# FORTRAN 90 L e c ```FORTRAN 90
Lecturer : Rafel Hekmat Hameed
University of Babylon
Subject : Fortran 90
College of Engineering
Year : Second B.Sc.
Mechanical Engineering Dep.
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Maattrriicceess
 A square matrix is said to be upper triangular when all the elements below
the diagonal are zero. The matrix below is an upper triangular matrix.
In an upper triangular matrix, not all elements above the diagonal need to be
non-zero.
 A square matrix is said to be lower triangular, when all the elements above
the diagonal are zero. The matrix below is a lower triangular matrix.
In a lower triangular matrix, not all elements below the diagonal need to be
non-zero.
 A square matrix is said to be diagonal, if all elements are zero, except those
in the diagonal. The matrix below is a diagonal matrix.
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 A diagonal matrix is called a scalar matrix, if a11 = a22 = a33 = … = ann = k
where k is a scalar. The matrix D below is a scalar matrix with k=4.
A scalar matrix with k=1 , is called an identity matrix . Shown below are
2 2, 33 , and 44 identity matrices.
 The transpose of a matrix A, denoted as AT, is the matrix that is obtained
when the rows and columns of matrix A are interchanged.
 A symmetric matrix A , is one such that AT=A , that is, the transpose of a
matrix A is the same as A . An example of a symmetric matrix is shown
below.
EXAMPLE
Write a fortran 90 program to read two square matrices A &amp; B with rank 2 and
shape (/3,3/) by using data statement, then do the following:
1. Construct [C] = [A]T . By using subroutine program.
2. Construct [D] = [C]  [B]. By using subroutine program.
3. Find the largest element in the minor diagonal of [D].
Use format statement to print your results in output file.
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B =൥ͳͲ
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implicit none
integer,parameter::n=3
integer,dimension(n,n)::a ,b , c, d
integer,dimension(n)::e
integer::i,j,max
data a/4,2,9,8,5,1,2,7,6/
data b/1,10,8,3,2,5,9,6,4/
! construct C
call trans (a,c)
write(*,6)((c(i,j),j=1,n),i=1,n)
PRINT*,&quot;****************&quot;
! construct D
call multi (c,b,d)
write(*,6)((d(i,j),j=1,n),i=1,n)
PRINT*,&quot;****************&quot;
!
larger element in the minor diagonal
do i=1,n
;
do j=1,n
if (i+j==n+1) e(i)=d(i,j)
enddo ; enddo
max=e(1)
do i=2,n
if(e(i).gt.max)then
max=e(i) ; endif
enddo
write(*,9) max
9 format(2x,&quot;max =&quot;,i5)
6 format(3(2x,i3))
;
End
subroutine trans (a,c)
implicit none
integer,parameter::n=3
integer,dimension(n,n)::a,c
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integer i,j
do i=1,n
;
do j=1,n
c(i,j)= a(j,i)
enddo ; enddo
end
subroutine multi (c,b,d)
implicit none
integer,parameter::n=3
integer,dimension(n,n)::b,c,d
integer i,j,k
do i=1,n
do j=1,n
d(i,j)=0
do k=1,n
d(i,j)=d(i,j)+c(i,k)*b(k,j)
enddo ; enddo ; enddo
end
EXAMPLE
Two matrix A with rank 2 and shape (/4,2/) and matrix B with rank 2 and shape
(/2,4/) read these matrices by using reading statement from input file and do the
following
1.
2.
3.
4.
Construct [C] = [A]  [B]. By using subroutine program
Transpose [C]
Construct [D], which is a diagonal matrix of [C]
Construct [E], which is an lower triangular matrix has the element of [C]T
Use format statement to print all generated matrices.
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B=ቂ
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implicit none
integer,parameter::n=4,m=2
integer,dimension(n,m)::a
integer,dimension(m,n)::b
integer,dimension(n,n)::c,d,e
integer::i,j,z
open(unit=10,file=&quot;rh.dat&quot;)
! construct C
call multi (a,b,c)
write(*,6)((c(i,j),j=1,n),i=1,n)
PRINT*,&quot;****************&quot;
! Transpose C
do i=1,n
;
do j=i,n
z=c(i,j)
;
c(i,j)=c(j,i)
enddo ; enddo
;
;
c(j,i)=z
write(*,6) ((c(i,j),j=1,n),i=1,n)
PRINT*,&quot;****************&quot;
! construct D
do i=1,n
;
do j=1,n
if(i==j) then
d(i,j)=c(i,j)
else
d(i,j)=0
endif
enddo ;
enddo ;
PRINT*,&quot;****************&quot;
write(*,6)((d(i,j),j=1,n),i=1,n)
! construct E
do i=1,n
;
do j=1,n
;
do j=i+1,n
e(i,j)=c(i,j)
enddo ; enddo
do i=1,n-1
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e(i,j)=0
;
enddo ;
enddo
write(*,6)((e(i,j),j=1,n),i=1,n)
6 format (4(2x,i3))
end
subroutine multi (a,b,c)
implicit none
integer,parameter::n=4,m=2
integer,dimension(n,m)::a
integer,dimension(m,n)::b
integer,dimension(n,n)::c
integer i,j,k
do i=1,n
do j=1,n
c(i,j)=0
do k=1,m
c(i,j)=c(i,j)+a(i,k)*b(k,j)
enddo ; enddo ; enddo
end
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