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My dude will not use excersies from previous exams.

Ludde127 2024-03-04

Will be things from everything in this course.

We will have less tasks this time OMG>OMOMOMGOMG LVOE HIM

Remember floats!!!!!!! My dude will be trole

Rounding errors, error thingy.

Extremely important to know fixed points.

  • Errors
    • Absolute error
    • Relative error
    • forward error
    • Backwards error
    • Round off error - not more than \(\frac{e_{mach}}{2}\) - good check
  • IEEE
    • If given number, calculate binary and put in IEEE 754
    • abstract and add
      • ex \(fl(0.25)=+1.0\dots 000\cdot_{2}^{-2}\)
      • \((0.25)_{10}=0.01\)
  • Taylor series
  • Be able to write in Horners scheme
  • Bracket method
  • Root esitance
  • Convergence
  • Convergence order
  • Banach's fixed point
    • assumption
  • modified newtons
  • Requirements for invertability, \(\det A\neq 0\)
  • Back and forward substitution
  • Know the rules for sum
    • ex \(\sum_{i=0}^ni=\frac{n(n+1)}{2}\)
  • LU
  • Solve after LU
  • Norms
    • conditions for norm
    • def
    • matrix norm
    • equivalent norms
    • 1 matrix norm
    • ∞ matrix norm
  • conditioning
  • Pertubation analysis
  • Condition number
    • properties
  • premise of pivoting
  • permutation matrix
  • Newton method
    • convergence
    • what it is
    • how to calculate with it
  • Interpolation
    • Know what the purpose is
  • What is meant by polynomial of degree N
  • Basis
    • Lagrange basis
    • Vandermond basis
    • monomial basis
    • newton basis
    • Advantages between each
  • Newton Divided difference - how to calculate
  • Interpolation error formula
    • Haven't asked yet
    • prolly not coming but ye might come
  • Chebychev not coming
  • Transformation on interval
    • See p.241
    • How to transform nodes given on certain interval to any interval
  • Piece wise interpolation
    • Why?
  • Spline
    • Know what it is
    • The definition
    • Why we use it
  • Cubic spline
    • Conditions - p.252
    • Types of cubic splines
      • Cheat sheet
    • We do not need to know system on p.256
  • [[bernstein-polynomial]]
    • Should have seen it but iffy on if we should know the formula
    • "Not bad to know it"
    • We do not need to know the properties
    • Not super important
  • [[bezier-curve]]
    • We should be able to create it given points
    • Therefore [[bernstein-polynomial]] needed
    • Knowing some of the properties,
      • what happens with the slope at the endpoints
    • How to create composite [[bezier-curve]]
  • Householder reflection matrix does not come but we should know what a projection matrix is.
  • Should be able to solve least squares problems with QR-factorization.
  • Should know what an eigenvalue problem is. Complete and partial.
  • Know the notation for eigenvalue and eigenvectors.
  • Know what an characteristic polynomial is.
  • Know how many eigenvalues a matrix has.
  • Know the Rayleigh quotient. IMPORTANT
  • Know properties of eigenvalues. Be able to prove the lemmas (maybe not the A^T has same eigenvalues as A). Especially the ones in assignment.
  • Know how the power method and the shifted inverted power method works.
  • For which shifts does the shifted inverted power method converge to different eigenvalues? Be careful when the shift is exactly the eigenvalue.
  • Know what similar matrices are and that they have the same spectrum.
  • Know what Real Schur form is.
  • Know how the QR method works. Know why they are similar ie why A = QR, show in algebra.
  • Convergence is not needed (a bit unclear)
  • Should know what singular values and singular value decomposition is.
  • Know what the low-rank approximation is and how to use it.
  • Know the numerical differentiation methods, how they work and how to use them.
  • Be able to calculate error analysis for numerical differentiation (will likely not be on exam).
  • Be able to do numerical differentiation in matrix form! (This is pretty important.)
  • Know what a quadrature is.
  • know newton-cotes formula, the exactness and how to use it.
  • know trapezoidal rule by heart.
  • Be able to calculate the error using interpolation error ("why not")
  • Simpsons rule may come.
  • Change of interval, transformations between different intervals (Use offsets and such).
  • Newton cotes closed and open? Wtf is this. We should know it
  • Know about weights
  • Be able to apply composite quadrature.
  • Maybe derive the error formula.
  • Composite simpsons rule.
  • We should be able to handle adaptive quadrature (he gave us the look that this will be on exam get ready)
  • Be able to show that there is no quadrature that is more exact than some that i missed.
  • Maybe gauss-legrendre "its not bad to know, it may help", know how the weights are computed and the properties for it. Okay seams like it will come he's sussy.
  • Matrix notation of DFT might come.
  • Know DFT.
  • Might need to know about FFT complexity not more tho, probably not this.

Thinks that will not come

Be aware that this course has removed some things such as ODE's.

Going trough exam

Be able to prove or disprove if some things are a norm. Know the criteria of a norm and how to check them.

Know the jacobi iterative method.

Know what information that is given from having R or Q in A = QR, What is size of a, rank of a, or what A^T * A is. MIGHT COME THIS TIME ASWELL DO THIS OR DIE