In The Name Of God

Course:

Numerical methods in linear algebra

Home Work:

Series 1

Student:

Milad Naderi

Student No:

92130901

.

"Every human activity, good or bad, except mathematics, must come to an end."

Paul Erdos

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program Jacobi !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Solving linear system (AX=b) by Jacobi iterative method !! !! 2) Investigation on effect of number of iteratives !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 04/17/14 MiladNaderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Sepecificaton of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Allocatable::A(:,:),b(:),Xk(:),Xk1(:),dif(:) !Real,Allocatable:: Real::tol,maxdif,dummy,len integer::i,j,k,n character::con,y ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write& Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Read (*,*) tol !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input) Open(Unit=4,File='X0.dat',status='OLD', action='READ') !availabe initial value vector (input) Open(Unit=5,File='Results.dat') !Results file contains iterative number,maximumdiffrence and xi at each level. ! Read A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) allocate (A(n,n)) allocate (Xk(n));allocate (Xk1(n));allocate (dif(n)) allocate (b(n))

8

doi=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2) ! Read B vector doi=1,n read(3,*) b(i) end do !close(3) ! Read X0 vector k=0 doi=1,n read(4,*) Xk(i) end do !close(4) Write(5,100) (i,i=1,n) 100 FORMAT (((10x,'K|',1000000(3x,'X('I10')|')))) write(5,101) k,(XK(i),i=1,n) 101 FORMAT (7x,I4,1000000('|',F16.4)) 300 dummy=0;len=0 If (k>0 .AND. MOD(k,50)==0) then Write (5,200) k 200 FORMAT(/,'After'I10'steps, solution was not convergence',/) Write (*,400) k 400 FORMAT(/,'After'I10'steps, solution was not convergence',/) END IF Doi=1,n Do j=1,n if ( i/=j) then len=A(i,j)*Xk(j) dummy=len+dummy end if end do Xk1(i)=(1./A(i,i))*(b(i)-dummy) dummy=0 END Do k=k+1 write(5,102) k,(XK1(i),i=1,n) 102 FORMAT (7x,I4,1000000('|',F16.4)) doi=1,n dif(i)=Xk1(i)-XK(i) if( abs(dif(i))>tol) then XK=XK1 go to 300 end if end do Write (5,*) 'solution is convergence' write(*,*) 'END OF JACOBI PROGRAM' END Program

9

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730.2 7666.1 6698.1 2.2 2477.1 1

5200.0 3953.0 2991.0 2000.0- 1481.0 2

498.1 3185.1 957.1 689.1 1427.1 3

9051.0 9154.0 5803.0 8111.0 8413.0 4

1677.1 305.1 2056.1 2738.1 1136.1 5

7272.0 7925.0 204.0 5922.0 8214.0 6

7076.1 3434.1 3855.1 3817.1 6445.1 7

1573.0 7595.0 484.0 1533.0 7494.0 8

4875.1 4473.1 1084.1 9716.1 964.1 9

4264.0 1256.0 2555.0 1724.0 8465.0 01

4894.1 6223.1 2314.1 9135.1 9304.1 11

2735.0 5007.0 8616.0 5605.0 2526.0 21

3924.1 8772.1 7553.1 854.1 8743.1 31

5106.0 1247.0 76.0 575.0 3776.0 41

7963.1 3932.1 3603.1 4493.1 5992.1 51

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5407.0 8808.0 2557.0 8486.0 7067.0 81

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solution is convergence

Tol=0.005 0 2,0,1,0,1TX 22

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program GS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Solving linear system (AX=b) by Gauss sidel iterative method !! !! 2) Investigation on effect of number of iteratives !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 04/17/14 MiladNaderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Sepecificaton of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Allocatable::A(:,:),b(:),Xk(:),Xk1(:),dif(:) !Real,Allocatable:: Real::tol,maxdif,dummy,len integer::i,j,k,n character::con,y ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write& Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Read (*,*) tol !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input)

02

Open(Unit=4,File='X0.dat',status='OLD', action='READ') !availabe initial value vector (input) Open(Unit=5,File='Results.xls') !Results file contains iterative number,maximumdiffrence and xi at each level. ! Read A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) allocate (A(n,n)) allocate (Xk(n));allocate (Xk1(n));allocate (dif(n)) allocate (b(n)) doi=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2) ! Read B vector doi=1,n read(3,*) b(i) end do !close(3) ! Read X0 vector k=0 doi=1,n read(4,*) Xk(i) end do !close(4) Write(5,100) (i,i=1,n) 100 FORMAT (((10x,'K|',1000000(3x,'X('I10')|')))) write(5,101) k,(XK(i),i=1,n) 101 FORMAT (7x,I4,1000000('|',F16.4)) 300 dummy=0;len=0 If (k>0 .AND. MOD(k,50)==0) then Write (5,200) k 200 FORMAT(/,'After'I10'steps, solution was not convergence',/) Write (*,400) k 400 FORMAT(/,'After'I10'steps, solution was not convergence',/) END IF Doi=1,n Do j=1,i-1 len=len+A(i,j)*Xk1(j) End do Do j=i+1,n len=len+A(i,j)*Xk(j) end Do dummy=len+dummy Xk1(i)=(1./A(i,i))*(b(i)-dummy) dummy=0 len=0 END Do k=k+1 write(5,102) k,(XK1(i),i=1,n)

30

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9089.0 2199.0 6230.1 2510.1 5699.0 4

