• --: Get Qual study material from Eric
• --: Read part 3 of Numerical Linear Algebra.
• --: Read part 4 of NLA.
• --: Read part 5 of NLA
• --: Read part 6 of NLA.

• --: Read a chapter in Numerical Recipies
• --: Read a chapter in Trefethlen and bau
• --: Schedule 1 hour qual study time.

Books I need to have finished by january:

• numerical recipies
• chapter on QR iteration from Mitya
• numerical linear algebra
• counter examples in real analysis
• baby rudin
• ODEs and DAEs
• PDEs book - emphasis on Fourier analysis.
• Baess and Strang?
• Axelson and Barker
• Interpolation book
• Saad

2007-09-14

Qual practice with Dmitri

What is stability?

• Applies to algorithms not as much as problems

• bound relating input error to output error, small bound leads to good conditioning

if you have %$ Ax=b $%, A spd

• what is spd?
• how would you solve
• derive the Cholesky factorization.
• when does A have an LU factorization? When is it unique?
  • Proof of uniqueness: LU &=& MN \M^{-1}L &=& NU^{-1} \ &=& I

When is this matrix a zero matrix?

• all the eigenvalues are zero.
• all singular values are zero.
• largest singular value is zero.
• if there exists an a for which Ax=0
• if %$ A^2=0 $% use [0 1; 0 0]
• if %$ A^T A = 0 $%

Can A be orthogonal and symmetric? Besides identity. and symmetric permutation matrix? All reflection matricies orthogonal any symmetric? %$ I-2\frac {vv^T}{v^T v} $%

are reflection and rotation matricies ever the same? identity.

given a unitary matrix, prove all eigenvalues are of length 1. (Av,Av)&=&|\lambda^2|(v,v) \(v, A^H Av) &=& \|\lambda^2| &=& 1

Write down Cauche Schwartz inequality. Is there a case where it is an equality? Prove %$ |ab^T| \leq |a||b|$% in 2 norm

Can you bound the 1 norm with the infinity norm for vectors? For matricies? (Consider [\epsilon 0; 1 1])

Assume matrix is not symmetric, all positive eigenvalues, defective, is matrix positive definite? No, [1 alpha; 1 1]

What is the difference between gram schmidt and modified gram schmidt?

derive the weak form of a pde.

given bilinear galerikin functional, prove SPD and SPD of finite distrectizations.

Gauss siedel converges for spd? It's true, prove.

when does jacobi converge?