(4, 4, {(1, 1) = 0.+0.*I, (1, 2) = 1.00000+0.*I, (1, 3) = 0.+0.*I, (1, 4) = 0.+0.*I, (2, 1) = 0.+0.*I, (2, 2) = 0.+0.*I, (2, 3) = 1.00000+0.*I, (2, 4) = 0.+0.*I, (3, 1) = 0.+0.*I, (3, 2) = 0.+0.*I, (3, 3) = 0.+0.*I, (3, 4) = 1.00000+0.*I, (4, 1) = -48841.00000+0.*I, (4, 2) = 8840.00000+0.*I, (4, 3) = -842.00000+0.*I, (4, 4) = 40.00000+0.*I})

(1.3.6)

Digits := 100;
LinearAlgebra:-Eigenvalues( A32 );

Digits := 100

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881324648198)

(1.3.7)

Digits := 250;
LinearAlgebra:-Eigenvalues( A32 );

Digits := 250

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881327288182)

(1.3.8)

 

 

Example 4

 

A4 := Matrix(8, 8, [[0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1], [-1050625/20736, 529925/1296, -15417673/10368, 3622249/1296, -55468465/20736, 93265/108, -1345/8, 52/3]]);

Matrix(8, 8, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (1, 6) = 0, (1, 7) = 0, (1, 8) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (2, 8) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = 1, (3, 5) = 0, (3, 6) = 0, (3, 7) = 0, (3, 8) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 1, (4, 6) = 0, (4, 7) = 0, (4, 8) = 0, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0, (5, 6) = 1, (5, 7) = 0, (5, 8) = 0, (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = 0, (6, 7) = 1, (6, 8) = 0, (7, 1) = 0, (7, 2) = 0, (7, 3) = 0, (7, 4) = 0, (7, 5) = 0, (7, 6) = 0, (7, 7) = 0, (7, 8) = 1, (8, 1) = -1050625/20736, (8, 2) = 529925/1296, (8, 3) = -15417673/10368, (8, 4) = 3622249/1296, (8, 5) = -55468465/20736, (8, 6) = 93265/108, (8, 7) = -1345/8, (8, 8) = 52/3})

(1.4.1)

LinearAlgebra:-Eigenvalues( A4 );

CharacteristicPolynomial: working on determinant of minor 2
CharacteristicPolynomial: working on determinant of minor 3

CharacteristicPolynomial: working on determinant of minor 4
CharacteristicPolynomial: working on determinant of minor 5
CharacteristicPolynomial: working on determinant of minor 6
CharacteristicPolynomial: working on determinant of minor 7
CharacteristicPolynomial: working on determinant of minor 8

 

Vector(8, {(1) = 1/3-(1/4)*I, (2) = 1/3+(1/4)*I, (3) = 4-5*I, (4) = 4+5*I, (5) = 1/3-(1/4)*I, (6) = 1/3+(1/4)*I, (7) = 4-5*I, (8) = 4+5*I})

(1.4.2)

A41 := Matrix( op( 1, A4 ),( i,j ) -> evalf( A4[i,j] ), ':-datatype' = ':-sfloat' );

Matrix(8, 8, {(1, 1) = 0., (1, 2) = 1.00000, (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (1, 6) = 0., (1, 7) = 0., (1, 8) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 1.00000, (2, 4) = 0., (2, 5) = 0., (2, 6) = 0., (2, 7) = 0., (2, 8) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 0., (3, 4) = 1.00000, (3, 5) = 0., (3, 6) = 0., (3, 7) = 0., (3, 8) = 0., (4, 1) = 0., (4, 2) = 0., (4, 3) = 0., (4, 4) = 0., (4, 5) = 1.00000, (4, 6) = 0., (4, 7) = 0., (4, 8) = 0., (5, 1) = 0., (5, 2) = 0., (5, 3) = 0., (5, 4) = 0., (5, 5) = 0., (5, 6) = 1.00000, (5, 7) = 0., (5, 8) = 0., (6, 1) = 0., (6, 2) = 0., (6, 3) = 0., (6, 4) = 0., (6, 5) = 0., (6, 6) = 0., (6, 7) = 1.00000, (6, 8) = 0., (7, 1) = 0., (7, 2) = 0., (7, 3) = 0., (7, 4) = 0., (7, 5) = 0., (7, 6) = 0., (7, 7) = 0., (7, 8) = 1.00000, (8, 1) = -50.66671, (8, 2) = 408.89275, (8, 3) = -1487.04408, (8, 4) = 2794.94522, (8, 5) = -2674.98384, (8, 6) = 863.56481, (8, 7) = -168.12500, (8, 8) = 17.33333})