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ecnegrevnoc si noitulos

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program SOR

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Solving linear system (AX=b) by Successive over relaxation method !! !! 2) Investigation on effect of number of iterative !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 04/17/14 MiladNaderi Original code !! !! !!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Sepecificaton of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Allocatable::A(:,:),b(:),Xk(:),Xk1(:),dif(:) !Real,Allocatable:: Real::tol,maxdif,dummy,len,w integer::i,j,k,n character::con,y ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write& Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Write(*,*) 'Please enter tolerance,tol= ' Read (*,*) tol Write(*,*) 'Please enter relaxation factor,w= ' Read (*,*) w !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input) Open(Unit=4,File='X0.dat',status='OLD', action='READ') !availabe initial value vector (input) Open(Unit=5,File='Results.dat') !Results file contains iterative number,maximumdiffrence and xi at each level. ! Read A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) allocate (A(n,n)) allocate (Xk(n));allocate (Xk1(n));allocate (dif(n)) allocate (b(n)) doi=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2)

05

! Read B vector doi=1,n read(3,*) b(i) end do !close(3) ! Read X0 vector k=0 doi=1,n read(4,*) Xk(i) end do !close(4) Write(5,100) (i,i=1,n) 100 FORMAT (((10x,'K|',1000000(3x,'X('I10')|')))) write(5,101) k,(XK(i),i=1,n) 101 FORMAT (7x,I4,1000000('|',F16.4)) 300 dummy=0;len=0 If (k>0 .AND. MOD(k,50)==0) then Write (5,200) k 200 FORMAT(/,'After'I10'steps, solution was not convergence',/) Write (*,400) k 400 FORMAT(/,'After'I10'steps, solution was not convergence',/) END IF Doi=1,n Do j=1,i-1 len=len+A(i,j)*Xk1(j) End do Do j=i,n len=len+A(i,j)*Xk(j) end Do dummy=len+dummy Xk1(i)=( XK(i)) +(( w / A(i,i)) * (b(i) - dummy)) dummy=0 len=0 END Do k=k+1 write(5,102) k,(XK1(i),i=1,n) 102 FORMAT (7x,I4,1000000('|',F16.4)) doi=1,n dif(i)=Xk1(i)-XK(i) if( abs(dif(i))>tol) then XK=XK1 go to 300 end if end do Write (5,*) 'solution is convergence' write(*,*) 'END OF SOR PROGRAM' END Program

Tol=0.005 0,0,0,0,0TX SOR 1.2

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Doolittle LU decomposition

program LU !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Dollitle LU Factorization with partial pivoting !! !! 2) Solving linear system (AX=b) !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 05/02/14 Milad Naderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Specification of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real*8,Allocatable::A(:,:),b(:),x(:),L(:,:),U(:,:),Uinv(:,:),d(:,:),P(:,:),v(:,:) Integer,Allocatable:: ri(:),II(:,:) Real*8::dummy,vas integer::i,j,k,n,rk ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((L)) is Lower triangular matrix with unit diagonal elements ! ((U)) is Upper triangular matrix ! ((n)) is size of matrix !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write & Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input) Open(Unit=4,File='U.dat') !Upper triangular matrix Open(Unit=5,File='L.dat') !Lower triangular matrix with unit diagonally Open(Unit=6,File='X.dat') !Solution of linear system Open(Unit=7,File='P.dat') !Permutation matrix ! Read A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) allocate (A(n,n)) allocate (U(n,n));allocate (L(n,n));allocate (X(n)); allocate (b(n));allocate (d(1,n));allocate (ri(n-1));allocate(II(n,n));allocate(P(n,n));allocate(V(n,n));allocate(Uinv(n,n)) do i=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2) ! Read b vector

52

do i=1,n read(3,*) b(i) end do !close(3) U=A ! U matrix calculation do k=1,n-1 call maxfound(U,rk,k,n) ri(k)=rk do j=k,n d(1,j)=U(k,j);U(k,j)=U(rk,j);U(rk,j)=d(1,j) end do do i=k+1,n U(i,k)=-1.*(U(i,k)/U(k,k)) end do do i=k+1,n do j=k+1,n U(i,j)=U(i,j)+U(i,k)*U(k,j) end do end do end do do i=1,n do j=1,n if ( i>j ) then U(i,j)=0. end if end do end do ! Write Upper triangular matrix in U.dat! do 2 i = 1,n do 1 j = 1,n 1 write(4,'(F10.3,$)') U(i,j) 2 write(4,*) ! Identity matrix Formation for Permutation matrix creation! !write(5,*) (ri(i),i=1,n-1) do i=1,n do j=1,n if (i==j) then II(i,j)=1 else II(i,j)=0 end if end do end do V=II;P=II do k=1,n-1 if (ri(k) /= k) then d(1,:)=II(k,:) P(k,:)=II(ri(k),:) P(ri(k),:)=d(1,:) end if V=MATMUL(P,V) P=II end do do 2000 i = 1,n do 1000 j = 1,n 1000 write(7,'(F10.5,$)') V(i,j) 2000 write(7,*)

