@tomleslie thank you for your response. I have extend your idea but not attained satisfactory results. Please see the highlighted (red) in the attached file

Buitin.mw

@acer thanks for your response

@Carl Love I want to convert the following relation

[Vector(5, {(1) = 1, (2) = 2, (3) = 3, (4) = 4, (5) = 5}), Vector(5, {(1) = 1, (2) = 1/2, (3) = 1/3, (4) = 1/4, (5) = 1/5}), Vector(4, {(1) = -1/2, (2) = -1/6, (3) = -1/12, (4) = -1/20}), Vector(3, {(1) = 1/3, (2) = 1/12, (3) = 1/30}), Vector(2, {(1) = -1/4, (2) = -1/20}), Vector(1, {(1) = 1/5})]

into the matrix form as,

[[[1,1, , , , ],[ , ,-1/(2), , , ],[2,1/(2), ,1/(3), , ],[ , ,-1/(6), ,-1/(4), ],[3,1/(3), ,1/(12),,1/(5)],[ , ,-1/(12), ,-1/(20), ],[4,1/(4), ,1/(30), , ],[ , ,-1/(20), , , ],[5,1/(5), , , , ]]]

Then how can I convert it into the following table using printf/other command

Thanks in advance.

@Carl Love Thanks. Let me check it!

@tomleslie thank you for your answer. Yes, I need the results as you obtained in the vector form. But, it is more efficient if you use the following logic to make the table

printf("_______________________________________________________________________________\\n");  
for i from 0 by 1 while i<=2 N do:  
if irem(i,2)=0 then
printf("%2.1f%16.7f%16.7f\\n",i,x[i/(2)],y[i/(2)]);
else  
printf("%2.1f%16.7f%16.7f\\n",i,,); fi;od; printf("_______________________________________________________________________________\\n")
I tried to make the table but find some errors. Kindly have a look. Thanks

@Preben Alsholm thanks for highlighting the mistake. I am sending you modify version. But facing same problem. Please have a look and give me some positive response.

Compare_coeff.mw

@Kitonum Thanks its very useful. 

@tomleslie thanks for your answer. The piecewise function are not shown as you mentioned before Eq. (1) and (2). The appropriate form of piecewise function that I need presented in the file. Kindly help me to fix the problem. Thanks

pwFunc.mw

@acer Sorry its missing. L is the maximum value of abs(diff(f, y)) for all a ≤ x ≤ b, -∞ < y < ∞ for any function f(x,y) or

abs(diff(f, y))<=L for all a ≤ x ≤ b, -∞ < y < ∞

M is the maximum value of abs(diff(y, x, x, x)) for all a ≤ x ≤ b, -∞ < y < ∞ for any function of y(x) or

abs(diff(y, x, x, x))<=M for all a ≤ x ≤ b, -∞ < y < ∞

For example for f:=y-x^(2)+1; 0 ≤ x ≤ 2, -∞ < y < ∞ and y:=(x+1)^2-.5*exp(x); 0 ≤ x ≤ 2, -∞ < y < ∞

the values of L = 1 and M = 0.5exp(2)

@acer thanks for your reply. I have one more question. In the beginning of the following file I declared Digits:=10. But some of values like Y[1,1,1], Y[1,2,1], Y[1,2,2]... have 9 digits after the point decimal (.) like 1.213233616, 1.213873330, 1.214086568 respectively. I need the 10 digits after point decimal (.) which command I have to use instead of Digits?

Refine_Extrapolation.mw

@Kitonum thanks for your answer. I tried to make the above table (Table 5.17 without character y[1,1],y[2,1],y[3,1] etc.) in the following file can you fix it? Because it is more convenient for me. 

I have one more question. In the beginning of the following file I declared Digits:=10. But some of values like Y[1,1,1], Y[1,2,1], Y[1,2,2]... have 9 digits after the point (.) like 1.213233616, 1.213873330, 1.214086568 respectively. I need the 10 digits after point (.) which command I have to use instead of Digits?

Refine_Extrapolation.mw

@tomleslie , In addition, I want to recall a vector B in file Q3 at highlighted portion whereas B matrix is present in file Q4. Which is the command to recall or import a B matrix in file Q3 at highlighted portion from another file Q4.

I shall be very grateful for your support.

Q3.mwQ4.mw

@tomleslie thanks. I fix my problem by doing some change in your above shared file (genMAt2). Now I want to know the solution of one more querry. 

In file Q2 we have a Vector b with some entries which are actually the values of chi[1, 1, 1], chi[1, 1, 2],...,chi[2, 2, 2] in such a following way

(Vector(8, {(1) = chi[1, 1, 1], (2) = chi[1, 1, 2], (3) = chi[1, 2, 1], (4) = chi[1, 2, 2], (5) = chi[2, 1, 1], (6) = chi[2, 1, 2], (7) = chi[2, 2, 1], (8) = chi[2, 2, 2]})) = (Matrix(8, 1, {(1, 1) = 0, (2, 1) = 0, (3, 1) = 0, (4, 1) = 0, (5, 1) = 1.000000000, (6, 1) = 1.000000000, (7, 1) = 1.000000000, (8, 1) = .3678794412}))

Matrix B given in file Q2 have entires which are the linear combination of chi[1, 1, 1], chi[1, 1, 2],...,chi[2, 2, 2]. I want to evaluate matrix B at the values of chi[1, 1, 1], chi[1, 1, 2],...,chi[2, 2, 2] which are obtained in vector b.

