Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/January 2007

Problem 1

Let $x_{1},x_{2},\ldots ,x_{n}$ be distinct real numbers, and consider the matrix

V=\left({\begin{array}{ccccc}1&x_{1}&x_{1}^{2}&\cdots &x_{1}^{n-1}\\1&x_{2}&x_{2}^{2}&\cdots &x_{2}^{n-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{n}&x_{n}^{2}&\cdots &x_{n}^{n-1}\\\end{array}}\right)

Prove that $V^{-1}$ can be expressed as

V^{-1}=\left({\begin{array}{c}\\\\\alpha ^{(1)},\alpha ^{(2)},\alpha ^{(3)},\cdots ,\alpha ^{(n)}\\\\\\\end{array}}\right)

where the column vector $\alpha ^{(j)}=(\alpha _{1}^{(j)},\alpha _{2}^{(j)},\cdots ,\alpha _{n}^{(j)})^{T}$ generates a polynomial

p_{j}(x)=\sum _{i=1}^{n}\alpha _{i}^{(j)}x^{i-1}

satisfying

p_{j}(x)={\frac {\prod _{i=1,i\neq j}^{n}(x-x_{i})}{\prod _{i=1,i\neq j}^{n}(x_{j}-x_{i})}}

You may cite anything you know about polynomial interpolation to justify that $V$ is nonsingular.

Solution 1

We want to show that

\,\!VV^{-1}=I

or equivalently the $\,\!i$ th, $\,\!j$ th entry of $\,\!VV^{-1}$ is $\,\!1$ for $\,\!i=j$ and $\,\!0$ for $\,\!i\neq j$ i.e.

\{VV^{-1}\}_{ij}=\delta _{ij}=\left\{{\begin{array}{cc}1&{\mbox{ if }}i=j\\0&{\mbox{ if }}i\neq j\end{array}}\right.

First notice that

\,\!\{VV^{-1}\}_{ij}={\begin{bmatrix}1+x_{i}^{1}+x_{i}^{2}+\cdots +x_{i}^{n-1}\end{bmatrix}}{\begin{bmatrix}\alpha _{1}^{(j)}\\\alpha _{2}^{(j)}\\\vdots \\\alpha _{n}^{(j)}\end{bmatrix}}=\sum _{k=1}^{n}\alpha _{k}^{(j)}x_{i}^{k-1}

Also notice that

\,\!p_{j}(x_{i})=\sum _{k=1}^{n}\alpha _{k}^{(j)}x_{i}^{k-1}=\delta _{ij}

Hence,

\,\!\{VV^{-1}\}_{ij}=p_{j}(x_{i})=\delta _{ij}

Problem 2

Let $A\,\!$ be a real matrix of order $n\,\!$ with eigenvalues $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\,\!$ and $n\,\!$ linearly independent eigenvectors $v_{1},v_{2},\ldots ,v_{n}\,\!$ .

Suppose you want to find an eigenvector $v_{j}\,\!$ corresponding to the eigenvalue $\lambda _{j}\,\!$ and you are given $\sigma \,\!$ such that

|\lambda _{j}-\sigma |<|\lambda _{i}-\sigma |{\mbox{ for all }}i\neq j\,\!

Specify a numerical algorithm for finding $v_{j}\,\!$ and give a convergence proof for this algorithm demonstrating that it is convergent under appropriate circumstances

Solution 2

Shifted Inverse Power Method

Let

{\begin{aligned}{\tilde {A}}&=(A-\sigma I)^{-1}\\\end{aligned}}

Then,

{\tilde {\lambda _{i}}}={\frac {1}{|\lambda _{i}-\sigma |}}

which implies

{\tilde {\lambda _{j}}}>{\tilde {\lambda }}_{i}{\mbox{ for all }}i\neq j

Since shifting the eigenvalues and inverting the matrix does not affect the eigenvectors,

{\tilde {A}}v_{i}={\tilde {\lambda _{i}}}v_{i}\quad \quad i=1,2,\ldots ,n

Assume $\|v_{i}\|=1$ for all $\!\,i$ . Generate $w_{0},w_{1},w_{2},\ldots$ to find $\,\!v_{j}$ . Start with arbitrary $\,\!w_{0}$ such that $\,\!\|w_{0}\|=1$ .