(1.4.3)

Digits := 74;
LinearAlgebra:-Eigenvalues( A41 );

Digits := 74

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881317242630)

(1.4.4)

Digits := 75;
LinearAlgebra:-Eigenvalues( A41 );

Digits := 75

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_dgeevx_

 

Vector[column](%id = 18446745881317239134)

(1.4.5)

A42 := Matrix( op( 1, A4 ),( i,j ) -> evalf( A4[i,j] ), ':-datatype' = ':-complex'( ':-sfloat' ) );

Matrix(8, 8, {(1, 1) = 0.+0.*I, (1, 2) = 1.00000+0.*I, (1, 3) = 0.+0.*I, (1, 4) = 0.+0.*I, (1, 5) = 0.+0.*I, (1, 6) = 0.+0.*I, (1, 7) = 0.+0.*I, (1, 8) = 0.+0.*I, (2, 1) = 0.+0.*I, (2, 2) = 0.+0.*I, (2, 3) = 1.00000+0.*I, (2, 4) = 0.+0.*I, (2, 5) = 0.+0.*I, (2, 6) = 0.+0.*I, (2, 7) = 0.+0.*I, (2, 8) = 0.+0.*I, (3, 1) = 0.+0.*I, (3, 2) = 0.+0.*I, (3, 3) = 0.+0.*I, (3, 4) = 1.00000+0.*I, (3, 5) = 0.+0.*I, (3, 6) = 0.+0.*I, (3, 7) = 0.+0.*I, (3, 8) = 0.+0.*I, (4, 1) = 0.+0.*I, (4, 2) = 0.+0.*I, (4, 3) = 0.+0.*I, (4, 4) = 0.+0.*I, (4, 5) = 1.00000+0.*I, (4, 6) = 0.+0.*I, (4, 7) = 0.+0.*I, (4, 8) = 0.+0.*I, (5, 1) = 0.+0.*I, (5, 2) = 0.+0.*I, (5, 3) = 0.+0.*I, (5, 4) = 0.+0.*I, (5, 5) = 0.+0.*I, (5, 6) = 1.00000+0.*I, (5, 7) = 0.+0.*I, (5, 8) = 0.+0.*I, (6, 1) = 0.+0.*I, (6, 2) = 0.+0.*I, (6, 3) = 0.+0.*I, (6, 4) = 0.+0.*I, (6, 5) = 0.+0.*I, (6, 6) = 0.+0.*I, (6, 7) = 1.00000+0.*I, (6, 8) = 0.+0.*I, (7, 1) = 0.+0.*I, (7, 2) = 0.+0.*I, (7, 3) = 0.+0.*I, (7, 4) = 0.+0.*I, (7, 5) = 0.+0.*I, (7, 6) = 0.+0.*I, (7, 7) = 0.+0.*I, (7, 8) = 1.00000+0.*I, (8, 1) = -50.66671+0.*I, (8, 2) = 408.89275+0.*I, (8, 3) = -1487.04408+0.*I, (8, 4) = 2794.94522+0.*I, (8, 5) = -2674.98384+0.*I, (8, 6) = 863.56481+0.*I, (8, 7) = -168.12500+0.*I, (8, 8) = 17.33333+0.*I})

(1.4.6)

Digits := 100;
LinearAlgebra:-Eigenvalues( A42 );

Digits := 100

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881317227806)

(1.4.7)

Digits := 250;
LinearAlgebra:-Eigenvalues( A42 );

Digits := 250

 

Eigenvalues: calling external function
Eigenvalues: initializing the output object
Eigenvalues: using software external library
Eigenvalues: CLAPACK sw_zgeevx_

 

Vector[column](%id = 18446745881356880102)

(1.4.8)

 

 

 

 

 

 

 

 

 

 

``


 

Download Problems_LinearAlgebra_Eigenvalues.mw

How to get the functional form of interpolation in the given example below

 

GP.mw

Hi, 

The procedure Statistics:-ChiSquareSuitableModelTest returns wrong or stupid results in some situations.
The stupid answer can easily be avoided if the user is careful enough.
The wrong answer is more serious: the standard deviation (in the second case below) is not correctly estimated.