Q2.mw

@tomleslie Thanks for your answer. The above reply of my post is not appropriate. In my previous question using some polynomials you generated a square matrix using collocation points but here I need to collocate a vector to generate a square matrix. Let me try to explain what I want to do. Consider the following code for any values of M1, M2 and M3

restart; with(LinearAlgebra); M1 := 3; M2 := 3; M3 := 3;
rho[1] := 1; rho[2] := 1; rho[3] := 1; alpha[1] := 0; alpha[2] := 0; alpha[3] := 0;
X := Vector(M1, proc (n) options operator, arrow; x^(n-1) end proc);
Y := Vector(M2, proc (n) options operator, arrow; y^(n-1) end proc);
Z := Vector(M3, proc (n) options operator, arrow; t^(n-1) end proc);
KroneckerProduct(X, Y); XYZ := KroneckerProduct(%, Z);
IntX := Vector(M1, proc (n) options operator, arrow; x^(n-alpha[1]*rho[1])*Beta(1-alpha[1], n/rho[1])/rho[1] end proc);
IntY := Vector(M2, proc (n) options operator, arrow; y^(n-alpha[2]*rho[2])*Beta(1-alpha[2], n/rho[2])/rho[2] end proc);
IntZ := Vector(M3, proc (n) options operator, arrow; t^(n-alpha[3]*rho[3])*Beta(1-alpha[3], n/rho[3])/rho[3] end proc); KroneckerProduct(X, Y);
IntXYZ1 := convert(KroneckerProduct(%, IntZ), Vector);KroneckerProduct(IntX, IntY);
IntXYZ2 := convert(KroneckerProduct(%, Z), Vector);

Here,  I want to collocate Vector IntXYZ1 and IntXYZ2 as highlighted by red color and generate squares matrices corresponds to IntXYZ1 and IntXYZ2, in such a way that the first M1M2 + 2(M3-1)M2 + 2(M3-1)(M1-2) rows of desired squares matrices are zero and other rows are obtained by collocating the vectors IntXYZ1/IntXYZ2 at x=(i-1)/(M1-1), y=(j-1)/(M2-1), t=(k-1)/(M3-1) for i = 2,3,…,M1-1, j = 2,3,…,M2-1, k = 2,3,…,M3 as given as XX in above for M1=M2=M3=3. 

In your response your wrote a general expression (t^p)*((1/2)^q)*((1/2)^r)/p to generate the other nonzero rows rather than collocating the vectors IntXYZ1/IntXYZ2. Hope you understand what I want to do. I am waiting for your response. Thanks

@Kitonum Thanks for your answer. Need some further modification of above matrix and want to construct the matrix as given in attached file for any values of M1 and M2. Let me explain it.

It is the matrix formulation for M1=M2=4 like this way

A=b

where A is formulated as,

1. First four rows (highlighted as red) are generated by collocating G’s at t=0 and x=0,1/3,2/3 and 1. For any values of M1 and M2, first M1 rows should generated by collocating G’s at t=0 and x=0,1/M1-1,2/M1-1,…,1.
2. Next, three rows (highlighted as green) are generated by collocating G’s at x=0 and t=1/3,2/3 and 1. For any values of M1 and M2, next M2-1 rows should generated by collocating G’s at x=0 and t=1/M2-1,2/M2-1,…,1.
3. Next, three rows (highlighted as blue) are generated by collocating G’s at x=1 and t=1/3,2/3 and 1. For any values of M1 and M2, next M2-1 rows should generated by collocating G’s at x=1 and t=1/M2-1,2/M2-1,…,1.
4. Finally, the last six rows (highlighted as black) are generated by collocating G’s at x=1/3,2/3 and t=1/3,2/3 and 1. For any values of M1 and M2, next M1M2-M1-2*M2+2 rows should generated by collocating G’s at x=1/M1-1,2/M1-1,…,M1-2/M1-1 and t=1/M2-1,2/M2-1,…,1.

In the similar way, b also generated with the help of functions phi(x), f1(x), f2(x) and g(x,t)

Take phi(x)=sin(x), f1=exp(-t), f2=sin(1)exp(-t) and g(x,t)=sin(x)exp(-2t)+sin(x)

Further how we can split A matrix into 3 matrices with same dimension M1M2 y M1M2 as:

A1+A2+A3=A

where A1 has red highlighted portion and rest rows are zero

A2 has green and blue highlighted portions and rest rows are zero

and A3 has black highlighted portion and other rows are zero.

Same procedure for spliting vector f=f1+f2+f3