For  $\,\!k=0,1,2,\ldots$ 
   $\,\!{\hat {w}}_{k+1}={\tilde {A}}w_{k}$ 
   $\,\!w_{k+1}={\frac {{\hat {w}}_{k+1}}{\|{\hat {w}}_{k+1}\|}}$ 
   $\lambda _{j}^{(k+1)}={\frac {({\hat {w}}_{k+1},w_{k})}{(w_{k},w_{k})}}={\frac {({\tilde {A}}w_{k},w_{k})}{(w_{k},w_{k})}}$  (Rayleigh quotient)
End

Convergence of Power Method

If $\,\!|\lambda _{1}|>|\lambda _{i}|$ , for all $\!\,i\neq 1$ , then $\,\!A^{k}w_{0}$ will be dominated by $\,\!v_{1}$ .

Since $v_{1},v_{2},\ldots ,v_{n}$ are linearly independent, they form a basis of $R^{n}\,\!$ . Hence,

w_{0}=\alpha _{1}v_{1}+\alpha _{2}v_{2}+\ldots +\alpha _{n}v_{n}\,\!

From the definition of eigenvectors,

{\begin{aligned}Aw_{0}&=\alpha _{1}\lambda _{1}v_{1}+\alpha _{2}\lambda _{2}v_{2}+\ldots +\alpha _{n}\lambda _{n}v_{n}\\A^{2}w_{0}&=\alpha _{1}\lambda _{1}^{2}v_{1}+\alpha _{2}\lambda _{2}^{2}v_{2}+\ldots +\alpha _{n}\lambda _{n}^{2}v_{n}\\&\vdots \\A^{k}w_{0}&=\alpha _{1}\lambda _{1}^{k}v_{1}+\alpha _{2}\lambda _{k}^{2}v_{2}+\ldots +\alpha _{n}\lambda _{n}^{k}v_{n}\\\end{aligned}}

To find a general form of $w_{k}\,\!$ , the approximate eigenvector at the kth step, examine a few steps of the algorithm:

{\begin{aligned}{\hat {w}}_{1}&=Aw_{0}\\w_{1}&={\frac {{\hat {w}}_{1}}{\|{\hat {w}}_{1}\|}}={\frac {Aw_{0}}{\|Aw_{0}\|}}\\{\hat {w}}_{2}&=Aw_{1}={\frac {A^{2}w_{0}}{\|Aw_{0}\|}}\\w_{2}&={\frac {{\hat {w}}_{2}}{\|{\hat {w}}_{2}\|}}={\frac {\frac {A^{2}w_{0}}{\|Aw_{0}\|}}{\frac {\|A^{2}w_{0}\|}{\|Aw_{0}\|}}}={\frac {A^{2}w_{0}}{\|A^{2}w_{0}\|}}\\{\hat {w}}_{3}&={\frac {A^{3}w_{0}}{\|A^{2}w_{0}\|}}\\w_{3}&={\frac {\frac {A^{3}w_{0}}{\|A^{2}w_{0}\|}}{\frac {\|A^{3}w_{0}\|}{\|A^{2}w_{0}\|}}}={\frac {A^{3}w_{0}}{\|A^{3}w_{0}\|}}\\\end{aligned}}

From induction,

w_{k}={\frac {A^{k}w_{0}}{\|A^{k}w_{0}\|}}

Hence,

{\begin{aligned}w_{k}&={\frac {A^{k}w_{0}}{\|A^{k}w_{0}\|}}\\&={\frac {\alpha _{1}\lambda _{1}^{k}v_{1}+\alpha _{2}\lambda _{2}^{k}v_{2}+\ldots +\alpha _{n}\lambda _{n}^{k}v_{n}}{\|A^{k}w_{0}\|}}\\&={\frac {\lambda _{1}^{k}(\alpha _{1}v_{1}+\alpha _{2}({\frac {\lambda _{2}}{\lambda _{1}}})^{k}v_{2}+\ldots +\alpha _{n}({\frac {\lambda _{n}}{\lambda _{1}}})^{k}v_{n})}{\|A^{k}w_{0}\|}}\end{aligned}}

