NLA MasterMath  Weekly Schedule 2018
Week 5
This week we looked at the consequences of symmetry of the system matrix A.
This implies symmetry of the upper Hessenberg matrix in the Arnoldi method
and leads to threeterm recurrence relations.
We covered:
 Chapter 6: Krylov Subspace Methods
 Section 6.6 (without 6.6.2)
 Section 6.7 (without Section 6.7.3)
 Section 6.8
Suggested exercises are:
 6.20, 6.21, 6.23 (only for GMRES and FOM)
Week 4
We will start with a short test that will count toerds your final grade
for the course
We will then cover:
 Chapter 6: Krylov Subspace Methods
 Sections 6.1, 6.2, 6.3
 6.4 (without 6.4.2)
 6.5 (without 6.5.6 and 6.5.8 and 6.5.9)
In particular, we derived in detail:
 The Full Orthogonalization Method
 The Generamized Minimal Residual Method
Suggested exercises are:
Week 3
We covered
 Chapter 5: Projection Methods
 Excluding Sect 5.4
The following concepts should be understood:
 Subspace methods with L=K and L=AK
 Ainner products and norms
 A*Ainner product and norms
 Algorithm Minimal Residuals Sect 5.3.2
 Algorithm Steepest Descent Sect 5.3.1
Suggested exercises are:
 5.1
 5.4, 5.8 (related to Ths 5.7 and 5.10)
 5.12 (random search direction)
 5.10 and 5.11
 5.16 and 5.17
Week 2
We covered
 Chapter 4: Basic Iterative Methods
 Excluding: 4.1.1: overlapping block methods
 Exclusing: 4.2.5, 4.3
The following concepts should be understood:
 splittings of a matrix
 fixed point iterations for affine transformations
 convergent if and only if spectral radius of iteration
matrix smaller than one
 errors versus residuals,
 are powers of a matrix times initial error / residual
 preconditioning: either to improve condition
number or to reduce the spectral radius of iteration matrix
 sufficient conditions for convergent iterations:
 regular splittings for inverse nonnegative matrices
 matrix A strictly diagonally dominant
 A is SPD
 basic methods as update & correction algorithms
Suggested exercises are:
 4.1, 4.4
 4.7 (but `consistently ordered matrices' will not be tested)
Further exercises were given on the blackboard:
1) give N&S conditions for Jacobi and GaussSeidel to converge
for an arbitrary invertible 2x2 matrix
2) apply the Jacobi method to an upper triangular system
3) implement the random minimal residual method (which selects
a random update direction of current approximation and minimizes the residual along that direction)
4) With A = [1 p q ; 1 1 r ; 0 1 1] give N&S conditions for
GaussSeidel to converge.
Week 1
We covered
 Chapter 3: Sparse Matrices
 Sections 3.1, 3.2, 3.3 only
The following concepts should be understood:
 adjacency graph
 row and column permuations of matrices
 symmetric permutation of a matrix
 reordering of equations and unknowns in a linear system
 LU and Cholesky decomposition depend on reorderings
 Cuthill McKee and Reverse CMK
 reoderings based on independent set and multicolorings
Suggested exercises are:
 3.4, 3.5, 3.6, 3.7, 3.8, 3.10, 3.11, 3.12
See alse the pdfsection for Lecture Notes of:
 Linear Algebra 2
 Introduction to Numerical Mathematics: SVD
which are (part of) two firstyear courses.
