4.1/

The derivative

  1. Derivative:
  2. The forward/backward difference formula:

    Derivative formulat [at \(x = x_0\)]

    This formula is known as the forward-difference formula if
    and the backward-difference formula if

    • Derivation:

    Error Bound:

  3. The \((n + 1)\)-point formula to approximate \(f'(x_j)\):
    • Derivation:
  4. Three-point Formula:

    for each \(j = 0, 1, 2\), where the notation \(\zeta_j\) indicates that this point depends on \(x_j\).

    • Derivation:

Three-Point Formulas

  1. Equally Spaced nodes:

    The formulas from Eq. (4.3) become especially useful if \(x_1 =\)
    \(x_2 =\)

  2. Three-Point Endpoint Formula:

    The approximation in Eq. (4.4) is useful at

    Because:

    Errors: the errors in both Eq. (4.4) and Eq. (4.5) are

    On the interval:

  3. Three-Point Midpoint Formula:

    Errors:

    This is because Eq. (4.5) uses data on \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) and Eq. (4.4) uses data \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    Note also that f needs to be evaluated at \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) in Eq. (4.5), whereas in Eq. (4.4) it \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

    On the interval:

Five-Point Formulas

  1. What?
  2. Why?

  3. Error:
    The error term for these formulas is

  4. Five-Point Midpoint Formula:

    Used for approximation at

    On the interval:

  5. Five-Point Endpoint Formula:

    Used for approximation at

    Left-endpoint approximations are found using this formula with \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) and right-endpoint approximations with \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\).

    The five-point endpoint formula is particularly useful for \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

    On the interval:

Approximating Higher Derivatives

  1. Approximations to Second Derivatives:

    On the interval:

    • Derivation:
      • Why does the error bound change?
  2. Second Derivative Midpoint Formula:

    On the interval:

    • Derivation:

    Error Bound: If \(f^{(4)}\) is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) on \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) it is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) , and the approximation is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) .

Round-Off Error Instability

  1. Form of Error:
    • We assume that our computations actually use the values \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • which are related to the true values \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) by:

      \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) &
      \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

  2. The total error:

    It is due to \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

  3. Error Bound:

    • ASSUMPTION: \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

    • ERROR:

  4. Reducing Truncation Error:
    • How? \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • Effect of reducing \(h\):
  5. Conclusion:

4.2/

Extrapolation

  1. What?
    • Extrapolation is used to
    • Extrapolation can be applied whenever
    • Suppose that for each number \(h \neq 0\) we have a formula \(N_1(h)\) that approximates an unknown constant \(\ \ \ \ \ \ \ \\), and that the truncation error involved with the approximation has the form,
      \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • The truncation error is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
      where,
      (1)
      (2)

    and, in general,
    \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

    • The object of extrapolation is
  2. Why?
  3. The \(\mathcal{O}(h)\) formula for approximating \(M\):

    The First Formula:

    The Second Formula:

  4. The \(\mathcal{O}(h^2)\) approximation formula for M:
    • Derivation:
  5. When to apply Extrapolation?
    (1)

    (2)

  6. The \(\mathcal{O}(h^4)\) formula for approximating \(M\):
  7. The \(\mathcal{O}(h^6)\) formula for approximating \(M\):
  8. The \(\mathcal{O}(h^{2j})\) formula for approximating \(M\):
    • Derivation:
  9. The Order the Approximations Generated:
  10. How to actually calculate a derivative using the Extrapolation formula:

Deriving n-point Formulas with Extrapolation

  1. Deriving Five-point Formula:

4.3/

Numerical Quadrature

  1. What?
  2. How?
  3. Based on:
  4. Method:
    • Derivation:
  5. The Quadrature Formula:
  6. The Error:

The Trapezoidal Rule

  1. What?
  2. Precision
  3. The Trapezoidal Rule:
    • Derivation:
  4. Error:

Simpson’s Rule

  1. What?
  2. Simpson’s Rule:
    • Derivation:
  3. Precision
  4. Error:

Measuring Precision

  1. What?
  2. Precision [degree of accuracy]:
  3. Precision of Quadrature Formulas:
    • The degree of precision of a quadrature formula is \(\mathcal{O}\)
    • The Trapezoidal and Simpson’s rules are examples of \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • Types of Newton-Cotes formulas:

