Jacobi Method. Solution of a linear system via iterations with Jacobi pre-conditioner
discussed in class may be re-expressed as follows. Let Mx_()=b_() be the linear system to
be solved. If all of the diagonal entries of M_() are non-zero, then we may re-express the
system as
D_(_())E_(_())x_(_())=b_(_())
where D_() is the diagonal matrix containing the diagonal entries of M_() and E_() contains the
off-diagonal entries of M_(_()). Jacobi iterations may be executed with the following:
x_(^())(k+1)=D_(^())(-1)(b_(_())-Ex_(_())^(k))
where superscript k is the iteration count. Note that D_(^())(-1) is simply the diagonal matrix
whose diagonal entries are the reciprocals of the corresponding entries of D_().
Now, consider the system
[[5,2,-1],[3,7,3],[1,-4,6]][[x],[y],[z]]=[[2],[-1],[1]]
Solve using the Jacobi iterations previously described with x_(^())(0)=[[0,0,0]]^(T) as the initial
guess. Solve until epsi _(an: )<0.1, where
epsi _(and )=|x_(^())(k+1)-x_(^())(k)|
Note that the previous tolerance is defined through the norm |n|=(a^(2)+b^(2)+c^(2))^((1)/(2)) with
a,b and c being the entries of vector n_(). How many iterations are required to converge to
the specified tolerance?
Note: Show all of your work. You may not use inherent (intrinsic) functions in packages
such as Matlab that execute the Jacobi method directly. However, you may use intrinsic
operations such as addition and multiplication of matrices.
1. Jacobi Method. Solution of a linear system via iterations with Jacobi pre-conditioner
be solved. If all of the diagonal entries of M are non-zero, then we may re-express the
system as
Dx+Ex=
where D is the diagonal matrix containing the diagonal entries of M and E contains the off-diagonal entries of M . Jacobi iterations may be executed with the following:
x*=D(b-Ex)
where superscript k is the iteration count. Note that D- is simply the diagonal matrix
whose diagonal entries are the reciprocals of the corresponding entries of D
Now, consider the system
52-1x][2 373 [1-46z][1
Solve using the Jacobi iterations previously described with x=000 as the initia guess. Solve until <0.1, where
=*-|
Note that the previous tolerance is defined through the norm = (+ ba+c) with , b and c being the entries of vector t . How many iterations are required to converge to the specified tolerance?
Note: Show all of your work. You may not use inherent (intrinsic) functions in packages such as Matlab that execute the Jacobi method directly. However, you may use intrinsic operations such as addition and multiplication of matrices.