Comparing a weighted $w_{k}\!\,$ and $v_{1}\!\,$ ,

{\begin{aligned}\left\|{\frac {\|A^{k}w_{0}\|}{\lambda _{1}^{k}}}w_{k}-\alpha _{1}v_{1}\right\|&=\left\|\alpha _{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}v_{2}+\ldots +\alpha _{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}v_{n}\right\|\\&\leq \alpha _{2}\left|{\frac {\lambda _{2}}{\lambda _{1}}}\right|^{k}+\ldots +\alpha _{n}\left|{\frac {\lambda _{n}}{\lambda _{1}}}\right|^{k}\end{aligned}}

since $\|v_{i}\|=1$ by assumption.

The above expression goes to $\,\!0$ as $k\rightarrow \infty$ since $\,\!|\lambda _{1}|>|\lambda _{i}|$ , for all $\!\,i\neq 1$ . Hence as $\!\,k$ grows, $\!\,w_{k}$ is parallel to $\!\,v_{1}$ . Because $\|w_{k}\|=1$ , it must be that $w_{k}\rightarrow \pm v_{1}$ .

Problem 3

Suppose A is an upper triangular, nonsingular matrix. Show that both Jacobi iterations and Gauss-Seidel iterations converge in finitely many steps when used to solve $A\mathbf {x} =\mathbf {b}$ .

Solution 3

Derivation of Iterations

Let $A=D+L+U\!\,$ where $D\!\,$ is a diagonal matrix, $L\!,$ is a lower triangular matrix with a zero diagonal and $U\!\,$ is an upper triangular matrix also with a zero diagonal.

The Jacobi iteration can be found by substituting into $Ax=b\!\,$ , grouping $L+U\!\,$ , and solving for $x\!\,$ i.e.

${\begin{aligned}Ax&=b\\(D+L+U)x&=b\\Dx+(L+U)x&=b\\Dx&=b-(L+U)x\\x&=D^{-1}(b-(L+U)x)\end{aligned}}$

Since $L=0\!\,$ by hypothesis, the iteration is

x^{(i+1)}=D^{-1}(b-Ux^{(i)})\!\,

Similarly, the Gauss-Seidel iteration can be found by substituting into $Ax=b\!\,$ , grouping $D+L\!\,$ , and solving for $x\!\,$ i.e.

${\begin{aligned}Ax&=b\\(D+L+U)x&=b\\(D+L)x+Ux&=b\\(D+L)x&=b-Ux\\x&=(D+L)^{-1}(b-Ux)\end{aligned}}$

Since $L=0\!\,$ by hypothesis, the iteration has identical from as Jacopi:

x^{(i+1)}=D^{-1}(b-Ux^{(i)})\!\,

Convergence in Finite Number of Steps

Jacobi and Gauss-Seidel are iterative methods that split the matrix $A\in \mathbb {R} ^{n\times n}\!\,$ into $D\!\,$ , $U\!\,$ , and $L\!\,$ : Diagonal, Upper (everything above the diagonal), and Lower (everything below the diagonal) triangular matrices, respectively. Their iterations go as such

x^{(i+1)}=(D+L)^{-1}(b-Ux^{(i)})\!\,

(Gauss-Seidel)

x^{(i+1)}=D^{-1}(b-(U+L)x^{(i)})\!\,

(Jacobi)

In our case $A\!\,$ is upper triangular, so $L\!\,$ is the zero matrix. As a result, the Gauss-Seidel and Jacobi methods take on the following identical form

x^{(i+1)}=D^{-1}(b-Ux^{(i)})\!\,

Additionally, $x\!\,$ can be written

x=D^{-1}(b-Ux)\!\,

Subtracting $x\!\,$ from $x^{(i+1)}\!\,$ we get

${\begin{aligned}e^{(i+1)}&=x^{(i+1)}-x\\&=D^{-1}(b-Ux^{(i)})-D^{-1}(b-Ux)\\&=D^{-1}U(-x^{(i)}+x)\\&=D^{-1}Ue^{(i)}\\&=D^{-1}U(D^{-1}Ue^{(i-1)})=(D^{-1}U)^{2}e^{(i-1)}\\&\vdots \\&=(D^{-1}U)^{i+1}e^{(0)}\end{aligned}}$

In our problem, $D^{-1}\!\,$ is diagonal and $U\!\,$ is upper triangular with zeros along the diagonal. Notice that the product $D^{-1}U\!\,$ will also be upper triangular with zeros along the diagonal.

