These computer exercises are small variants of those used at the teaching of Fortran at Linköping University, both for the students of technology and for the students in the mathematics and science program.
When I write course directory below it is intended as an advice to the local teacher to make the required files available to the students. Of course, it is also possible to type them from the textbook, extract them from this HTML-file, or get them electronically from our course directory. Using UNIX it is easy to introduce symbolic names, so I am calling the course directory $KURSBIB instead of the complete old name /undv/mai/numt/fort/kursbib or the new name /mailocal/lab/numt/TANA70/
Locally is both Fortran 77 (f77) and Fortran 90 (f90) available on our workstations. Both the workstations and the file servers are from Digital Equipment Corporation, they are DEC Alpha stations running Digital UNIX. Fortran 77 is the DEC compiler, Fortran 90 is also a DEC compiler. Previously we used DEC MIPS stations, running DEC Fortran 77 and NAG Fortran 90.
Students of technology are supposed to solve the exercises 2 to 5, while mathematicians are supposed to solve the exercises 1, 2, 3, 4a and 5.
Further information on the course (including experiences from earlier students) is available (partly in Swedish, partly in English) on URL=http://www.nsc.liu.se/~boein/edu/edu.html
The computers used for teaching at the Department of Mathematics have been replaced during 1996, from DEC Ultrix on MIPS machines to DEC UNIX on Alpha machines. The new computers also mean a change in location of the course directory. You define the course directory by writing
setenv KURSBIB /mailocal/lab/numt/TANA70/A suitable environment is obtained with the TANA70setup command.
Further information on the DEC Fortran 90, and its system parameters, hints on compilation, and some examples are available.
The following material shall be handed to the teacher:
Number of steps Step length 1 1 2 0.5 4 0.25 8 0.125For those who do not know Pascal the recommended alternative is to get an elementary textbook on numerical methods and code Runge-Kutta directly in Fortran.
program RK1; (* A simple program in Pascal for Runge-Kutta's method for a first order differential equation. dy/dx = x^2 + sin(xy) y(1) = 2 *) var number, i : integer; h, k1, k2, k3, k4, x, y : real; function f(x,y : real) : real; begin f := x*x + sin(x*y) end; begin number := 1; while number > 0 do begin x := 1.0; y := 2.0; writeln(' Give the number of steps '); read(number); if number >= 1 then begin writeln(' Give the step length '); read(h); writeln(' x y'); writeln(x, y); for i := 1 to number do begin k1 := h*f(x,y); k2 := h*f(x+0.5*h,y+0.5*k1); k3 := h*f(x+0.5*h,y+0.5*k2); k4 := h*f(x+h,y+k3); x := x + h; y := y + (k1+2*k2+2*k3+k4)/6; writeln(x, y); end; end; end; end.If you wish to run this program you have to replace the first line
program RK1;with the following extended line
program RK1(INPUT, OUTPUT);when running on a DEC UNIX system, but not on a Sun Solaris system.
When the program is believed to be correct (test with the equation x*x + 2 = 0), the second assignment is to modify the program so that it can accept data from either the keyboard or from a file $KURSBIB/lab21.dat. The modifications are required to be done in such a way that the user can select if data are supposed to be input directly from the keyboard or from a file, in which case the user also will be required to give the name of the file.
Finally, check that the program also works for the file $KURSBIB/lab22.dat.
Note that the program uses complex numbers, and that such values are input as pairs within parenthesis.