53

! L matrix Formation! Call inverse(U,Uinv,n) L=MATMUL(V,MATMUL(A,Uinv)) do 200 i = 1,n do 100 j = 1,n 100 write(5,'(F10.5,$)') L(i,j) 200 write(5,*) !Solving Linear system equation by inverse matrix of L and U' !Comment: Since we MUST use Inverse subroutine for L matrix calculation ( at above procedure); we use here matrix inversion for ! Finding of system solution instead using of Forward and Backward Subsituation ! b=MATMUL(V,b) Call inverse(L,P,n) X=MATMUL(P,b) X=MATMUL(Uinv,X) write(6,*) 'X(i) |------Result------|' do i=1,n Write(6,30) i,X(i) 30 Format (/,x,'X(',I3,')',2x,'|',F10.5) end do END Program Subroutine maxfound(U,rk,k,n) !============================================================ ! Max Found at one array ! Method: Based on Comparing entires in array ! Milad Naderi 2014 !----------------------------------------------------------- ! input ... ! U(n,n) - array of coefficients for matrix U ! n - size of matrix ! k - dummy ! output ... ! rk - number of row including maximum absolute value !=========================================================== Implicit None Integer::k,n,rk,i Real*8::max Real*8,Dimension(n,n)::U rk=k max=abs(U(k,k)) do i=k+1,n IF( max < abs(U(i,k)) ) then max=abs(U(i,k)) rk=i end if end do RETURN ; END SUBROUTINE subroutine inverse(a,c,n) !============================================================ ! Inverse matrix ! Method: Based on Doolittle LU factorization for Ax=b ! Alex G. December 2009 !----------------------------------------------------------- ! input ... ! a(n,n) - array of coefficients for matrix A ! n - dimension ! output ...

54

! c(n,n) - inverse matrix of A ! comments ... ! the original matrix a(n,n) will be destroyed ! during the calculation !=========================================================== implicit none integer:: n Real*8,Dimension(n,n)::a,c,U,L Real*8,Dimension(n)::b,d,x !double precision L(n,n), U(n,n), b(n), d(n), x(n) Real*8::coeff integer::i,j,k ! step 0: initialization for matrices L and U and b ! Fortran 90/95 aloows such operations on matrices L=0.0 U=0.0 b=0.0 ! step 1: forward elimination do k=1, n-1 do i=k+1,n coeff=a(i,k)/a(k,k) L(i,k) = coeff do j=k+1,n a(i,j) = a(i,j)-coeff*a(k,j) end do end do end do ! Step 2: prepare L and U matrices ! L matrix is a matrix of the elimination coefficient ! + the diagonal elements are 1.0 do i=1,n L(i,i) = 1.0 end do ! U matrix is the upper triangular part of A do j=1,n do i=1,j U(i,j) = a(i,j) end do end do ! Step 3: compute columns of the inverse matrix C do k=1,n b(k)=1.0 d(1) = b(1) ! Step 3a: Solve Ld=b using the forward substitution do i=2,n d(i)=b(i) do j=1,i-1 d(i) = d(i) - L(i,j)*d(j) end do end do ! Step 3b: Solve Ux=d using the back substitution x(n)=d(n)/U(n,n) do i = n-1,1,-1 x(i) = d(i) do j=n,i+1,-1 x(i)=x(i)-U(i,j)*x(j) end do x(i) = x(i)/u(i,i) end do ! Step 3c: fill the solutions x(n) into column k of C do i=1,n c(i,k) = x(i) end do b(k)=0.0 end do Return; end subroutine

55

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56

-1 ) SSOR

program SSOR !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Solving linear system (AX=b) by Symmetric Successive over relaxation method(SSOR) !! !! 2) Investigation on effect of number of iterates !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 04/05/14 Milad Naderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Specification of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Allocatable::A(:,:),b(:),Xk(:),Xk5(:),dif(:),Xk1(:) !Real,Allocatable:: Real::tol,maxdif,dummy,len,w integer::i,j,k,n character::con,y ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk5)) is unknowns vector for k+1/2 step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write & Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Write(*,*) 'Please enter tolerance,tol= ' Read (*,*) tol Write(*,*) 'Please enter relaxation factor,w= ' Read (*,*) w !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input) Open(Unit=4,File='X0.dat',status='OLD', action='READ') !availabe initial value vector (input) Open(Unit=5,File='Results.dat') !Results file contains iterative number,maximum diffrence and xi at each level. ! Read A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) allocate (A(n,n)) allocate (Xk(n));allocate (Xk1(n));allocate (dif(n));allocate (Xk5(n)); allocate (b(n)) do i=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2)

57

! Read B vector do i=1,n read(3,*) b(i) end do !close(3) ! Read X0 vector k=0 do i=1,n read(4,*) Xk(i) end do !close(4) Write(5,100) (i,i=1,n) 100 FORMAT (((10x,'K|',1000000(3x,'X('I10')|')))) write(5,101) k,(XK(i),i=1,n) 101 FORMAT (7x,I4,1000000('|',F16.4)) 300 dummy=0;len=0 If (k>0 .AND. MOD(k,50)==0) then Write (5,200) k 200 FORMAT(/,'After'I10'steps, solution was not convergence',/) Write (*,400) k 400 FORMAT(/,'After'I10'steps, solution was not convergence',/) END IF ! Forward SOR ! Do i=1,n Do j=1,i-1 len=len+A(i,j)*Xk5(j) End do Do j=i+1,n dummy=dummy+A(i,j)*Xk(j) end Do dummy=len+dummy Xk5(i)=((1-w)* XK(i)) + (( w / A(i,i)) * (b(i) - dummy)) dummy=0 len=0 END Do ! Backward SOR ! Do i=n,1,-1 Do j=1,i-1 len=len+A(i,j)*Xk5(j) End do Do j=i+1,n dummy=dummy+A(i,j)*Xk1(j) end Do dummy=len+dummy Xk1(i)=((1-w)* XK5(i)) + (( w / A(i,i)) * (b(i) - dummy)) dummy=0 len=0 END Do k=k+1 write(5,102) k,(XK1(i),i=1,n) 102 FORMAT (7x,I4,1000000('|',F16.4)) do i=1,n dif(i)=Xk1(i)-XK(i) if( abs(dif(i))>tol) then

