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

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GS76
GS76 on 17 Jan 2020
Commented: GS76 on 19 Jan 2020
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.
  2 Comments
GS76
GS76 on 17 Jan 2020
The condition number is determined in relation to the all the matrix coefficients and the diagonal coefficients, as follows:
Condition.PNG

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Accepted Answer

GS76
GS76 on 19 Jan 2020
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

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