The following material shall be handed to the teacher:
****************************************************************************** ** This program calculates all roots (real and/or complex) to an ** ** N:th-degree polynomial with complex coefficients. (N <= 10) ** ** ** ** n n-1 ** ** P(z) = a z + a z + ... + a z + a ** ** n n-1 1 0 ** ** ** ** The execution terminates if ** ** ** ** 1) Abs (Z1-Z0) < EPS ==> Root found = Z1 ** ** 2) ITER > MAX ==> Slow convergence ** ** ** ** The program sets EPS to 1.0E-7 and MAX to 30 ** ** ** ** The NEWTON-RAPHSON method is used: z1 = z0 - P(z0)/P'(z0) ** ** The values of P(z0) and P'(z0) are calculated with HORNER'S SCHEME. ** ** ** ** The array A(0:10) contains the complex coefficients of the ** ** polynomial P(z). ** ** The array B(1:10) contains the complex coefficients of the ** ** polynomial Q(z), ** ** where P(Z) = (z-z0)*Q(z) + P(z0) ** ** The coefficients to Q(z) are calculated with HORNER'S SCHEME. ** ** ** ** When the first root has been obtained with the NEWTON-RAPHSON ** ** method, it is divided away (deflation). ** ** The quotient polynomial is Q(z). ** ** The process is repeated, now using the coefficients of Q(z) as ** ** input data. ** ** As a starting approximation the value STARTV = 1+i is used ** ** in all cases. ** ** Z0 is the previous approximation to the root. ** ** Z1 is the latest approximation to the root. ** ** F0 = P(Z0) ** ** FPRIM0 = P'(Z0) ** ****************************************************************************** COMPLEX A(0:10), B(1:10), Z0, Z1, STARTV INTEGER N, I, ITER, MAX REAL EPS DATA EPS/1E-7/, MAX /30/, STARTV /(1,1)/ ****************************************************************************** 20 WRITE(6,*) 'Give the degree of the polynomial' READ (5,*) N IF (N .GT. 10) THEN WRITE(6,*) 'The polynomial degree is maximum 10' GOTO 20 WRITE (6,*) 'Give the coefficients of the polynomial,', , ' as complex constants' WRITE (6,*) 'Highest degree coefficient first' DO 30 I = N, 0, -1 WRITE (6,*) 'A(' , I, ') = ' READ (5,*) A(I) 30 CONTINUE WRITE (5,*) ' The roots are',' Number of iterations' ****************************************************************************** 40 IF (N GT 0) THEN C ****** Find the next root ****** Z0 = (0,0) ITER = 0 Z1 = STARTV 50 IF (ABS(Z1-Z0) .GE. EPS) THEN C ++++++ Continue the iteration until two estimates ++++++ C ++++++ are sufficiently close. ++++++ ITER = ITER + 1 IF (ITER .GT. MAX) THEN C ------ Too many iterations ==> TERMINATE ------ WRITE (6,*) 'Too many iterations.' WRITE (6,*) 'The latest approximation to the root is ',Z1 GOTO 200 ENDIF Z0 = Z1 HORNER (N, A, B, Z0, F0, FPRIM0) C ++++++ NEWTON-RAPHSON'S METHOD ++++++ Z1 = Z0 - F0/FPRIM0 GOTO 50 ENDIF C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 100 WRITE (6,*) Z1, ' ',Iter C ****** The root is found. Divide it away and seek the next one ***** N = N - 1 FOR I = 0 TO N DO A(I) = B(I+1) GOTO 40 ENDIF 200 END SUBROUTINE HORNER(N, A, B, Z, F, FPRIM) ****** For the meaning of the parameters - see the main program ****** ****** BI and CI are temporary variable. ****** INTEGER N, I COMPLEX A(1:10), B(0:10), Z, F, FPRIM, BI, CI BI = A(N) B(N) = BI CI = BI DO 60 I = N-1, 1, -1 BI = A(I) + Z*BI CI = BI + Z*CI B(I) = BI C ++++++ BI = B(I) for the calculation of P(Z) ++++++ C ++++++ CI for the calculation of P'(Z) ++++++ 60 CONTINUE F = A(0) + Z*BI FPRIM = CI RETURN END ****** END HORNER'S SCHEME ****** ****** This program is composed by Ulla Ouchterlony 1984 ****** ****** The program is translated by Bo Einarsson 1995 ******Comments. I recommend the following process at the solution of this exercise.
f77 -u lab2_e.f a.outUsing a modern system, for example Fortran 90, you put the statement IMPLICIT NONE first in booth the main program and the subroutine.
f77 -C lab2_e.f a.out
The explicit DO-loop below will write seven lines, each one with one value of X
DO I = 1, 13, 2 WRITE(*,*) X(I) END DO
The implicit DO-loop below will write only one line, but with seven values of X
WRITE(*,*) ( X(I), I = 1, 13, 2)
Make a test run and print a table with x and J0(x) for x = 0.0, 0.1, ... 1.0. Make the output to look nice!
