BSITG

This function evaluates the integral of a spline, given its B-spline representation.

Function Return Value

BSITG — Value of the integral of the spline from A to B. (Output)

Required Arguments

A — Lower limit of integration. (Input)

B — Upper limit of integration. (Input)

KORDER — Order of the spline. (Input)

XKNOT — Array of length KORDER + NCOEF containing the knot sequence. (Input)

XKNOT must be nondecreasing.

XKNOT must be nondecreasing.

NCOEF — Number of B-spline coefficients. (Input)

BSCOEF — Array of length NCOEF containing the B-spline coefficients. (Input)

FORTRAN 90 Interface

Generic: BSITG (A, B, KORDER, XKNOT, NCOEF, BSCOEF)

Specific: The specific interface names are S_BSITG and D_BSITG.

FORTRAN 77 Interface

Single: BSITG (A, B, KORDER, XKNOT, NCOEF, BSCOEF)

Double: The double precision function name is DBSITG.

Description

The function BSITG computes the integral of a spline given its B-spline representation. Specifically, given the knot sequence t = XKNOT, the order k = KORDER, the coefficients a = BSCOEF , n = NCOEF and an interval [a, b], BSITG returns the value

This routine uses the identity (22) on page 151 of de Boor (1978), and it assumes that t1 = … = tk and tn + 1 = … = tn + k.

Comments

1. Workspace may be explicitly provided, if desired, by use of B2ITG/DB2ITG. The reference is:

CALL B2ITG(A, B, KORDER, XKNOT, NCOEF, BSCOEF, TCOEF, AJ, DL, DR)

The additional arguments are as follows:

TCOEF — Work array of length KORDER + 1.

AJ — Work array of length KORDER + 1.

DL — Work array of length KORDER + 1.

DR — Work array of length KORDER + 1.

2. Informational errors

Type | Code | Description |
---|---|---|

3 | 7 | The upper and lower endpoints of integration are equal. |

3 | 8 | The lower limit of integration is less than XKNOT(KORDER). |

3 | 9 | The upper limit of integration is greater than XKNOT(NCOEF + 1). |

4 | 4 | Multiplicity of the knots cannot exceed the order of the spline. |

4 | 5 | The knots must be nondecreasing. |

Example

We integrate the quartic (k = 5) spline that interpolates x3 at the points {i/10 : i = −10, …, 10} over the interval [0, 1]. The exact answer is 1/4 since the interpolant reproduces cubic polynomials.

USE BSITG_INT

USE BSNAK_INT

USE BSINT_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER KORDER, NDATA, NKNOT

PARAMETER (KORDER=5, NDATA=21, NKNOT=NDATA+KORDER)

!

INTEGER I, NCOEF, NOUT

REAL A, B, BSCOEF(NDATA), ERROR, EXACT, F,&

FDATA(NDATA), FI, FLOAT, VAL, X, XDATA(NDATA),&

XKNOT(NKNOT)

INTRINSIC FLOAT

! Define function and integral

F(X) = X*X*X

FI(X) = X**4/4.0

! Set up interpolation points

DO 10 I=1, NDATA

XDATA(I) = FLOAT(I-11)/10.0

FDATA(I) = F(XDATA(I))

10 CONTINUE

! Generate knot sequence

CALL BSNAK (NDATA, XDATA, KORDER, XKNOT)

! Interpolate

CALL BSINT (NDATA, XDATA, FDATA, KORDER, XKNOT, BSCOEF)

! Get output unit number

CALL UMACH (2, NOUT)

!

NCOEF = NDATA

A = 0.0

B = 1.0

! Integrate from A to B

VAL = BSITG(A,B,KORDER,XKNOT,NCOEF,BSCOEF)

EXACT = FI(B) - FI(A)

ERROR = EXACT - VAL

! Print results

WRITE (NOUT,99999) A, B, VAL, EXACT, ERROR

99999 FORMAT (' On the closed interval (', F3.1, ',', F3.1,&

') we have :', /, 1X, 'Computed Integral = ', F10.5, /,&

1X, 'Exact Integral = ', F10.5, /, 1X, 'Error '&

, ' = ', F10.6, /, /)

END

Output

On the closed interval (0.0,1.0) we have :

Computed Integral = 0.25000

Exact Integral = 0.25000

Error = 0.000000