58

XK=XK1 go to 300 end if end do Write (5,*) 'solution is convergence' write(*,*) 'END OF SOR PROGRAM' END Program

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60

program LSOR !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Solving linear system (AX=b) by Line Successive over relaxation method(LSOR) !! !! 2) Investigation on effect of number of iteratives !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 06/05/14 Milad Naderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

implicit none !^^^^^^^^^^^^^^^^^^ Specification of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Allocatable::A(:,:),b(:),Xk(:),dif(:),Xk1(:),Aprim(:,:),bprim(:),Xkprim(:) !Real,Allocatable:: Real::tol,maxdif,dummy,len,w,f,e integer::i,j,k,n,NL,p,aa,bb,ia,jb,t,l,m,npl ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk5)) is unknowns vector for k+1/2 step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order ! ((NL)) is number of lines which divides coefficient matrix(A) to N/NL submatrix of NL order ! ((NPL)) is number of points in each line !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Write & Read Principal data ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Write(*,*) 'Please enter tolerance,tol= ' Read (*,*) tol Write(*,*) 'Please enter relaxation factor,w= ' Read (*,*) w Write(*,*) 'Please enter number of lines(NL),Note NL must be a integer multiplier of Matrix order(N),NL= ' Read (*,*) NL !^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ OPEN DAT FILES ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='b.dat',status='OLD', action='READ') !available b vector (input) Open(Unit=4,File='X0.dat',status='OLD', action='READ') !availabe initial value vector (input) Open(Unit=5,File='Results.dat') !Results file contains iterative number,maximum diffrence and xi at each level. ! INPUT A matrix n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) NPL=N/NL;ia=1;jb=NPL;e=0;f=0 allocate (A(n,n));allocate (APRIM(NPL,NPL)) allocate (Xk(n));allocate (Xk1(n));allocate (dif(n)) allocate (b(n));allocate (bprim(NPL));allocate (Xkprim(NpL))

62

do i=1,n read(2,*) ( A(i,j), j = 1, n ) end do !close(2) ! INPUT B vector do i=1,n read(3,*) b(i) end do !close(3) ! INPUT X0 vector k=0 do i=1,n read(4,*) Xk(i) end do !close(4) ! Write header in Results.dat ! Write(5,100) (i,i=1,n);100 FORMAT (((10x,'K|',1000000(3x,'X('I10')|'))));write(5,101) k,(XK(i),i=1,n); 101 FORMAT (7x,I4,1000000('|',F16.4)) ! First of loop! 300 dummy=0;len=0 ! Write a note text if number of iterations be large in Results.dat ! IF (k>0 .AND. MOD(k,50)==0) then;Write (5,200) k;200 FORMAT(/,'After'I10'steps, solution was not convergence',/);Write (*,400) k; 400 FORMAT(/,'After'I10'steps, solution was not convergence',/);END IF ! LSOR ! Do P=1,NL aa=((p-1)*NPL)+1;bb=P*NPL DO i=aa,bb DO j=1,n IF (j>bb) THEN f=f+A(i,j)*XK(j) ELSE IF ( (J =1) ) THEN e=e+A(i,j)*XK1(j) END IF END DO bprim(1+i-aa)=((1-w)*XK(i))+(w*b(i)/A(i,i))-e-f e=0;f=0 END DO Do L=1,NPL DO M=1,NPL IF (L==M) THEN APRIM(L,L)=1+w ELSE APRIM(L,M)=((w*A(aa+L-1,aa+m-1))/A(aa+L-1,aa+L-1)) END IF END DO END DO call GS (APRIM,Bprim,NPL,XKPRIM,tol) do t=1,NpL XK1(aa+t-1)=XKprim(t) END DO END DO

63

k=k+1 write(5,102) k,(XK1(i),i=1,n) 102 FORMAT (7x,I4,1000000('|',F16.4)) do i=1,n dif(i)=Xk1(i)-XK(i) if( abs(dif(i))>tol) then XK=XK1 go to 300 end if end do Write (5,*) 'solution is convergence' write(*,*) 'END OF SOR PROGRAM' END Program Subroutine GS(A,B,N,Xk1,tol) implicit none !^^^^^^^^^^^^^^^^^^ Sepecificaton of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real,Dimension(n,n)::A Real,Dimension(n)::b,Xk,Xk1,dif Real::tol,maxdif,dummy,len integer::i,j,k,n character::con,y ! ((A)) is coefficient square matrix (n*n) ! ((b)) is RHS of Ax=b and is a vector (n*1) ! ((Xk)) is unknowns vector for k step ! ((Xk1)) is unknowns vector for k+1 step ! ((Tolerance)) is allowable diffrence between Xk and Xk+1 (convergence factor) ! ((n)) is matrix order XK=0. k=0 300 dummy=0;len=0 If (k>0 .AND. MOD(k,50)==0) then Write (*,400) k 400 FORMAT(/,'After'I10'steps, solution was not convergence',/) END IF Do i=1,n Do j=1,i-1 len=len+A(i,j)*Xk1(j) End do Do j=i+1,n len=len+A(i,j)*Xk(j) end Do dummy=len+dummy Xk1(i)=(1./A(i,i))*(b(i)-dummy) dummy=0 len=0 END Do k=k+1 do i=1,n dif(i)=Xk1(i)-XK(i)

64

if( abs(dif(i))>tol) then XK=XK1 go to 300 end if end do write(*,*) 'END OF Gauss Sidel PROGRAM' END Subroutine;Return

)

.

L .

: .