At the use of the NAG library the NAG User Notes are very useful. They describe the system specific parts of the library. Some of these are available here as
The required Bessel function in the NAG library is S17AEF and has the arguments X and IFAIL and in this order, where X is the argument for the Bessel function in single or double precision, and where IFAIL is an error parameter (integer). The error parameter is given the value 1 before entry, and is checked at exit. If it is 0 at exit, everything is OK, if it is 1 at exit, the magnitude of the argument is too large.
If the NAG library is installed in the normal way it is linked with the command
f77 prog.f -lnagwhere prog.f is your program.
If you use Fortran 90 for your own program but has the NAG library available only in Fortran 77 special care is necessary, see chapter 15, Method 2.
On Sun you compile and link with
f90 prog.f -lnag -lF77and on DEC ULTRIX with
f90 prog.f -lnag -lfor -lutil -li -lots
On DEC UNIX -lfor -lutil -lots are available.
It is very important to note in which precision the NAG library is made available on your system. Using the wrong precision will usually give completely wrong results!
Local information: The NAG library is in double precision on the DEC system.
Write a program in Fortran 77 for tabulating the Bosse function Bo(x). Write the main program in such a way that it asks for an interval for which the function values are to be evaluated.
Make a test run and print a table with x and Bo(x) for x = 0.0, 0.1, ... 1.0. Make the output to look nice! Investigate also what happens with the arguments +800 and -900!
The Bosse function BO is in the library libbosse.a on the course directory and has the arguments X and IFAIL and in this order, where X is the argument for the Bosse function in single or double precision, and where IFAIL is an error parameter (integer). The error parameter is given the value 1 before entry, and is checked at exit. If it is 0 at exit, everything is OK, if it is 1 at exit, the argument is too large, if it is 2 at exit, the argument is too small. If the error parameter is zero on entry and an error occurs the program will stop with an error message from the library error handling subroutine, and not return to the user's program. The error handling is similar to the one in the NAG Fortran 77 library.
If the Bosse library is installed in the normal way it is linked with the command
f77 prog.f -lbosse -L$KURSBIBwhere prog.f is your program.
If you use Fortran 90 for your own program but you still have the Bosse library available only in Fortran 77 special care is necessary, see chapter 15, Method 2, see also the discussion above in exercise 3b.
To make life simpler we have however made also a Fortran 90 version of the Bosse library, which is used with
f90 prog.f90 -lbosse90 -L$KURSBIBIt is very important to note in which precision the Bosse library is made available on your system. Using the wrong precision will usually give completely wrong results!
Local information: The Bosse library is in double precision on the DEC system.
Compare with Exercise 3 a.
y'(t) = z(t) y(0) = 1 z'(t) = 2*y*(1+y^2) + t*sin(z) z(0) = 2Calculate an approximation to y(0.2) using the step size 0.04. Repeat the calculation with the step sizes 0.02 and 0.01. The functions representing the differential functions are to be given as internal functions. It is therefore not permitted (in this exercise) to use the usual external functions.
Starting values of t, y and z are to be given interactively, as well as the step size h.
Note that the requirement not to use external functions makes it a little more difficult to use a subprogram for the Runge-Kutta steps (in this case also the subprogram has to be internal). The aim of this requirement is to give the student an opportunity to use the new internal functions of Fortran 90.
The formulas for Runge-Kutta at systems are available here as an dvi-file and here as an HTML-file (which requires indices)!
