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11. Assignment 10
Due Wednesday, April 23
Gregory Beylkin, ECOT 323
(1) Implement the Crank-Nicolson scheme for the heat equation
= x (a(x) x ) + f (x, t),
t
with the initial condition
|t=0 = 0 ,
and the boundary condition (t, 0) = 0 and (t, 1

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7. Assignment 7
Due Wednesday, March 12
Gregory Beylkin, ECOT 323
(1) (This problem is due March 19. Note: you will need the code of this
assignment for another HW).
Implement trapezoidal rule to solve the initial value problem
y = f (t, y)
whe

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4. Assignment 4
Due Wednesday, February 19
Gregory Beylkin, ECOT 323
(1) Implement QR iteration for a real symmetric tridiagonal matrix. Demonstrate its performance on (at least two) examples. What is the complexity
of the algorithm?
(2) Suppos

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5. Assignment 5
Due Wednesday, February 26
Gregory Beylkin, ECOT 323
(1) Compute a QR step with the matrix
2
1
(a) without a shift
(b) with the shift = 1.
Which approach appears to be better?
(2) Show that Jacobis method for nding eigenvalues o

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3. Assignment 3
Due Wednesday, February 12
Gregory Beylkin, ECOT 323
(1) Prove that
(a) if all singular values of matrix A C nn are equal, then A = U ,
where U is a unitary matrix and is a constant
(b) if A C nn is nonsingular and is an eigenva

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2. Assignment 2
Due Wednesday, February 5
Gregory Beylkin, ECOT 323
(1) The power method.
(a) Show that the Hilbert matrix (see below) is positive denite. Hint:
use
1
1
=
xi+j2 dx .
i+j1
0
(b) Implement the power method for nding the dominant e

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6. Assignment 6
Due Wednesday, March 5
Gregory Beylkin, ECOT 323
(1) The linear system
y = Ay,
y(0) = y0 ,
where A is symmetric is solved by the explicit Eulers method.
Let en = yn y(nh), n = 0, 1, . . . , and prove that
|en |2 |y0 |2 max |(1 + h

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8. Assignment 8
Due Wednesday, March 19
Gregory Beylkin, ECOT 323
(1) Determine the order and the region of absolute stability of s-step AdamsBashforth methods for s = 2, 3,
3
1
s = 2 : yn+2 = yn+1 + h f (tn+1 , yn+1 ) f (tn , yn )
2
2
4
5
23
f

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10. Assignment 9
Due Wednesday, April 9
Gregory Beylkin, ECOT 323
(1) The best known explicit Runge-Kutta method is dened by the following
formulas:
k1 = hf (xn , yn )
1
k2 = hf (xn + 1 h, yn + 2 k1 )
2
1
1
k3 = hf (xn + 2 h, yn + 2 k2 )
k4 =

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12. Assignment 11
Due Wednesday, April 30
Gregory Beylkin, ECOT 323
(1) Solve the Poissons equation
u = f
on the square (x, y) : 0 x, y 1 with the homogeneous Dirichlet boundary conditions. Assume that the function f is well approximated by
N

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1. Assignment 1
Due Wednesday, January 29
Gregory Beylkin, ECOT 323
(1) Any square matrix may be represented in the form A = SV , where S is a
Hermitian nonnegative denite matrix and V is a unitary matrix. This is
the so-called polar decomposit