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Undergraduate practice chapters

1. Undergraduate · Chapter 621

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

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2. Undergraduate · Chapter 622

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

   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=-0.5.

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3. Undergraduate · Chapter 623

   Matrix conditioning · Small diagonal entry a=0.2; Final data error b=0.8

   Use the current perturbation δ=0.48, so Δb=(0,0.48). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.2; Final data error b=0.8.

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4. Undergraduate · Chapter 624

   Logistic differential equations · Capacity K=4; Growth rate r=0.8

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4; Growth rate r=0.8.

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5. Undergraduate · Chapter 625

   Euler stability · Decay rate a=2; Step size h=0.3

   Use n=6 updates, corresponding to time t=nh=1.8. 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=0.3.

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6. Undergraduate · Chapter 626

   Newton root iteration · Number whose root is sought A=6; 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=6; Positive starting guess b=5.5.

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7. Undergraduate · Chapter 627

   Orthogonal projections · Vector horizontal a=0.5; Vector vertical b=-3

   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=-3.

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8. Undergraduate · Chapter 628

   Matrix conditioning · Small diagonal entry a=0.25; 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=0.25; Final data error b=0.9.

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9. Undergraduate · Chapter 629

   Logistic differential equations · Capacity K=4.5; Growth rate r=1

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=4.5; Growth rate r=1.

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10. Undergraduate · Chapter 630

    Euler stability · Decay rate a=0.5; 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=0.5; Step size h=1.3.

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11. Undergraduate · Chapter 631

    Newton root iteration · Number whose root is sought A=7; 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=7; Positive starting guess b=6.

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12. Undergraduate · Chapter 632

    Orthogonal projections · Vector horizontal a=1; 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=1; Vector vertical b=-2.5.

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13. Undergraduate · Chapter 633

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

    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=2.5; Second eigenvalue b=0.5.

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14. Undergraduate · Chapter 634

    Matrix conditioning · Small diagonal entry a=0.3; 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=0.3; Final data error b=1.

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15. Undergraduate · Chapter 635

    Logistic differential equations · Capacity K=5; Growth rate r=1.2

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5; Growth rate r=1.2.

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16. Undergraduate · Chapter 636

    Euler stability · Decay rate a=0.75; 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=0.75; Step size h=1.4.

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17. Undergraduate · Chapter 637

    Newton root iteration · Number whose root is sought A=8; Positive starting guess b=6.5

    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=8; Positive starting guess b=6.5.

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18. Undergraduate · Chapter 638

    Orthogonal projections · Vector horizontal a=1.5; 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=1.5; Vector vertical b=-2.

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19. Undergraduate · Chapter 639

    Matrix conditioning · Small diagonal entry a=0.35; 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=0.35; Final data error b=0.1.

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20. Undergraduate · Chapter 640

    Logistic differential equations · Capacity K=5.5; Growth rate r=1.4

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=5.5; Growth rate r=1.4.

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