TA LL T

ij jiL L Do i=1,n

11

2 2

1

( )i

ii ii ik

k

L a L

1

1

1( ) 1, 2,...,

i

ji ij ik jk

kii

L a L L j i i nL

END DO

program CholeskiMilad !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Purposes: !! !! 1) Choleski Factorization in form of A=L*LT !! !! !! !! Notes: !! !! !! !! !! !! Record of revisions: !! !! Date Programmer Description of change !! !! ==== ========== ===================== !! !! 04/17/14 Milad Naderi Original code !! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! implicit none !^^^^^^^^^^^^^^^^^^ Sepecificaton of all parameters used in this program ^^^^^^^^^^^^^^^^^^^^ ! Real*8,Allocatable::A(:,:),L(:,:),LT(:,:) Real*8::dummy,dummy2 integer::i,j,k,n ! ((A)) is Symmetric Positive definite matrix (n*n) ! ((L)) is Lower triangular matrix created by choleski factorization ! ((LT)) is Transpose of Lower triangular matrix created by choleski factorization

65

! ((n)) is matrix order ! ((i,j,k)) are indices ! ((dummy and dummy2)) are dummy parameter in loops for calculate Sigma values. !** OPEN DAT FILES **! Open(Unit=2,File='A.dat',status='OLD', action='READ') !availabele A matrix (input) Open(Unit=3,File='L.dat') !Calculated L Matrix (output) Open(Unit=4,File='LT.dat') !Calculated LT Matrix (output) !** calculate size of A **! n=0 do read(2,*,end=20) dummy n=n+1 end do 20 rewind(2) !** Allocate A,L and LT matrices **! allocate (A(n,n));allocate (L(n,n));allocate (LT(n,n)) !** Read A **! do i=1,n read(2,*) ( A(i,j), j = 1, n ) end do !** First of calculations **! dummy=0.;dummy2=0. Do i=1,n Do k=1,i-1 dummy=dummy+(L(i,k))**2 END DO L(i,i)=sqrt((A(i,i)-dummy)) dummy=0 Do j=i+1,n Do k=1,i-1 dummy2=dummy2+(L(i,k)*L(j,k)) END DO L(j,i)=((A(j,i)-dummy2)/(L(i,i))) dummy2=0 END DO END DO DO 2000 i = 1,n DO 1000 j = 1,n 1000 write(3,'(F10.5,$)') L(i,j) 2000 write(3,*) DO i=1,n DO j=1,n LT(i,j)=L(j,i) END DO END DO DO 200 i = 1,n DO 100 j = 1,n 100 write(4,'(F10.5,$)') LT(i,j) 200 write(4,*) write(*,*) 'END OF Choleski decomposition PROGRAM' END Program

66

:

16 4 4 -4

4 10 4 2

4 4 6 -2

-4 2 -2 4

A

4.00000 0.00000 0.00000 0.00000

1.00000 3.00000 0.00000 0.00000

1.00000 1.00000 2.00000 0.00000

-1.00000 1.00000 -1.00000 1.00000

L

4.00000 1.00000 1.00000 -1.00000

0.00000 3.00000 1.00000 1.00000

0.00000 0.00000 2.00000 -1.00000

0.00000 0.00000 0.00000 1.00000

TL

2 :

25 15 -5

15 18 0

-5 0 11

A

5.00000 0.00000 0.00000

3.00000 3.00000 0.00000

-1.00000 1.00000 3.00000

L

5.00000 3.00000 -1.00000

0.00000 3.00000 1.00000

0.00000 0.00000 3.00000

TL

67

2 ) (

Linear algebra .

2 ) (

Linear algebra :

Linear Algebra

MATLAB Linear Algebra Functions

Matrices in the MATLAB Environment

Systems of Linear Equations

Inverses and Determinants

Factorizations

Powers and Exponentials

Eigenvalues

Singular Values

Category Function Description

Matrix analysis norm Matrix or vector norm

normest Estimate the matrix 2-norm

rank Matrix rank

det Determinant

trace Sum of diagonal elements

null Null space

orth Orthogonalization

rref Reduced row echelon form

subspace Angle between two subspaces

68

Category Function Description

Linear equations \ and / Linear equation solution

inv Matrix inverse

cond Condition number for inversion

condest 1-norm condition number

estimate

chol Cholesky factorization

cholinc Incomplete Cholesky

factorization

linsolve Solve a system of linear

equations

lu LU factorization LU

ilu Incomplete LU factorization LU

luinc Incomplete LU factorization LU

qr Orthogonal-triangular

decomposition

lsqnonneg Nonnegative least-squares

pinv Pseudoinverse

lscov Least squares with known

covariance

Eigenvalues and

singular values

eig Eigenvalues and eigenvectors

svd Singular value decomposition SVD

eigs A few eigenvalues

svds A few singular values

poly Characteristic polynomial

polyeig Polynomial eigenvalue problem

condeig Condition number for

69

Category Function Description

eigenvalues

hess Hessenberg form

qz QZ factorization QZ

schur Schur decomposition

Matrix functions expm Matrix exponential Exponential

logm Matrix logarithm

sqrtm Matrix square root

funm Evaluate general matrix

function

17

:AMELP arbegla raeniL

sdnammoC sisaB

laimonyloPcitsiretcarahC

tcudorPssorC

woReteleD

tnanimreteD

noisnemiD

tcudorPtoD

seulavnegiE

srotcevnegiE

noitanimilEnaissuaG

serauqStsaeL

evloSraeniL

b=X.A

paM

esrevnIxirtaM

ylpitluMralacSxirtaM

ecapSlluN

)secapslluN) (sisab (

irtaMmodnaRx

mroFnolehcEwoRdecudeR

xirtaMbuS

esopsnarT

07

:ELPAM

>

5*5

>

. :

>

2

>

>

>

>

27

V M

>

>

>

37

( ) 3

.

evlos - nork leknah dnoc ruhcs dvsg giedro sdvs dvs lohc rq uli ul

yrellag

( ) 3

:ul

:

tinu a htiw L xirtam ralugnairt rewol a ,U ni xirtam ralugnairt reppu na snruter )A(ul = ]P,U,L[

.A*P = U*L taht hcus ,P xirtam noitatumrep a dna ,lanogaid

:

74

ilu:

.

