## Numerical Computing with Modern Fortran## Richard J. Hanson and Tim Hopkins |

# Chapter 7: Case Study: Toward A Modern QUADPACK Routine

Source Code:

These are examples of numerical quadrature. The underlying integrator is a version of the routine qag, found in the classic Quadpack program suite. This has been modernized to allow for multiple integrals, vector integrals, and to facilitate importing user data into the integrand evaluation function. The names of the scalar and vector integration routines are generic, qag2003, in all examples.

Each examples uses the module quadpack2003. As a first step, compile set_precision.f90 and then quadpack2003.f90. Have these modules accessible for the compilation of the examples.

There are five driver programs that may be built using the makefile.

- TestEx1fs uses testEx1fs.f90, testModEx1fs.f90
A quadrature computes the values of a Bessel function with a real argument. The base derived type quadpackbase is extended to include a parameter, the function argument. This is passed to the evaluation routine for the integrand.

- TestEx1fv uses testEx1fv.f90, testModEx1fs.f90
Computes the values of an array of Bessel functions. The base derived type quadpackbase is extended as in example TestEx1fs. An array of these extended types is allocated for the 101 values of the array of arguments. A do loop evaluates the array values, returning the Bessel function values as the result. OpenMP is used on this loop, since each array component is independent of the others.

- TestEx2 uses testEx2.f90, testModEx2.f90
A multiple integral, for two dimensions, is evaluated. The algorithm is based on recursive calls to the one-dimensional routine. The outer integral evaluates the integrand of the inner integral by calls to the one-dimensional routine.

The inner integral has variable limit functions. The base derived type is extended to include procedure pointers for the limit functions, the integrand evaluation routine itself, and problem data. Use of the ASSOCIATE construct is illustrated.

- TestEx3 uses testEx3.f90, testModEx3.f90
A complex-valued line integral is evaluated by casting it as a vector integral with two components, the first one for the real part and the second for the imaginary part.

- extraTests uses extraTests.f90, testextraTests.f90
This main program illustrates six quadrature problems.

There is performance information obtained for evaluating the same problem as in TestEx1fs. The number of function evaluations, and array or vector evaluations of the integrand, are printed. An additional scalar is included in the extended type. This is used in an evaluation of a real Laplace transform integral, both for a scalar and a vector function. This is for problems five and six.

### Sample output from TestEx1fs

Integral (PI*J_0(100)) is = 0.062787

### Sample output from TestEx1fv

Integral (PI*J_0(i)) is = 3.141593, with i = 0 Integral (PI*J_0(i)) is = 2.403939, with i = 1 Integral (PI*J_0(i)) is = 0.703374, with i = 2 Integral (PI*J_0(i)) is = -0.816977, with i = 3 Integral (PI*J_0(i)) is = -1.247683, with i = 4 Integral (PI*J_0(i)) is = -0.557937, with i = 5 Integral (PI*J_0(i)) is = 0.473266, with i = 6 Integral (PI*J_0(i)) is = 0.942727, with i = 7 Integral (PI*J_0(i)) is = 0.539257, with i = 8 Integral (PI*J_0(i)) is = -0.283791, with i = 9 Values for i=10,95 are not shown. Integral (PI*J_0(i)) is = 0.145564, with i = 96 Integral (PI*J_0(i)) is = -0.097875, with i = 97 Integral (PI*J_0(i)) is = -0.249292, with i = 98 Integral (PI*J_0(i)) is = -0.171136, with i = 99 Integral (PI*J_0(i)) is = 0.062787, with i = 100

### Sample output from TestEx2

ALPHA, BETA, MY_PHI(ALPHA,BETA) = 1.000000 1.000000 2.257974

### Sample output from TestEx2

Complex Value of ALPHA = 0.500000 0.000000 Upper Unit Circle Integral of 1/z**ALPHA is = -2.000 + I * 2.000 IER from line integration = 0 Line integral max. error estimate = 2.2204E-14

### Sample Output from extraTests

Integral (PI*J_0(100)) is = 6.278740049149326E-002 Result flag from integration = 0 Formula class, evaluations = 6 , 1525 Formula class, vector evaluations = 6 , 25 A multiple - or double - integral has an exact value that is known. This provides a test case for the algorithm. The base type is extended in a similar manner for the case TestEx2. Double Integral (4*log(2+sqrt(3)-2*PI/6) is = 3.17343648530607 Relative error is = 0.000000000000000E+000 Result flag from integration = 0 Formula class, evaluations = 6 , 61 The previous integrand function is modified so that the inner integral has variable limits that describe a curved sub-region. Double Integral over curved region is = 2.25797353331653 Result flag from integration = 0 Formula class, evaluations = 6 , 61 A one-dimensional infinite integral is evaluated. The infinite upper limit is truncated to a finite value. Infinite Integral is = -0.361689424516046 Result flag from integration = 0 Vector evaluations = 173 A one-dimensional infinite integral is evaluated. The infinite upper limit is truncated to a finite value. Laplace Transform = 0.500000000000000 Result flag from integration = 0 A vector of one-dimensional infinite integrals is evaluated. The infinite upper limit is truncated to a finite value. An alternate error norm is used for the vector result. A procedure pointer in the base type points to a root-mean-square norm, overriding the default max norm. The evaluation routine returns a matrix of function values and the number of these evaluations is printed. Laplace Transform = 0.500000000000000 0.249999999999998 0.124999999999981 Result flag from integration = 0 Matrix evaluations = 2