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Undergraduate practice chapters

1. Undergraduate · Chapter 601

   Newton root iteration · Number whose root is sought A=2; 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=2; Positive starting guess b=3.5.

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2. Undergraduate · Chapter 602

   Two-state Markov chains · Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.7

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.8; Transition 2 → 1: q=0.7.

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3. Undergraduate · Chapter 603

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

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4. Undergraduate · Chapter 604

   Diagonal eigenmodes · First eigenvalue a=0; 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; Second eigenvalue b=-2.

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5. Undergraduate · Chapter 605

   Matrix conditioning · Small diagonal entry a=0.05; Final data error b=0.5

   Use the current perturbation δ=0.3, so Δb=(0,0.3). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.05; Final data error b=0.5.

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6. Undergraduate · Chapter 606

   Logistic differential equations · Capacity K=2.5; Growth rate r=0.2

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=2.5; Growth rate r=0.2.

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7. Undergraduate · Chapter 607

   Euler stability · Decay rate a=1.25; 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=1.25; Step size h=1.5.

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8. Undergraduate · Chapter 608

   Newton root iteration · Number whose root is sought A=3; 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=3; Positive starting guess b=4.

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9. Undergraduate · Chapter 609

   Two-state Markov chains · Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.8

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.9; Transition 2 → 1: q=0.8.

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10. Undergraduate · Chapter 610

    Orthogonal projections · Vector horizontal a=-1; 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=-1; Vector vertical b=2.

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11. Undergraduate · Chapter 611

    Diagonal eigenmodes · First eigenvalue a=0.5; 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=0.5; Second eigenvalue b=-1.5.

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12. Undergraduate · Chapter 612

    Matrix conditioning · Small diagonal entry a=0.1; Final data error b=0.6

    Use the current perturbation δ=0.36, so Δb=(0,0.36). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.1; Final data error b=0.6.

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13. Undergraduate · Chapter 613

    Logistic differential equations · Capacity K=3; Growth rate r=0.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3; Growth rate r=0.4.

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14. Undergraduate · Chapter 614

    Euler stability · Decay rate a=1.5; 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=1.5; Step size h=0.1.

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15. Undergraduate · Chapter 615

    Newton root iteration · Number whose root is sought A=4; 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=4; Positive starting guess b=4.5.

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16. Undergraduate · Chapter 616

    Orthogonal projections · Vector horizontal a=-0.5; 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=-0.5; Vector vertical b=2.5.

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17. Undergraduate · Chapter 617

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

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18. Undergraduate · Chapter 618

    Matrix conditioning · Small diagonal entry a=0.15; Final data error b=0.7

    Use the current perturbation δ=0.42, so Δb=(0,0.42). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.15; Final data error b=0.7.

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19. Undergraduate · Chapter 619

    Logistic differential equations · Capacity K=3.5; Growth rate r=0.6

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=3.5; Growth rate r=0.6.

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20. Undergraduate · Chapter 620

    Euler stability · Decay rate a=1.75; Step size h=0.2

    Use n=6 updates, corresponding to time t=nh=1.2. 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=0.2.

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