.

ilu(A,setup) returns L+U-speye(size(A)), where L is a unit lower triangular matrix and U

is an upper triangular matrix.

[L,U] = ilu(A,setup) returns a unit lower triangular matrix in L and an upper triangular

matrix in U.

[L,U,P] = ilu(A,setup) returns a unit lower triangular matrix in L, an upper triangular

matrix in U, and a permutation matrix in P.

ilu works on sparse square matrices only

qr:

The qr function performs the orthogonal-triangular decomposition of a matrix. This factorization is

useful for both square and rectangular matrices. It expresses the matrix as the product of a real

complex unitary matrix and an upper triangular matrix.

[Q,R] = qr(A) produces an upper triangular matrix R of the same dimension as A and a unitary

matrix Q so that A = Q*R. For sparse matrices, Q is often nearly full

:

A = [ 1 2 3

4 5 6

7 8 9

10 11 12 ]

[Q,R] = qr(A)

Q =

-0.0776 -0.8331 0.5444 0.0605

-0.3105 -0.4512 -0.7709 0.3251

-0.5433 -0.0694 -0.0913 -0.8317

-0.7762 0.3124 0.3178 0.4461

R =

-12.8841 -14.5916 -16.2992

0 -1.0413 -2.0826

0 0 0.0000

0 0 0

75

chol:

.

R = chol(A) produces an upper triangular matrix R from the diagonal and upper triangle of matrix

A, satisfying the equation R'*R=A where R' is transpose of R. The lower triangle is assumed to be

the (complex conjugate) transpose of the upper triangle. Matrix A must be positive definite.

L = chol(A,'lower') produces a lower triangular matrix L from the diagonal and lower triangle

of matrix A, satisfying the equation L*L'=A

:

A =

1 -1 -1 -1 -1

-1 2 0 0 0

-1 0 3 1 1

-1 0 1 4 2

-1 0 1 2 5

C=chol(A)

ans =

1 -1 -1 -1 -1

0 1 -1 -1 -1

0 0 1 -1 -1

0 0 0 1 -1

0 0 0 0 1

isequal(C'*C,A)

ans =

1

svd:

( Singular Value Decomposition)

s = svd(X) returns a vector of singular values. .

[U,S,V] = svd(X) produces a diagonal matrix S of the same dimension as X, with nonnegative

diagonal elements in decreasing order, and unitary matrices U and V so that X = U*S*V'.

[U,S,V] = svd(X,0) produces the "economy size" decomposition. If X is m-by-n with m > n, then

svd computes only the first n columns of U and S is n-by-n.

[U,S,V] = svd(X,'econ') also produces the "economy size" decomposition. If X is m-by-n with

m >= n, it is equivalent to svd(X,0). For m < n, only the first m columns of V are computed and S is

m-by-m.

76

:

X =

1 2

3 4

5 6

7 8

[U,S,V] = svd(X)

U =

-0.1525 -0.8226 -0.3945 -0.3800

-0.3499 -0.4214 0.2428 0.8007

-0.5474 -0.0201 0.6979 -0.4614

-0.7448 0.3812 -0.5462 0.0407

S =

14.2691 0

0 0.6268

0 0

0 0

V =

-0.6414 0.7672

-0.7672 -0.6414

[U,S,V] = svd(X,0)

U =

-0.1525 -0.8226

-0.3499 -0.4214

-0.5474 -0.0201

-0.7448 0.3812

S =

14.2691 0

0 0.6268

V =

-0.6414 0.7672

-0.7672 -0.6414

svds:

.

.

s = svds(A) computes the six largest singular values and associated singular vectors of

matrix A. If A is m-by-n, svds(A) manipulates eigenvalues and vectors returned by eigs(B),

where B = [sparse(m,m) A; A' sparse(n,n)], to find a few singular values and vectors

of A. The positive eigenvalues of the symmetric matrix B are the same as the singular values

of A.

s = svds(A,k) computes the k largest singular values and associated singular vectors of

matrix A.

77

s = svds(A,k,sigma) computes the k singular values closest to the scalar shift sigma. For

example, s = svds(A,k,0) computes the k smallest singular values and associated singular

vectors.

ordeig:

E = ordeig(T) takes a quasitriangular Schur matrix T, typically produced by schur, and

returns the vector E of eigenvalues in their order of appearance down the diagonal of T.

E = ordeig(AA,BB) takes a quasitriangular matrix pair AA and BB, typically produced by qz,

and returns the generalized eigenvalues in their order of appearance down the diagonal of AA-

*BB.

