Fortran Programming Tutorials (Revised) : 039 : Cramer's rule Video Lecture | Introduction to Fortran Programming (Basic Level) - Database Management

hello guys my name is Aaron and welcome
to my channel this series is a series of
tutorials and whorin programming now in
this tutorial we'll be looking at an
example for the determinant of a matrix
now the last tutorial we saw how to find
the determinant of a matrix using the
slice operation and recursive recursive
operations and all okay now
in this tutorial we'll be using the
determinant the property of finding the
determinant of a matrix okay to find a
solution for a system of simultaneous
equations now to explain you guys what's
happening suppose if you have a system
of simultaneous equations that you can
express like ax equals B something of
that sort
okay I'll just show you guys and just
show you guys and chrome saw solving
system of linear equations instead
there's a system of linear equations
okay I have your link and Viki this will
be really easy to follow okay then why
did this load I'll just show you guys
suppose if you have a system of linear
equations okay you can express that as a
matrix X and as a vector matrix a square
matrix a and the vector X and which
which will be equal to which will be
equal to a right hand side matrix B if
you can express this you can express not
a set of equations like this then you
see if you know the determinant of a
okay and if you are able to you you plug
in the values of B in different called
different rows of a in different columns
of a in a in a particular particular
manner and find the determinant by
plugging these okay using those
determinant values and the determinant
value of a it is possible to find it is
possible to find the values of each and
every variable easily now this is the
system of Simon
just don't leave any questions there
will be okay yeah yeah now the equations
are coming you just took a little time I
suppose yeah that's just okay
supposing our system of equations like
this like this one like 3x plus 2y minus
Z equals 1/2 X minus 2i plus 4z equals
-2 like that what you can do is that you
can represent this you can represent
this equation set of equations ok in the
matrix form and now what I'm doing is
I'm representing this this 3 3 here 2 2
here minus 1 this coefficient here 2
minus 2 4 minus 1 point 5 and minus 1 ok
on the right hand side
I'm representing this 1 minus 2 0 as 1
minus 2 0 here now this way this becomes
your a matrix and your X matrix becomes
area variables X Y inside okay and then
your right hand side where your B matrix
is this 1 2 3 now now if you are now
using this canvas rule states that
dramas rule states that if you are able
to replace the first column with be okay
and then the second column with the
third column with me like this suppose
you have a matrix yeah you have the same
hat with a and then in the first column
instead of the actual values like 3 2 y
-1 and replace them with the matrix B
like this I replace them with the matrix
B values like this ok I get a pneumatic
scholars X a similarly keeping that as
keeping that aside with a thing I'm
taking the same matrix a in that in this
row second column second column
replacing that with matrix B I will get
a new - colors xB I keep it aside
similarly in the third matrix a in this
matrix I give the same at taking the
same a matrix and in that I replace the
third column with the matrix B ok
and let me name this X C now what does
if you are able to do that now what your
variables x y&z are here x1 y1 z1 x2 x3
their values are eat nothing but the
determinant of the determinant of this
matrix here by a like that first well
first will be the determinant of X my
first variables answer will be
determinant of XA
you know by determinant of a men that
will be determinant of B here by
determinant determinant of this matrix a
by determinant of B a and then you have
determinant of sexy near a very big
amount of it but this is how it happen
now this is what we this is what you are
following here now this is the program
I'll just walk you guys through 1/2 what
happens here this is the implicit man
this is where I said the order of the
matrix out of the manner just like the
previous example I am setting it for 3 I
just need this to I trading variables I
comma J for the loops ok now this is the
mattress where and I get all the get all
the no there's where I store all the a
matrix and stuff if you guys notice
instead of 2 dimensional matrix I took I
am taking a 3 dimensional matrix because
the instead of since the defining
separate mattresses separate mattresses
for you guys look at the explanation in
steps after writing everything in
separate mattresses like X a xB XC at
all ok what I do is that I just define
that as a three dimensional matrix now
this plus one stands for you know this
plus one is for including the matrix
itself now if if I there are three
variables let's say then I'm other than
method say I'll have three more
mattresses so if you guys notice if I'm
at the club of this and has over three
rows three columns three columns and if
one three columns and three heights kind
of lecture okay there x-axis will be
three units the y-axis will be three
and access will be four units kind of so
you know guys notice this man Texas of
three and your no to include that kind
of a feature what I'm doing is I'm just
I'm just putting a plus 1 this plus one
well this for putting a whereas all the
remaining n values is for storing all
the mattresses for X Y and Z's and so on
I hope you guys understood hope you guys
understand okay
and then I define two matrices with B
and X okay now just a little careless