The first subprogram LASMAT is used to interactively input the floating point values of a quadratic matrix. The subprogram is required to ask for the dimension and give the user the choice of providing from the keyboard either all the values of the matrix, or only those which are different from zero, The matrix shall then be stored in a file, using maximal accuracy and minimal storage space. The possible sparsity of the matrix is however not permitted to be exploited at this storage process. The subprogram shall ask for the name of the file, but the file type (extension) will be .mat. The user is not permitted to give the file type.
The second subprogram LASVEK is used to interactively input the floating point values of a vector. The subprogram is required to ask for the dimension and give the user the choice of providing from the keyboard either all the values of the vector, or only those which are different from zero, The vector shall then be stored in a file, using maximal accuracy and minimal storage space. The possible sparsity of the vector is however not permitted to be exploited at this storage process. The subprogram shall ask for the name of the file, but the file type (extension) will be .vek. The user is not permitted to give the file type.
The third subprogram MATIN reads a matrix from the file with the given name and stores it as an array (matrix) in the subprogram. The user is not permitted to give the file type.
The fourth subprogram VEKIN reads a vector from the file with the given name and stores it as an array (vector) in the subprogram. The user is not permitted to give the file type.
The fifth subprogram MATUT prints a matrix on paper, with rows and columns in the normal way if the dimension is at most 10, and in an easily understandable way if the dimension is larger than 10. The available 132 positions of a typical line printer are to be utilized as well as possible. Output on paper is under UNIX not done directly, so you will have to first write to a file, which is then moved to paper with the UNIX print command lpr or preferably fpr or asa, see the end of Appendix 2. In order to get 132 positions you can usually add -w132 to the print command.
The sixth subprogram VEKUT prints a vector on paper, preferably in the transposed form (line-wise).
The seventh subprogram LOS solves the system of equations through a call to the provided routine SOLVE_LINEAR_SYSTEM, which is given in double precision. No changes are permitted in that routine! If the dimension of the matrix and the vector do not agree, then the subprogram LOS or the main program is required to give an error message. An error message is also required if the routine SOLVE_LINEAR_SYSTEM detects that the matrix is singular.
The main program HUVUD utilizes LASMAT, LASVEK, MATIN and VEKIN in order to get the matrix A and the vector b, calls the solver subprogram LOS and prints the matrix A , the right hand side b and the solution x with the subprograms MATUT and VEKUT.
The dimension in all the programs shall use the dynamic possibilities of Fortran 90. One alternative is to use pointers at the specification, see chapter 12. Another possibility, suggested by Arie ten Cate is to introduce SAVEd arrays in a MODULE. A simple example of this is available.
The program is required to return to the starting point after each calculation, and ask for a new matrix and/or vector. It has to be possible to use a stored set of matrix and vector, without manual input of a lot of values.
In the exercise it is included the essential task of performing any additional specifications, test the reasonableness of the given values, and to construct suitable test matrices and vectors.
A compulsory test example follows
33 16 72 -359 A = -24 -10 -57 b = 281 -8 -4 -17 85The students are requried to hand in a complete program listing, and results from actual runs with matrices of the order 3, 10, and 12, including input of a sparse matrix.
I assume that the main program is on the file huvud.f90, the subroutine LASMAT on the file lasmat.f90, and so on, and that a possible complete program is on the file labb5.f90.
f90 huvud.f90 lasmat.f90 lasvek.f90 ... los.f90 solve.f90Execute with
f90 -c huvud.f90 f90 -c lasmat.f90 f90 -c lasvek.f90 ... f90 -c los.f90 f90 -c solve.f90 f90 huvud.o lasmat.o lasvek.o ... los.o solve.oExecute with
a.outWhen one routine has been changed, now only that one has to be recompiled, followed by linking of all the routines. This method is much faster!
f90 labb5.f90Execute with
a.outThis method is very slow!