:

A= rand(10)

A =

Columns 1 through 8

0.8223 0.3262 0.6153 0.5815 0.0702 0.7337 0.1239 0.7284

0.7229 0.9730 0.5831 0.9377 0.0693 0.6505 0.4674 0.4068

0.9259 0.3650 0.6983 0.0478 0.1360 0.5163 0.6567 0.9383

0.4926 0.3091 0.0293 0.0540 0.7889 0.3264 0.2902 0.2554

0.6549 0.1209 0.5279 0.0206 0.0924 0.6618 0.7545 0.5332

0.8901 0.9158 0.0321 0.6815 0.2379 0.1176 0.5581 0.9548

0.5385 0.1355 0.8271 0.5986 0.2436 0.1478 0.4278 0.2677

0.2822 0.3321 0.3400 0.1140 0.1048 0.0198 0.2672 0.2501

0.9760 0.8975 0.8467 0.7962 0.8584 0.9643 0.7537 0.9277

0.0364 0.4996 0.2461 0.6179 0.6982 0.9704 0.8984 0.0686

Columns 9 through 10

0.2994 0.1125

0.5916 0.5158

0.2033 0.8378

0.6359 0.9208

0.7984 0.4982

0.5017 0.2776

0.6508 0.6525

0.7960 0.9173

0.2334 0.5098

0.6008 0.9742

>> [U, T] = schur(A)

U =

Columns 1 through 7

87

1522.0 4930.0- 9022.0- 1303.0 5354.0- 9281.0- 5352.0-

4750.0 7282.0- 6361.0 4565.0 5061.0- 2352.0- 2463.0-

3330.0 3506.0- 4173.0- 4992.0- 0902.0 9722.0 3013.0-

2045.0 7514.0 7602.0 3901.0 3825.0 4201.0 2362.0-

4043.0- 3324.0 7714.0- 8710.0 0961.0 5954.0- 8982.0-

0426.0- 6350.0 4660.0 1153.0 6922.0 2974.0 7403.0-

9990.0- 8431.0 3151.0- 4844.0- 8650.0- 3052.0- 8872.0-

7361.0 6943.0- 6160.0 4410.0- 3534.0 2882.0- 6132.0-

6562.0 6632.0 5251.0- 0751.0- 7463.0- 3184.0 9054.0-

3102.0- 6030.0- 6717.0 8073.0- 8691.0- 8741.0- 0553.0-

01 hguorht 8 snmuloC

5352.0 3266.0 9620.0-

9034.0- 5793.0- 0560.0-

2292.0- 5771.0 0013.0

5542.0- 0542.0 2040.0

9740.0 5921.0- 9234.0

2011.0 1891.0 3132.0-

2792.0- 6660.0 3517.0-

3066.0 4181.0- 6722.0-

8152.0 8934.0- 9820.0

6740.0 3851.0 6303.0

= T

7 hguorht 1 snmuloC

1201.0- 9133.0 9091.0- 5711.0- 7012.0- 2014.0 7101.5

1960.0 0190.0- 4031.0- 5275.0 1423.0 7728.0- 0

5320.0- 4801.0 3713.0 8665.0- 7728.0- 0672.0- 0

0885.0 4790.0- 7510.0 4577.0 0 0 0

6643.0- 2278.0 9954.0 0 0 0 0

1122.0- 9954.0 4613.0- 0 0 0 0

2783.0- 0 0 0 0 0 0

9057.0- 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

01 hguorht 8 snmuloC

6343.0- 1829.0- 7325.0

3721.0 9950.0 6841.0-

5783.0 1093.0- 7991.0

1790.0 0870.0- 9612.0-

0475.0- 7515.0- 2012.0

3270.0 2421.0- 4270.0

7940.0 4811.0 8535.0

7951.0 1530.0 2783.0-

4904.0 9731.0 0

9731.0 6091.0- 0

)T(giedro >>

79

ans =

5.1017

-0.8277 + 0.2991i

-0.8277 - 0.2991i

0.7754

0.4599 + 0.5253i

0.4599 - 0.5253i

-0.3872 + 0.6343i

-0.3872 - 0.6343i

0.1379 + 0.2793i

gsvd:

[U,V,X,C,S] = gsvd(A,B) returns unitary matrices U and V, a (usually) square matrix X, and

nonnegative diagonal matrices C and S so that

A = U*C*X'

B = V*S*X'

C'*C + S'*S = I

A and B must have the same number of columns, but may have different numbers of rows. If A

is m-by-p and B is n-by-p, then U is m-by-m, V is n-by-n and X is p-by-q where q =

min(m+n,p).

sigma = gsvd(A,B) returns the vector of generalized singular values,

sqrt(diag(C'*C)./diag(S'*S)).

:

A = reshape(1:15,5,3)

B = magic(3)

A =

1 6 11

2 7 12

3 8 13

4 9 14

5 10 15

B =

8 1 6

3 5 7

4 9 2

[U,V,X,C,S] = gsvd(A,B)

X =

2.8284 -9.3761 -6.9346

-5.6569 -8.3071 -18.3301

2.8284 -7.2381 -29.7256

81

C =

0.0000 0 0

0 0.3155 0

0 0 0.9807

0 0 0

0 0 0

S =

1.0000 0 0

0 0.9489 0

0 0 0.1957

[U,V,X,C,S] = gsvd(A,B,0)

U =

0.5700 -0.6457 -0.4279

-0.7455 -0.3296 -0.4375

-0.1702 -0.0135 -0.4470

0.2966 0.3026 -0.4566

0.0490 0.6187 -0.4661

C =

0.0000 0 0

0 0.3155 0

0 0 0.9807

The other three matrices, V, X, and S are the same as those obtained with the full

decomposition.

The generalized singular values are the ratios of the diagonal elements of C and S.

sigma = gsvd(A,B)

sigma =

0.0000

0.3325

5.0123

schur:

( Schur)

The schur command computes the Schur form of a matrix.

T = schur(A) returns the Schur matrix T.

T = schur(A,flag) for real matrix A, returns a Schur matrix T in one of two forms

depending on the value of flag:

'complex' T is triangular and is complex if A has complex eigenvalues.