little warning here guys
though B and X deep it can be
one-dimensional I am making them as a
two-dimensional array and I'm using them
as a matrix here by adding this one here
so I guess I could have just defined
like yeah that will be sufficient but
I'm intentionally putting it as income
r1 to explain you guys that you explain
you guys that there is a small
difference with there's a small
difference between matrix and an array
nothing much okay now what I do is I
just use this reshape command V shape
command to set the values of a now I'm
not I'm just taking the same this
example over here in the wikipedia to
solve it okay so I'm using that value
again that value here I'm not really
arranging it now if we honest for to
this this kind of initialization won't
work but no for planning this problem
setting like this and then be setting
the same V values same B values over
here simpler simple enough and then what
I'm going to do I'm just going to hear
that I'll come to this function come to
this subroutine called little later Here
I am printing all the matters matrix
values a matrix value a comma B here and
then I just print the values of matrix X
a then xB XC and finally the solution
now the major fan part comes in this
call I'm just calling a subroutine
called scram and I'm passing four
arguments in to namely X a B and end and
guys notice okay where do I have this if
I have this in this module now in this
module ignore these two I was
experimenting with a certain stuff but
it didn't work out properly as I thought
I just get I was getting some you know a
segmentation fault issues actually what
I did I thought you know instead of
setting like this
I'll just instead of you know assigning
like this here I thought okay I just
said this equal to some function and
that function will you know
automatically feed the values from the
user and be the values from the user and
set it here so that you know this kind
of stuff might be easy if you have
multiple simultaneous equations and all
for now if I just showing some
segmentation fault errors I am pretty
sure I'm pretty sure as to what exactly
is going on but anyway as I know you
ignore this ignore this now this is the
same recursive function recursive
function we used to calculate the
determinant in the previous four paths
and this is the slice subroutine we that
the same fly subroutine which we used in
the previous previous example to find
the slot to slice the matrices slice the
matrices and this is the new subroutine
we put this subroutine calculate the
subroutine calculates the solution
solution by by you know Mike and
continuously calling did this
determinant function determinant
function and but and imply implied Lee
calling you know slice slice slice this
slice is a subroutine through this
function okay now I'll explain what's
what's in this yes it's putting an
implicit none I'm just getting this
value as you know an integer everything
turned in condition and this is the
important part I'm getting this matrix a
as a dimension of the same order same
size as mention here but I am giving an
argument callers intent in out meaning
this value of a will come in with Satar
this matrix a or this variable a will
come in with certain answer will with
certain values and when it exits out
the subroutine it will and it will go
away with assert with a new entirely
different set of values okay and this
how the wireless will change depends on
what kind of what kind of an alteration
has been made to it by a series of
commands that execute all of it and then
I define B I define B and this time and
defining be a as an array i
one-dimensional array alone
okay this is this is that's the small
change but nothing nothing - nothing to
be different what and then I'm defining
the external external function that
tough for this is which is nothing but
the like you know determinant function
this function is external to this
subroutine some defining it here then
I'm defining this integer variable for
nitration process and then I'm defining
X and 2 which has a dimension n comma 1
okay this as our matrix 2 dimension
matrix and I'm setting it as intent out
meaning it comes or comes in without any
value and it it'll goes out with the
value simple as that now this is the do
part what I do is that firstly in this
operation what time it actually saying
is that see you guys notice only the
first
what do you call throw column height
whichever okay I call it height you know
you can call it anything you want but
okay the height 1 alone is actually
initialized here that has a high -
height 3 height 4 they are not
initialized
ok now but if you want to find XA xB XC
like that if you guys notice what do you
have to do is that copy this value
everywhere copy this value to all the
heights and then change only one set of
columns simple enough so I am doing the
same here
what I'm doing is that in this do loop
the first step I'm copying or copying
all the house and copying the first
Heights value to all the Heights so you
know copy to the height
then hide three and a height for like
that and all okay like that
you mean I'm using the value I'm using
the thumb high to express the third
quacks s but you know there you know you
can use something like page or book
shelf or paper whatever I mean a
different programming languages have a
different convention to know to name the
third axis so you guys could you guys
you know call it whatever you want and
as long as it makes sense okay then what
I do is that I change the ith column
right column of every page with the
matrix B now what this will do is that
okay in this step in this step what it
will do is that will just copy the
