[1] Sections 1.1 to 1.3.9

• Error, Accuracy, Stability:
  Absolute Error, Relative Error
  Data Error, Computational Error
  Truncation Error , Rounding Error
  Forward Error, Backward Error
  Well Conditioned, Stable, Well Posed
• Floating-Point Arithmetic:
  Underflow, Overflow, Machine Precision
  Subnormals, Gradual Underflow
  Cancellation
• Important Formula:
  Absolute Error, Relative Error
  Forward Error, Backward Error
  Condition Number
  Machine precision definitions on page 5
• Know how to compute:
  Forward/Backward Error
  Absolute/Relative Error
  fl(x+y), fl(x-y), fl(x*y), fl(x/y)
  E.g., when does 1 + a = 1 ?

[2] Sections 2.1 to 2.53

• Existence and Uniqueness:
  Nonsingular, Singular
• Conditioning:
  Vector and Matrix Norms
  Condition Number
  Error bounds, residual
• Solving:
  Upper and Lower Triangular
  Forward and Back Substitution
  Elementary Elimination Matrices
  Partial or Row Pivoting, Complete Pivoting
  LU Factorization, Gaussian Elimination
  Complexity of LU vs Inversion
  Complexity of solving modified systems
  Symmetric, Positive Definite, Banded, Sparse
  Cholesky
  Complexity for banded systems
• Important Formula:
  1-norm, 2-norm, inf-norm, p-norm for vectors
  1-norm and inf-norm for matrices
  Condition Number
  Properties of vector norms
  Properties of matrix norms
  Properties of the Condition Number
  Error bounds using condition number
  Elementary Elimination Matrices
• Know how to compute:
  Solution to triangular systems
  LU Factorizations (2x2 or 3x3 matrices)
  Norms and condition numbers of diagonal matrices
  Solve small linear systems of equations

[3] Sections 3.1 to 3.7 (exclude 3.4.2)

• Existence and Uniqueness:
  Span(A), Rank(A)
  Normal Equations
  Orthogonality of Span(A) with Residual
• Conditioning:
  Pseudoinverse and condition number
  Conditioning of the normal equations
• Solving:
  Normal Equations
  Triangular systems
  QR Factorization
  Householder, Givens
  Gram-Schmidt theory (not actual formula)
  Singular Value Decomposition
  2-norm and 2-norm condition number with SVD
  Pseudoinverse with SVD
  Comparison of methods (work vs stability)
• Important Formula:
  Normal Equations, Pseudoinverse, condition number
  Error bounds involving condition number
  Householder transformation
  Givens Rotation
  2-norm (condition number) using SVD
• Know how to compute:
  Setup and solution small least-squares problem (3x2)
  Normal equations
  Householder and Givens transformations
  Norms, condition numbers given the SVD

[4]Sections 4.1 to 4.7 (exclude 4.2.5, 4.5.9, 4.5.11, 4.6)

• Existence and Uniqueness:
  Characteristic Polynomial using determinant
  Companion Matrix
  Algebraic and Geometric multiplicity
  Invariant subspace using eigenvectors
  Diagonal, Tridiagonal, Triangular, Hessenberg
  Orthogonal, Unitary, Symmetric, Hermitian, Normal
  Which matrices are diagonalizable? have real eigenvalues?
  How close to diagonal using general similarity transforms?
  How about with orthogonal transforms? Unitary transforms?
• Conditioning:
  Bound involving cond of matrix of eigenvectors
  Bound involving angle between right/left eigenvectors
• Solving:
  Shift, Inversion, Powers, Polynomials of A
  Similarity Transformations
  Unitary and Orthogonal Similarity Transforms
  Eigenvalues of diagonal, triangular, block triangular
  Defective matrices are not diagonalizable
  Normalized power iteration with/without shifts
  Normalized inverse iteration with/without shifts
  Rayleigh quotient
  Rayleigh quotient iteration
  Deflation
  Jordon, Schur, and Real Schur forms
  QR iteration with/without shifts
  Preliminary reduction to Hessenberg or Tridiagonal
  Properties of Lanczos and Arnoldi
  Properties of Jacobi and Divide-and-Conquer
  Comparison of methods (stability vs work)
• Important Formula:
  Characteristic Polynomial
  Bounds on p. 167 and 168
  Eigenvalues after shift, inverse, powers, polynomial
  Rayleigh quotient
• Know how to compute:
  Eigenvalues of a small matrix (2x2)
  Eigenvalues of a diagonal or triangular matrix
  Eigenvectors of a small matrix (2x2)
  1-step of the normalized power iteration
  1-step of inverse iteration (for triangular matrix)
  Rayleigh quotient

[5] Sections 5.1 to 5.6.2 (remaining after midterm)

• Existence and Uniqueness:
  Scalar and Systems case
  Single and multiple roots
• Conditioning:
  Derivative and Jacobian
  Multiple roots
  Condition number
• Rates of convergence:
  Linear, superlinear, quadratic
  Number of correct digits per iteration
• Solving Scalar Problems:
  Bisection
  Stability, rate of convergence of bisection
  Fixed point iterations
  Conditions for convergence of fixed point iterations
  Rate of convergence of fixed point iterations
  Newton's method
  Secant method
  Inverse interpolation (general concept/properties)
  Linear fractional interpolation (general concept/properties)
  Safeguard methods
  Zeros of polynomials by Newton's method
  Zeros of polynomials by companion matrix
  General comparison of methods for scalar problems
• Newton's Method for Systems:
  Cost per step
  Convergence rate
• Important Formula:
  Condition number for root finding
  Convergence rate
  Convergence of fixed point iteration
• Know how to compute
  Jacobian of a system of equations
  Condition number for root finding
  1 or 2 steps of bisection
  1 step of a fixed point iteration
  1 step of Newton's method
  1 step of Secant method
  1 step of Newton's method for systems