Closed Newton-Cotes Formulas

  1. What?
    • It is called closed because:
  2. Form of the Formula:

    where,

  3. The Error Analysis:
  4. Degree of Preceision:
    • Even-n: the degree of precision is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • Odd-n: the degree of precision is
  5. Closed Form Formulas:
    • \(n = 1\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
    • \(n = 2\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
    • \(n = 3\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
    • n = 4:

Open Newton-Cotes Formulas

  1. What?
    • They \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • They use
    • This implies that
    • Open formulas contain \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
  2. Form of the Formula:

    where,

  3. The Error Analysis:
  4. Degree of Preceision:
    • Even-n:
    • Odd-n:
  5. Open Form Formulas:
    • \(n = 0\): [PUT NAME HERE]
    • \(n = 1\):
    • \(n = 2\):
    • n = 3:

4.4/

Composite Rules

  1. What?
  2. Why?
  3. Notice:

Composite Simpson’s rule

  1. Composite Simpson’s rule:
  2. Error in Comoposite Simpson’s Rule:

    Error:

  3. Theorem [Rule and Error]:
  4. Algorithm:

Composite Newton-Cotes Rules

  1. Composite Trapezoidal rule:
  2. Composite Midpoint rule:

Round-Off Error Stability

  1. Stability Property:
  2. Proof:

4.5/

Main Idea

  1. What?
  2. Why?
  3. Error in Composite Trapezoidal rule:

    This implies that \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

  4. Extrapolation Formula:

    Extrapolation then is used to produce \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) approximations by

    and according to this table,

    Calculate the Romberg table this way:

  5. Algorithm:

4.6/

Main Idea

  1. What?
  2. Why?

  3. Approximation Formula:
    \(\int_{a}^{b} f(x) dx =\)

    • Derivation:
  4. Error Bound:
    • Error relative to Composite Approximations:
    • Error relative to True Value:
    • ERROR DERIVATION:

    This implies

  5. Procedure:

    When the approximations in (4.38)

    Then we use the error estimation procedure to

    If the approximation on one of the subintervals fails to be within the tolerance \(\ \ \ \ \ \ \ \\), then

  6. Algorithm:
  7. Derivation:

4.7/

Main Idea

  1. What?

    • To Measure Accuracy:

    • The Coefficients \(c_1, c_2, ... , c_n\) in the approximation formula are \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) ,
      and,
      The Nodes \(x_1, x_2, ... , x_n\) are restricted by/to \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
      This gives,
      The number of Parameters to choose is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

    • If the coefficients of a polynomial are considered parameters,

      This, then, is

  2. Why?

Legendre Polynomials

  1. What?
  2. Why?
  3. Properties:
    3. The roots of these polynomials are:
  4. The first Legendre Polynomials:
    \(P_0(x) = \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_1(x) = \ \ \, \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_2(x) = \ \ \ \ \ \ \ \ \ \ ,\)
    \(P_3(x) = \ \ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_4(x) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
  5. Determining the nodes:
    • PROOF:

    The nodes \(x_1, x_2, ... , x_n\) needed to

Gaussian Quadrature on Arbitrary Intervals

  1. What?
  2. The Change of Variables:
  3. Gaussian quadrature [arbitrary interval]:

4.8/

Approximating Double Integral

  1. What?
  2. Why?
  3. Comoposite Trapezoidal Rule for Double Integral:
    \(\ \ \iint_R f(x,y) \,dA \ =\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
    • DERIVATION:
  4. Comoposite Simpsons’ Rule for Double Integral:
    • Rule:
    • Error:
    • Derivation:

Gaussian Quadrature for Double Integral Approximation

  1. What?
  2. Why?
  3. Example:

Non-Rectangular Regions

  1. What?

    Form:


    or,

  2. How?
    • We use
    • Step Size:
      • x:
      • y:
  3. Simpsons’ Rule for Non-Rect Regions:
  4. Simpsons’ Double Integral [Algorithm]:
  5. Gaussian Double Integral [Algorithm]:

Triple Integral Approximation

  1. What?
    • On what?
    • Form:
  2. Gaussian Triple Integral [Algorithm]: