Iterative Methods for Large Linear Systems · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Solving big systems by approaching

Mathematics · Linear Algebra · ages 23-24
Name ______________________   Date ____________

  1. Use one Jacobi step from (0, 0) on 5x + y = 11, x + 4y = 9. What is the new x?

    Answer: ______________

  2. Use one Jacobi step from (0, 0) on 5x plus y = 11, x plus 4y = 9. What is the new x?

    Answer: ______________

  3. The system here is strictly diagonally dominant, so Jacobi converges from any starting guess.

    Circle one:   True   False

  4. Use one Jacobi step from (0, 0) on the same system. What is the new y?

    Answer: ______________

  5. Ana says the system 5x + y = 11, x + 4y = 9 is strictly diagonally dominant, so Jacobi converges from any starting guess. Is Ana right?

    Circle one:   True   False

  6. Why does the conjugate gradient method demand a symmetric positive definite matrix?

    • It inverts only the diagonal of the matrix
    • It needs whole number matrix entries
    • It minimizes a quadratic energy needing symmetry and one minimum

  7. After the first step the guess is (2.2, 2.25). What is x after the second Jacobi step?

    • 2.2
    • 1.75
    • 2.25

  8. Continue Jacobi on the same system. After the first step the guess is (2.2, 2.25). What is x after the second step?

    • 1.75
    • 2.2
    • 2.25
    • 1.8

  9. What does a preconditioner change, and why does that help?

    • The exact solution of the system
    • The spectrum, clustered so iteration converges sooner
    • The number of unknowns

  10. Why does the conjugate gradient method demand a symmetric positive definite matrix?

    • It minimizes a quadratic energy step by step, which needs symmetry and definiteness
    • It inverts only the diagonal, which fails whenever off diagonal entries are negative
    • It needs whole number entries, which only symmetric positive definite matrices have
    • It squares the matrix first, which is possible only for symmetric matrices
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Solving big systems by approaching W1-mt_FKkxKYVGYJ-s1

  1. 2.2 · Jacobi reuses only old values, so x1 is 11 over 5.
  2. 2.2 · Jacobi reuses only old values, so x1 is 11 over 5.
  3. True · 5 exceeds 1 and 4 exceeds 1, which triggers the guarantee.
  4. 2.25 · Similarly y1 is 9 over 4, which is 2.25.
  5. True · Diagonal dominance (5 > 1 and 4 > 1) guarantees Jacobi converges from any start.
  6. It minimizes a quadratic energy needing symmetry and one minimum · The energy geometry needs symmetry and a true minimum.
  7. 1.75 · Feed y1 into the x formula: (11 minus 2.25) over 5 is 1.75.
  8. 1.75 · Feed y1 = 2.25 into the x formula: (11 minus 2.25) over 5 is 1.75.
  9. The spectrum, clustered so iteration converges sooner · Reshaped eigenvalues mean fewer effective modes to grind down.
  10. It minimizes a quadratic energy step by step, which needs symmetry and definiteness · Conjugate gradients minimize an energy whose geometry needs symmetry and a true minimum.
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