PS: the expression "CORRECT ANSWER" is a short for "POTENTIALLY CORRECT ANSWER" given that what ChiSquareSuitableModelTest really does is not documented
 

restart:

with(Statistics):

randomize():

N := 100:
S := Sample(Normal(0, 1), N):

infolevel[Statistics] := 1:

# 0 parameter to fit from the sample S  CORRECT ANSWER

ChiSquareSuitableModelTest(S, Normal(0, 1), level = 0.5e-1):
print():

Chi-Square Test for Suitable Probability Model
----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Bins:                    10
Degrees of freedom:      9
Distribution:            ChiSquare(9)
Computed statistic:      15.8
Computed pvalue:         0.0711774
Critical value:          16.9189774487099
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false

 

(1)

# 2 parameters (mean and standard deviation) to fit from the sample S  INCORRECT ANSWER

ChiSquareSuitableModelTest(S, Normal(a, b), level = 0.5e-1, fittedparameters = 2):


print():
# verification
m := Mean(S);
s := StandardDeviation(S);
t := sqrt(add((S-~m)^~2) / (N-1));

print():
error "the estimation of the StandardDeviation ChiSquareSuitableModelTest is not correct";
print():

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Model specialization:    [a = -.2143e-1, b = .8489]
Bins:                    10
Degrees of freedom:      7
Distribution:            ChiSquare(7)
Computed statistic:      3.8
Computed pvalue:         0.802504
Critical value:          14.0671405764057
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false

 

 

HFloat(-0.021425681632689854)

 

HFloat(0.8531979363682092)

 

HFloat(0.8531979363682094)

 

 

Error, the estimation of the StandardDeviation ChiSquareSuitableModelTest is not correct

 

(2)

# ONLY 1 parameter (mean OR standard deviation ?) to fit from the sample S  STUPID ANSWER
#
# A stupid answer: the parameter to fit not being declared, the procedure should return
# an error of the type "don(t know what is the paramater tio fit"
ChiSquareSuitableModelTest(S, Normal(a, b), level = 0.5e-1, fittedparameters = 1):


print():
WARNING("ChiSquareSuitableModelTest should return it can't fit a single parameter");
print():

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Model specialization:    [a = -.2143e-1, b = .8489]
Bins:                    10
Degrees of freedom:      8
Distribution:            ChiSquare(8)
Computed statistic:      3.8
Computed pvalue:         0.874702
Critical value:          15.5073130558655
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false

 

 

Warning, ChiSquareSuitableModelTest should return it can't fit a single parameter

 

(3)

ChiSquareSuitableModelTest(S, Normal(a, 1), level = 0.5e-1, fittedparameters = 1):  #CORRECT ANSWER
print():

# verification
m := Mean(S);
print():

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Model specialization:    [a = -.2143e-1]
Bins:                    10
Degrees of freedom:      8
Distribution:            ChiSquare(8)
Computed statistic:      16.4
Computed pvalue:         0.0369999
Critical value:          15.5073130558655
Result: [Rejected]
This statistical test provides evidence that the null hypothesis is false

 

 

HFloat(-0.021425681632689854)

 

(4)

ChiSquareSuitableModelTest(S, Normal(0, b), level = 0.5e-1, fittedparameters = 1):  #CORRECT ANSWER

print():
# verification
s := sqrt((add(S^~2) - 0^2) / N);
print():

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Model specialization:    [b = .8492]
Bins:                    10
Degrees of freedom:      8
Distribution:            ChiSquare(8)
Computed statistic:      6.4
Computed pvalue:         0.60252
Critical value:          15.5073130558655
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false

 

 

HFloat(0.8491915633531496)

 

(5)

 


 

Download ChiSquareSuitableModelTest.mw

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