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Undergraduate practice chapters

1. Undergraduate · Chapter 521

   Newton root iteration · Number whose root is sought A=6; Positive starting guess b=5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=6; Positive starting guess b=5.

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2. Undergraduate · Chapter 522

   Two-state Markov chains · Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.3

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.3.

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3. Undergraduate · Chapter 523

   Orthogonal projections · Vector horizontal a=-1; Vector vertical b=1.5

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-1; Vector vertical b=1.5.

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4. Undergraduate · Chapter 524

   Diagonal eigenmodes · First eigenvalue a=-0.5; Second eigenvalue b=2.5

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-0.5; Second eigenvalue b=2.5.

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5. Undergraduate · Chapter 525

   Matrix conditioning · Small diagonal entry a=0.45; Final data error b=0.2

   Use the current perturbation δ=0.12, so Δb=(0,0.12). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.45; Final data error b=0.2.

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6. Undergraduate · Chapter 526

   Logistic differential equations · Capacity K=2; Growth rate r=1.8

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2; Growth rate r=1.8.

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7. Undergraduate · Chapter 527

   Euler stability · Decay rate a=1.75; Step size h=1.5

   Use n=6 updates, corresponding to time t=nh=9. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.75; Step size h=1.5.

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8. Undergraduate · Chapter 528

   Newton root iteration · Number whose root is sought A=7; Positive starting guess b=5.5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=7; Positive starting guess b=5.5.

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9. Undergraduate · Chapter 529

   Two-state Markov chains · Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.4

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.4.

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10. Undergraduate · Chapter 530

    Orthogonal projections · Vector horizontal a=-0.5; Vector vertical b=2

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=-0.5; Vector vertical b=2.

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11. Undergraduate · Chapter 531

    Diagonal eigenmodes · First eigenvalue a=0; Second eigenvalue b=3

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=0; Second eigenvalue b=3.

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12. Undergraduate · Chapter 532

    Matrix conditioning · Small diagonal entry a=0.5; Final data error b=0.3

    Use the current perturbation δ=0.18, so Δb=(0,0.18). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.5; Final data error b=0.3.

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13. Undergraduate · Chapter 533

    Logistic differential equations · Capacity K=2.5; Growth rate r=2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2.5; Growth rate r=2.

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14. Undergraduate · Chapter 534

    Euler stability · Decay rate a=2; Step size h=0.1

    Use n=6 updates, corresponding to time t=nh=0.6. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=2; Step size h=0.1.

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15. Undergraduate · Chapter 535

    Newton root iteration · Number whose root is sought A=8; Positive starting guess b=6

    Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=8; Positive starting guess b=6.

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16. Undergraduate · Chapter 536

    Two-state Markov chains · Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.5

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.5.

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17. Undergraduate · Chapter 537

    Orthogonal projections · Vector horizontal a=0; Vector vertical b=2.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=0; Vector vertical b=2.5.

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18. Undergraduate · Chapter 538

    Diagonal eigenmodes · First eigenvalue a=0.5; Second eigenvalue b=-2

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=0.5; Second eigenvalue b=-2.

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19. Undergraduate · Chapter 539

    Matrix conditioning · Small diagonal entry a=0.55; Final data error b=0.4

    Use the current perturbation δ=0.24, so Δb=(0,0.24). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.55; Final data error b=0.4.

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20. Undergraduate · Chapter 540

    Logistic differential equations · Capacity K=3; Growth rate r=0.2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3; Growth rate r=0.2.

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