matrix a to all the how the heights in
this step it will just copy it will just
copy the value of B and we'll replace
the ith column everywhere for instance
for for instance B will be copied to the
first column in the second height and
then the second column of the third
height and the for third column of the
fourth right and so on so this is
exactly what we want this exactly what
do you want we just want the first
column
now the second height to be filled with
a B the second column of the third
height to be filled with B and the third
column the fourth height to be filled
with B now I'm to explain the same
that's why I wrote this statement here
yay come a-callin comma colon comma 1 is
the first height we're in it's a
two-dimensional matrix that is the value
of a and then to fire 2 for X say what
you have to do is at D place this this
this column with B so in second height
since all of them I initially equal made
equal to this the second height I just
remove the first column values and
replace them with P now with the third
height what I have to do is that if you
want to get this the third head what if
I want do I get this I just have to copy
all this 2/3 height
tapping 2/3 height and then instead of
second column this value I had to
replace this with B and that's why I
have it so that's why it's second column
like like that yeah I have to copy
everything to this value copy everything
to the fourth height and replace the
third column with be simple as that
similarly Simon day similarly suppose if
there are like 15 rows or 15 columns
like that this do loop algorithm does
this do loop will take care of it and
then finally this is where we are this
is where the value of XY ends the values
of the solution this solutions are
calculated what it simply does is that
it just calculates the determinant of
that that particular height under
consideration okay and and in a device
and it returns the value here and that
value is divided by the determinant of a
simple enough simple enough fine Oris
what we can do is that you know to make
this job a little simple okay all I can
do is that I can just type real 8 BL 8
that a that take the time to square a
and I can set this as dead time to score
a equals this because it's this and then
M type dead time to square a simply not
so that you know I'm not running this
again and again and again okay
once I calculate this and I'm just I
will be using this here so that they're
not this this slow doesn't put so much
overhead and during the recursion
recursion process okay and that's that's
it that's it now it will calculate it
will by cameras roll it will calculate
the data this is this is exactly what
will be followed here it will just
calculate the determinant of the first
second page divided by determinant of
the first page and then determined
kanpei third place inevitable determine
in the first page like that when
essentially the German term XA by a
determine the same idea to be
determinant of X determinant of a
terminal X by determinant of a
tournament of exceeded by a determinant
of a so on and so forth okay now this
value in this subroutine well by doing
all this operations
your a is change B remains unchanged a
and remains on stage and your excrements
has changed so that way when you call
when you call the subroutine crammed in
this function your X gets changed a gets
changed not the first page but not the
first height but the all the remaining
Heights gets changed B remains unchanged
and remains unchanged and then I'm just
printing up all these values all right
now and to run all this simple manner
what I do I have written a simple class
shell function when shell script just
like in the previous tutorial and I made
it okay and I'm a and I kind of made it
and I kind of made it an executable if
you guys notice I made it as an
executable here so all I have to do is
that bash okay grammar as such
okay we run this click enter and it will
work fine okay it just lists out all the
files that are available remove mod
function or any mod filter dot o files
or dot exe files if there are any and
then it'll just list all the files then
produce Brody compares these two files
compare these two files and program
builds it to produce an executable
called scram or dot exe and then runs
the cram sorry runs the executable runs
the executable and because of formatting
features available here because of the
formatting which is I kind of introduced
here okay it becomes a little fancy the
output becomes a little fancy okay now
the matrix a and this pie
symbol I'm using it to represent
separate that from be okay now this is
this part is matrix a and this is
actually matrix B now this is Madras
exhale this is Madras XA and xB and I'm
a taxi and the solution is actually at
the bottom now if you guys look at this
is where matter PA gets printed matrix B
gets printed here using this cord
similarly for matrix B so this is matrix
a so this matter XA its matter xB this
is matter it's XC and then the solution
gets printed here
simple enough all right as usual I
actually tried to explain run this by
explaining how my typing all this and
explain you guys and you know we have
these are getting pretty much messy and
complicated so by the time I finished it
took me so much time so I that's why I'm
explaining ice like this hope you guys
understand okay if you guys have any
questions or doubts you can just ask me
in the comment section I'll I'll do my
best to answer all the questions
possible okay that's all I have for you
guys in this tutorial thank you guys for
watching and see you guys in the next
tutorial bye and and one and one more
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