Determining the number of rows or columns of a lower triangular matrix, maintaining constant diagonal coefficients

Hi all,

This problem has stumped me due to my insufficient mathematical knowledge.

I am looking to determining the number of rows or columns of a lower triangular matrix, maintaining constant diagonal coefficients, for the minimum condition number.

Please see the attached PDF which describes what I am trying to do.

The best I have come up with is to check 5 different matrices, each with its number of rows/columns (solved using the finite element method to determine the number of steps {rows/columns}). I have attached the MAT files for these values for each matrix.

I did this with the following code. I then compare it manually.

% Diagonal elements of the "a" calibration matrix.
a_pn_d=diag(aij,0);
% Maximum and minimum diagonal elements of the "a" calibration matrix.
a_pn_d_max=max(a_pn_d);
a_pn_d_min=min(a_pn_d);
% Ratio of the minimum to maximum diagonal elements of the "a" calibration
% matrix.
a_pn_d_R=a_pn_d_min/a_pn_d_max;
% Relative error of each row.
a_sum=sum(aij,2);
a_sd=a_sum./a_pn_d;
% Condition number with a p-norm of 1, for the "a" calibration matrix. 
% https://blogs.mathworks.com/cleve/2017/07/17/what-is-the-condition-number-of-a-matrix/#a3219326-029f-4d0b-bf53-12917948c5f2
a_cond_1=cond(aij,1);

Is there a method to optimise this problem, looking for the number of steps (rows/columns) of a lower triangular matrix, while maintaining constant diagonal coefficients as well as the minimum condition number for the matrix?

I am open to any suggestions and/or assistance in this regard.

[GS76]

I hope this example helps with my question.

Marked in "yellow". The diagonal coefficients should be constant or as close as possible.

The condtion number for this matrix is 51.3.

[GS76]

The condition number is determined in relation to the all the matrix coefficients and the diagonal coefficients, as follows:

Answer 1

[GS76]

I have the following code for the above equation, but I do not think it is right as the answer is large.

I have attached the *.mat for reference.

Any assistance would be much appreciated.

% Condition number for the optimal distribution of the hole-drilling depth increments, base on the "Integral Method" for the non-uniform hole-drilling residual stress measurement technique

% Determination of the condition number for the optimal distribution of the hole-drilling depth increments,
% base on the "Integral Method" for the non-uniform hole-drilling residual stress measurement technique.

% The following equation as referenced from this source (see below) is used 
% for this determination.

% https://link.springer.com/article/10.1007/BF02331114
 
% Develop the terms of the equation
% The individual terms of the equation will be developed individually and then brought together.
% Loading the files
% Loading the required variables.

% Loading of the required *.mat file and then the variable.

anp = load("191231_GS_0295_10.mat","aij");
aij = anp.aij
 
% Length of array
N = length(aij)

% Summation of matrix

aij_2 = aij^2
y1 = sum(aij_2(:))

% Product of the matrix

aii = diag(aij)
aii_2 = aii.^2

% Product of the square of the diagonal of the matrix

y2 = 4*prod(aii_2,"all")

% The complete equation is as follows:

K_A = (y1 + (y1.^2-y2).^0.5)./y2

[GS76]

Based on the above I thought it is best to ask this question seperately, as I had deviated from the original question above.

The question is at the following link:

https://www.mathworks.com/matlabcentral/answers/500975-solving-an-equation-with-terms-that-require-double-summation-and-products