'real' T has the real eigenvalues on the diagonal and the complex eigenvalues in 2-by-2

blocks on the diagonal. 'real' is the default.

80

If A is complex, schur returns the complex Schur form in matrix T. The complex Schur form

is upper triangular with the eigenvalues of A on the diagonal

[U,T] = schur(A,...) also returns a unitary matrix U so that A = U*T*U' and

U'*U = eye(size(A)).

:

H is a 3-by-3 eigenvalue test matrix:

H = [ -149 -50 -154

537 180 546

-27 -9 -25 ]

Its Schur form is

schur(H)

ans =

1.0000 -7.1119 -815.8706

0 2.0000 -55.0236

0 0 3.0000

The eigenvalues, which in this case are 1, 2, and 3, are on the diagonal. The fact that the off-

diagonal elements are so large indicates that this matrix has poorly conditioned eigenvalues;

small changes in the matrix elements produce relatively large changes in its eigenvalues.

cond:

.

. 1 .

c = cond(X) returns the 2-norm condition number, the ratio of the largest singular value of X

to the smallest.

c = cond(X,p) returns the matrix condition number in p-norm:

norm(X,p) * norm(inv(X),p

If p is... Then cond(X,p) returns the...

1 1-norm condition number

2 2-norm condition number

'fro' Frobenius norm condition number

28

...eht snruter )p,X(dnoc nehT ...si p fI

rebmun noitidnoc mron ytinifnI fni

:dnocr

. k )y(

1

1 1

1 1y

A A k

y 1 y

.

:leknah

. leknaH

)c(leknah = H

)r,c(leknah = H

.

. dne c Pr] [: 2 j i P j i H)1 ( ) , (

esohw dna c si nmuloc tsrif esohw xirtam leknaH erauqs eht snruter )c(leknah = H

.lanogaid-itna tsrif eht woleb orez era stnemele

fI .r si wor tsal esohw dna c si nmuloc tsrif esohw xirtam leknaH a snruter )r,c(leknah = H

.sliaverp c fo tnemele tsal eht ,r fo tnemele tsrif eht morf sreffid c fo tnemele tsal eht

:

;01:7 = r ;3:1 = c

)r,c(leknah = h

= h

8 3 2 1

9 8 3 2

01 9 8 3

]01 9 8 3 2 1[ = p

38

:nork

.

)Y,X(nork = K

. Kq n p m, Yq p, Xn m,

:

)3(dnar=A >>

= A

5872.0 4319.0 7418.0

9645.0 4236.0 8509.0

5759.0 5790.0 0721.0

)2(dnar=B >>

= B

6079.0 9469.0

2759.0 6751.0

)B,A(nork >>

= sna

3072.0 7862.0 5688.0 3188.0 8097.0 1687.0

6662.0 9340.0 3478.0 0441.0 8977.0 4821.0

8035.0 7725.0 8316.0 2016.0 2978.0 0478.0

5325.0 2680.0 3506.0 7990.0 0768.0 8241.0

3929.0 9329.0 7490.0 1490.0 3321.0 5221.0

5619.0 9051.0 4390.0 4510.0 5121.0 0020.0

:evlos

.

)qe(evlos

)rav,qe(evlos

)nqe,...,2qe,1qe(evlos

)nrav,...,2rav,1rav,nqe,...,2qe,1qe(evlos = g

84

:

A = solve('3*x+5*y-6*z=4', '5*x-3*z=4', '2*x+6*y-2*z=0)'

A =

x: [1x1 sym]

y: [1x1 sym]

z: [1x1 sym]

>>A.x

ans =

16/53

>>A.y

ans =

-20/53

>>A.z

ans =

-44/53

gallery:

.

[A,B,C,...] = gallery(matname,P1,P2,...) returns the test matrices specified by the

quoted string matname. The matname input is the name of a matrix family selected from the table

below. P1,P2,... are input parameters required by the individual matrix family. The number of

optional parameters P1,P2,... used in the calling syntax varies from matrix to matrix..

:

85

:

tridiag Tridiagonal matrix (sparse)

A = gallery('tridiag',c,d,e) returns the tridiagonal matrix with subdiagonal c,

diagonal d, and superdiagonal e. Vectors c and e must have length(d)-1.

A = gallery('tridiag',n,c,d,e), where c, d, and e are all scalars, yields the Toeplitz

tridiagonal matrix of order n with subdiagonal elements c, diagonal elements d, and

superdiagonal elements e. This matrix has eigenvalues

d + 2*sqrt(c*e)*cos(k*pi/(n+1))

where k = 1:n.

A = gallery('tridiag',n) is the same as A = gallery('tridiag',n,-1,2,-1), which

is a symmetric positive definite M-matrix (the negative of the second difference matrix).

68

:

)3,2,1,6,'gaidirt'(yrellag = A

= A

2 )1,1(

1 )1,2(

3 )2,1(

2 )2,2(

1 )2,3(

3 )3,2(

2 )3,3(

1 )3,4(

3 )4,3(

2 )4,4(

1 )4,5(

3 )5,4(

2 )5,5(

1 )5,6(

3 )6,5(

2 )6,6(

87

: 1. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics (Second Edition), Springer 2007.

2. B.N. Datta, Numerical Linear Algebra And Applications, Northern Illinois University.

3. G.H. Golub, C.F. Van Loan, Matrix Computations (Third Edition), The Johns Hopkins University

Press, 1996.

4. T.J.Chung, Computational Fluid Dynamics (Second Edition), Cambridge University Press, 2010.