function x $=$ gauss$($A$,$B$)$

$\%$The sizes of matrices A$,$B are supposed to be NA x NA and NA x NB$.$

$\%$This function solves Ax $=$ B by Gauss elimination algorithm.

NA $=$ size$($A$,2)$; $[$NB$1,$NB$]=$ size$($B$)$;

if NB$1$ ~$=$ NA$,$ error$(\text{'}$A and B must have compatible dimensions'$)$; end

N $=$ NA $+$ NB; AB $=[$A$(1$:NA$,1$:NA$)$ B$(1$:NA$,1$:NB$)]$; $\%$ Augmented matrix

epss $=$ eps$*$ones$($NA$,1)$;

for k $=1$:NA

$\%$Scaled Partial Pivoting at AB$($k$,$k$)$ by Eq$.(2\mathrm{.}2\mathrm{.}20)$

$[$akx$,$kx$]=$ max$($abs$($AB$($k:NA$,$k$))./...$

max$($abs$([$AB$($k:NA$,$k $+1$:NA$)$ epss$(1$:NA $-$ k $+1)]\text{'}))\text{'})$;

if akx $<$ eps, error$(\text{'}$Singular matrix and No unique solution'$)$; end

mx $=$ k $+$ kx $-1$;

if kx $>1\%$ Row change if necessary

tmp$\_$row $=$ AB$($k$,$k:N$)$;

AB$($k$,$k:N$)=$ AB$($mx$,$k:N$)$;

AB$($mx$,$k:N$)=$ tmp$\_$row;

end

$\%$ Gauss forward elimination

AB$($k$,$k $+1$:N$)=$ AB$($k$,$k$+1$:N$)/$AB$($k$,$k$)$;

AB$($k$,$k$)=1$; $\%$make each diagonal element one

for m $=$ k $+1$: NA

AB$($m$,$k$+1$:N$)=$ AB$($m$,$k$+1$:N$)-$ AB$($m$,$k$)*$AB$($k$,$k$+1$:N$)$; $\%$Eq$.(2\mathrm{.}2\mathrm{.}5)$

AB$($m$,$k$)=0$;

end

end

$\%$backward substitution for a upper$-$triangular matrix eqation

$\%$ having all the diagonal elements equal to one

x$($NA$,$:$)=$ AB$($NA$,$NA$+1$:N$)$;

for m $=$ NA$-1$: $-1$:$1$

x$($m$,$:$)=$ AB$($m$,$NA $+1$:N$)-$AB$($m$,$m $+1$:NA$)*$x$($m $+1$:NA$,$:$)$; $\%$Eq$.(2\mathrm{.}2\mathrm{.}7)$

end

modify this MATLAB code to solve Gaus Jordan