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Undergraduate practice chapters

1. Undergraduate · Chapter 501

   Newton root iteration · Number whose root is sought A=3; Positive starting guess b=3.5

   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=3; Positive starting guess b=3.5.

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2. Undergraduate · Chapter 502

   Two-state Markov chains · Transition 1 → 2: p=0.2; Transition 2 → 1: q=0.9

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.2; Transition 2 → 1: q=0.9.

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3. Undergraduate · Chapter 503

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

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

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4. Undergraduate · Chapter 504

   Diagonal eigenmodes · First eigenvalue a=-2; Second eigenvalue b=1

   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=-2; Second eigenvalue b=1.

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5. Undergraduate · Chapter 505

   Matrix conditioning · Small diagonal entry a=0.3; Final data error b=0.9

   Use the current perturbation δ=0.54, so Δb=(0,0.54). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.3; Final data error b=0.9.

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6. Undergraduate · Chapter 506

   Logistic differential equations · Capacity K=6; Growth rate r=1.2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=6; Growth rate r=1.2.

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7. Undergraduate · Chapter 507

   Euler stability · Decay rate a=1; Step size h=1.2

   Use n=6 updates, corresponding to time t=nh=7.2. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1; Step size h=1.2.

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8. Undergraduate · Chapter 508

   Newton root iteration · Number whose root is sought A=4; Positive starting guess b=4

   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=4; Positive starting guess b=4.

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9. Undergraduate · Chapter 509

   Two-state Markov chains · Transition 1 → 2: p=0.3; Transition 2 → 1: q=0.1

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.3; Transition 2 → 1: q=0.1.

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10. Undergraduate · Chapter 510

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

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

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11. Undergraduate · Chapter 511

    Diagonal eigenmodes · First eigenvalue a=-1.5; Second eigenvalue b=1.5

    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=-1.5; Second eigenvalue b=1.5.

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12. Undergraduate · Chapter 512

    Matrix conditioning · Small diagonal entry a=0.35; Final data error b=1

    Use the current perturbation δ=0.6, so Δb=(0,0.6). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.35; Final data error b=1.

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13. Undergraduate · Chapter 513

    Logistic differential equations · Capacity K=1; Growth rate r=1.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1; Growth rate r=1.4.

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14. Undergraduate · Chapter 514

    Euler stability · Decay rate a=1.25; Step size h=1.3

    Use n=6 updates, corresponding to time t=nh=7.8. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.25; Step size h=1.3.

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15. Undergraduate · Chapter 515

    Newton root iteration · Number whose root is sought A=5; Positive starting guess b=4.5

    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=5; Positive starting guess b=4.5.

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16. Undergraduate · Chapter 516

    Two-state Markov chains · Transition 1 → 2: p=0.4; Transition 2 → 1: q=0.2

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.4; Transition 2 → 1: q=0.2.

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17. Undergraduate · Chapter 517

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

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

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18. Undergraduate · Chapter 518

    Diagonal eigenmodes · First eigenvalue a=-1; 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=-1; Second eigenvalue b=2.

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19. Undergraduate · Chapter 519

    Matrix conditioning · Small diagonal entry a=0.4; Final data error b=0.1

    Use the current perturbation δ=0.06, so Δb=(0,0.06). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.4; Final data error b=0.1.

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20. Undergraduate · Chapter 520

    Euler stability · Decay rate a=1.5; Step size h=1.4

    Use n=6 updates, corresponding to time t=nh=8.4. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.5; Step size h=1.4.

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