labb5: huvud.o lasmat.o lasvek.o matin.o vekin.o \ matut.o vekut.o los.o solve.o f90 -o labb5 huvud.o lasmat.o lasvek.o matin.o vekin.o \ matut.o vekut.o los.o solve.o huvud.o: huvud.f90 f90 -c huvud.f90 lasmat.o: lasmat.f90 f90 -c lasmat.f90 lasvek.o: lasvek.f90 f90 -c lasvek.f90 matin.o: matin.f90 f90 -c matin.f90 vekin.o: vekin.f90 f90 -c vekin.f90 matut.o: matut.f90 f90 -c matut.f90 vekut.o: vekut.f90 f90 -c vekut.f90 los.o: los.f90 f90 -c los.f90 solve.o: solve.f90 f90 -c solve.f90The above is used by moving the file makefile to your directory and when you wish to compile you write only make in the terminal window. Then only those files that have been changed as compared with the previous make command are recompiled (or all files if it is the first time), then all the object modules are linked to a program, ready to run with the command labb5.
If you have used other file names you have to make the appropriate modifications to the file makefile.
In principle make works in such a way that if an item which is after a colon : has a later time than that before the colon, the command on the next line is performed.
Remark. The backslash \ indicates a continuation line in UNIX.
The routine SOLVE_LINEAR_SYSTEM is available in single precision in the course directory with the name solve1.f90 and in double precision with the name solve.f90, and looks as follows. The single precision version is discussed at length in the Solution to Exercise (11.1).
SUBROUTINE SOLVE_LINEAR_SYSTEM(A, X, B, ERROR) IMPLICIT NONE ! Array specifications DOUBLE PRECISION, DIMENSION (:, :), INTENT (IN) :: A DOUBLE PRECISION, DIMENSION (:), INTENT (OUT) :: X DOUBLE PRECISION, DIMENSION (:), INTENT (IN) :: B LOGICAL, INTENT (OUT) :: ERROR ! The work area M is A extended with B DOUBLE PRECISION, DIMENSION (SIZE (B), SIZE (B) + 1) :: M INTEGER, DIMENSION (1) :: MAX_LOC DOUBLE PRECISION, DIMENSION (SIZE (B) + 1) :: TEMP_ROW INTEGER :: N, K ! Initiate M N = SIZE (B) M (1:N, 1:N) = A M (1:N, N+1) = B ! Triangularization phase ERROR = .FALSE. TRIANG_LOOP: DO K = 1, N - 1 ! Pivotering MAX_LOC = MAXLOC (ABS (M (K:N, K))) IF ( MAX_LOC(1) /= 1 ) THEN TEMP_ROW (K:N+1 ) = M (K, K:N+1) M (K, K:N+1) = M (K-1+MAX_LOC(1), K:N+1) M (K-1+MAX_LOC(1), K:N+1) = TEMP_ROW (K:N+1) END IF IF (M (K, K) == 0.0D0) THEN ERROR = .TRUE. ! Singular matrix A EXIT TRIANG_LOOP ELSE TEMP_ROW (K+1:N) = M (K+1:N, K) / M (K, K) M (K+1:N, K+1:N+1) = M (K+1:N, K+1:N+1) & - SPREAD( TEMP_ROW (K+1:N), 2, N-K+1) & * SPREAD( M (K, K+1:N+1), 1, N-K) M (K+1:N, K) = 0 ! These values are not used END IF END DO TRIANG_LOOP IF (M (N, N) == 0.0D0) ERROR = .TRUE. ! Singular matrix A ! Back substitution IF (ERROR) THEN X = 0.0D0 ELSE DO K = N, 1, -1 X (K) = M (K, N+1) - SUM (M (K, K+1:N) * X (K+1:N)) X (K) = X (K) / M (K, K) END DO END IF END SUBROUTINE SOLVE_LINEAR_SYSTEM
At the the specification of the array to store the matrix, you may chose either the method in a) or b) above, or perhaps invent a third method.
If the makefile is used, the reference to solve.f90 must be changed.