Let $R=D^{-1}U\!\,$ , $R\in \mathbb {R} ^{n\times n}\!\,$

R={\begin{pmatrix}0&*&*&\cdots &*\\0&0&*&\cdots &*\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &\cdots &0&*\\0&\cdots &\cdots &0&0\end{pmatrix}}

Also, let ${\tilde {R}}\in \mathbb {R} ^{n\times n}\!\,$ be the related matrix

{\begin{aligned}{\tilde {R}}&=\left({\begin{array}{ccc|cccc}0&\cdots &0&*&*&\cdots &*\\0&\cdots &0&0&*&\cdots &*\\\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&*\\\hline 0&\cdots &0&0&\cdots &0&0\\\vdots &\vdots &\vdots &\vdots &\cdots &\vdots &\vdots \\0&\cdots &0&0&\cdots &0&0\\\end{array}}\right)\\&=\left({\begin{array}{c|c}\mathbf {a} &T\\\hline \mathbf {0} &\mathbf {a} ^{T}\\\end{array}}\right)\end{aligned}}

Where $\mathbf {a} \!\,$ is $k\times (n-k)\!\,$ , $T\!\,$ is $k\times k\!\,$ , and $\mathbf {0} \!\,$ is $(n-k)\times (n-k)\!\,$ .

Finally, the product $R{\tilde {R}}\!\,$ (call it (1)) is

{\begin{aligned}R{\tilde {R}}&={\begin{pmatrix}0&*&*&\cdots &*\\0&0&*&\cdots &*\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &\cdots &0&*\\0&\cdots &\cdots &0&0\end{pmatrix}}\left({\begin{array}{ccc|cccc}0&\cdots &0&*&*&\cdots &*\\0&\cdots &0&0&*&\cdots &*\\\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&*\\\hline 0&\cdots &0&0&\cdots &0&0\\\vdots &\vdots &\vdots &\vdots &\cdots &\vdots &\vdots \\0&\cdots &0&0&\cdots &0&0\\\end{array}}\right)\\&=\left({\begin{array}{ccc|ccccc}0&\cdots &0&0&*&\cdots &\cdots &*\\0&\cdots &0&0&0&*&\cdots &*\\\vdots &\vdots &\vdots &\vdots &\cdots &\cdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &\cdots &0&*\\\hline 0&\cdots &0&0&\cdots &\cdots &0&0\\\vdots &\vdots &\vdots &\vdots &\cdots &\cdots &\vdots &\vdots \\0&\cdots &0&0&\cdots &\cdots &0&0\\\end{array}}\right)\\&=\left({\begin{array}{c|c}\mathbf {a} &{\tilde {T}}\\\hline \mathbf {0} &\mathbf {a} ^{T}\\\end{array}}\right)\end{aligned}}

Here ${\tilde {T}}\!\,$ is almost identical in structure to $T\!\,$ , except that its diagonal elements are zeros.

At this point the convergence in $n\!\,$ steps (the size of the starting matrix) should be apparent since $R\!\,$ is just ${\tilde {R}}\!\,$ where $k=0\!\,$ and each time $R\!\,$ is multiplied by ${\tilde {R}}\!\,$ , the k-th super-diagonal is zeroed out (where k=0 is the diagonal itself). After $n-1\!\,$ applications of $R\!\,$ , the result will be the zero matrix of size $n\!\,$ .

In brief, $R^{n}\!\,$ is the zero matrix of size $n\!\,$ .

Therefore $e^{(n)}=(D^{-1}U)^{n}e^{(0)}\equiv 0\!\,$ , i.e. the Jacobi and Gauss-Seidel Methods used to solve $A\mathbf {x} =b\!\,$ converge in $n\!\,$ steps when $A\in \mathbb {R} ^{n\times n}\!\,$ is upper triangular.