@tomleslie 

Thank you

@Carl Love 

Yes, this storage take little memory. There is no need to work with vectors.

Thank you very much.

@vv 

I have big Matrix(150000,150000) which consume high memory.

I think store the matrix as vectors representing the diagonal and non zero values is good idea.

How to use vectors instead matrix and solve the system for very big matrices

@Carl Love 

When I initialize the matrix with Matrix(150000, 150000) and work with it, the consumed memory is very high.

For this, I want to store the matrix as vectors and solve the system. The vectors are corresponding to non zero values and ther positions in row and colon.

@acer 

Your answer is very helpfull and I must read and develop your proposition.

Thank you 

@acer 

Thank you very much for your response which is very helpfull.

Although, how to know the values of A(i) by seeing the different colors of the 3d curve.

Is it possible to get with the plot a bar graphe (or legend) where there are all the colors of the plot with an indication of the values of colors.

Thanks

@acer 

Thank you for the answer.

I attached a part of my code. essai.mw

Replacing A with color is a good idea for a plot equivalent to density plot.

@vv 

Thank you for your answer.

My objective id to define a Fourier serie of type "a0+Sum(an(n)*cos(n*Pi/ln(x2/x1)*ln(x/x1))+ bn(n)*sin(n*Pi/ln(x2/x1)*ln(x/x1)),n=1..N)" in range [x1..x2]

However, int(C(n,x)*S(m,x)/x,x=x1..x2) is not null.

where:

C := (n,x) -> cos(n*Pi*ln(x/x1)/ln(x2/x1));

S := (n,x) -> sin(n*Pi*ln(x/x1)/ln(x2/x1));

Your above answer "Fact" is not simple for me to understand. 

I don't know if it's possible to get this type of serie.

Thanks

@Rouben Rostamian  

Thanks for your answer.

My problem is int(C(n,x)*S(m,x)/x,x=x1..x2) is not null.

@Rouben Rostamian  

Thank you for for your above answer.

The basis cos(n*Pi*ln(x)), sin(n*Pi*ln(x)) is orthogonal in the period 1..exp(2)

My goal if possible to get a basis (orthogonal) as

cos(n*Pi/ln(x2/x1)*ln(x/x1)) and sin(n*Pi/ln(x2/x1)*ln(x/x1)) in the range [x1 x2]

Thanks

@Rouben Rostamian  

Yes.

Although,

a0+Sum(an*cos(n*Pi*ln(x))+bn*sin(n*Pi*ln(x)),n=1..N) is ok

In the period 1 to exp(2) it is biorthogonal

Thank you very mach for this post.

I'm looking for more improvement for pdsolve as:

pde[25] with f(x,y) as source and not 0.

pde[23] with f(r,theta) as source and not 0.

and other improvements for Helmoltz equations

Thanks

@acer 

Thank you

@acer 

Yes, Tabulate do a good work

Thank you

@acer 

Many thanks for your great answer. I will read it carrefully. I am very greatfull. 

Have the best

Thank you