◆ Pchid()

Function Pchid	(	N As	Long,
		X() As	Double,
		F() As	Double,
		D() As	Double,
		IA As	Long,
		IB As	Long,
		Info As	Long,
		Optional Skip As	Boolean = `False`,
		Optional Idxfd As	Long = `0`,
		Optional Incfd As	Long = `1`
	)

Integral of piecewise cubic Hermite / cubic spline function (over an interval whose endpoints are data points)

Purpose: This routine evaluates the definite integral of a piecewise cubic Hermite function over an interval whose endpoints are data points. The interpolant is defined by N, X(), F() and D() computed by Pchim, Pchic, Pchsp or Pchse.

Returns: Double
Value of the requested integral.

Parameters

[in]	N	Number of data points. (N >= 2)
[in]	X()	Array X(LX - 1) (LX >= N) Independent variable values. The elements of X() must be strictly increasing.
[in]	F()	Array F(LF - 1) (LF >= Incfd(N - 1) + 1) Function values. F(Idxfd + IIncfd) is the value corresponding to X(I) (I = 0 to N - 1).
[in]	D()	Array D(LD - 1) (LD >= Incfd(N - 1) + 1) Derivative values. D(Idxfd + IIncfd) is the value corresponding to X(I) (I = 0 to N - 1).
[in]	IA	Index in X() for the lower limit of integration. (0 <= IA < IB <= N - 1)
[in]	IB	Index in X() for the upper limit of integration. (0 <= IA < IB <= N - 1)
[out]	Info	= 0: Successful exit. = -1: The argument N had an illegal value. (N < 2) = -2: The argument X() had an illegal value. (X() is not strictly increasing) = -3: The argument F() is invalid. = -4: The argument D() is invalid. = -7: The argument IA had an illegal value. (IA < 0 or IA > N-1) = -8: The argument IB had an illegal value. (IB < 0 or IB > N-1) = -10: The argument Incfd had an illegal value. (Incfd < 1)
[in]	Skip	(Optional) Logical variable which should be set to True if the user wishes to skip checks for validity of preceding parameters, or to False otherwise. This will save time in case these checks have already been performed (say, in Pchim, Pchic, Pchsp or Pchse). (defaualt = False)
[in]	Idxfd	(Optional) Index of the first data in F() and D(). (default = 0)
[in]	Incfd	(Optional) Increment between successive values in F() and D(). (Incfd >= 1) (default = 1)

Reference: SLATEC (PCHIP)

Example Program

Using the following table, compute S = integral of 1/(1 + x^2) dx [0, 4] (= atan(4)).

  x    1/(1 + x^2)
----- -------------
 -1        0.5
  0        1 
  1        0.5 
  2        0.2 
  3        0.1 
  4        0.05882
  5        0.03846
----- -------------

Sub Ex_Pchid()
    Const N = 7, IA = 1, IB = 5
    Dim X(N - 1) As Double, Y(N - 1) As Double, D(N - 1) As Double, S As Double
    Dim Info As Long, I As Long
    '-- Data
    X(0) = -1: Y(0) = 0.5
    X(1) = 0: Y(1) = 1
    X(2) = 1: Y(2) = 0.5
    X(3) = 2: Y(3) = 0.2
    X(4) = 3: Y(4) = 0.1
    X(5) = 4: Y(5) = 0.05882
    X(6) = 5: Y(6) = 0.03846
    '-- Spline interpolation
    Call Pchse(N, X(), Y(), D(), Info)
    If Info <> 0 Then
        Debug.Print "Error in Pchse: Info =", Info
        Exit Sub
    End If
    '-- Compute integral 1/(1 + x^2) dx [0, 4] (= atan(4))
    S = Pchid(N, X(), Y(), D(), IA, IB, Info)
    Debug.Print "S =", S, "S(true) =", Atn(4)
    Debug.Print "Info =", Info
End Sub

Example Results: S = 1.31070738095238 S(true) = 1.32581766366803

Info = 0