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IntervalExamplesFM.f95
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PROGRAM INTERVAL_EXAMPLES
USE FMZM
USE FM_INTERVAL_ARITHMETIC
! This program contains the code for the FM interval arithmetic examples that are discussed in
! the paper "A Multiple-Precision Interval Arithmetic Package".
IMPLICIT NONE
INTEGER :: I, J, J_MAX, J_MIN, K, N, NF, K_FUNCTION, N_ORDER, N_STEPS, KPRT, KW, K_GRAPH
TYPE (FM_INTERVAL) :: A, B, RESULT, S, TERM, PI, C, X, Y, V(5), DET
TYPE (FM_INTERVAL), ALLOCATABLE :: MATRIX(:,:), RHS(:), SOLN(:)
INTEGER, ALLOCATABLE :: KSWAP(:)
TYPE (FM) :: A_FM, B_FM, S_FM, PI_FM, C_FM, X_FM, Y_FM
TYPE (FM) :: ERROR_FM, WIDTH_FM, LEAST_WIDTH_FM
DOUBLE PRECISION :: RAND
! Set the FM precision to 50 digits.
CALL FM_SET(50)
! Write output to file IntervalExamplesFM.out.
! The FM_SETVAR call directs all output from within the FM package to unit 22 also.
OPEN(22,FILE='IntervalExamplesFM.out')
CALL FM_SETVAR(' KW = 22 ')
! Unit 31 will be used to write several files that give the interval width
! for each step of the different examples.
K_GRAPH = 31
! Example 1. Ignoring variable correlation in formulas causes interval arithmetic
! to get too wide a resulting interval.
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) " Example 1. Compute f(x) = X**2 - X + 3 using different formulas."
WRITE (22,*) ' '
X = TO_FM_INTERVAL( " -0.5 ", " 1.0 " )
WRITE (22,*) ' '
WRITE (22,*) ' X is the interval [ -0.5 , 1.0 ]'
WRITE (22,*) ' '
X_FM = 0.5
A_FM = X_FM**2 - X_FM + 3
X_FM = LEFT_ENDPOINT(X)
B_FM = X_FM**2 - X_FM + 3
LEAST_WIDTH_FM = B_FM - A_FM
Y = X**2
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A)") ' X**2 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]'
Y = X*X
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A)") ' X*X gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]'
WRITE (22,*) ' '
WRITE (22,*) ' '
Y = X**2 - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X**2 - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*X - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*X - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*(X - 1) + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*(X - 1) + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = (X - 0.5)**2 + 2.75
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' (X - 0.5)**2 + 2.75 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) ' '
! Results are not as bad when the interval doesn't include zero.
X = TO_FM_INTERVAL( " 0.1 ", " 1.0 " )
WRITE (22,*) ' '
WRITE (22,*) ' X is the interval [ 0.1 , 1.0 ]'
WRITE (22,*) ' '
X_FM = 0.5
A_FM = X_FM**2 - X_FM + 3
X_FM = RIGHT_ENDPOINT(X)
B_FM = X_FM**2 - X_FM + 3
LEAST_WIDTH_FM = B_FM - A_FM
Y = X**2 - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X**2 - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*X - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*X - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*(X - 1) + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*(X - 1) + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = (X - 0.5)**2 + 2.75
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' (X - 0.5)**2 + 2.75 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) ' '
! Results are better when the interval doesn't include a local minimum.
X = TO_FM_INTERVAL( " 0.9 ", " 1.0 " )
WRITE (22,*) ' '
WRITE (22,*) ' X is the interval [ 0.9 , 1.0 ]'
WRITE (22,*) ' '
X_FM = LEFT_ENDPOINT(X)
A_FM = X_FM**2 - X_FM + 3
X_FM = RIGHT_ENDPOINT(X)
B_FM = X_FM**2 - X_FM + 3
LEAST_WIDTH_FM = B_FM - A_FM
Y = X**2 - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X**2 - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*X - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*X - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*(X - 1) + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*(X - 1) + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = (X - 0.5)**2 + 2.75
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' (X - 0.5)**2 + 2.75 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) ' '
! Look at an even smaller interval
X = TO_FM_INTERVAL( " 0.99 ", " 1.0 " )
WRITE (22,*) ' '
WRITE (22,*) ' X is the interval [ 0.99 , 1.0 ]'
WRITE (22,*) ' '
X_FM = LEFT_ENDPOINT(X)
A_FM = X_FM**2 - X_FM + 3
X_FM = RIGHT_ENDPOINT(X)
B_FM = X_FM**2 - X_FM + 3
LEAST_WIDTH_FM = B_FM - A_FM
Y = X**2 - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X**2 - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*X - X + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*X - X + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = X*(X - 1) + 3
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' X*(X - 1) + 3 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
Y = (X - 0.5)**2 + 2.75
C_FM = (RIGHT_ENDPOINT(Y) - LEFT_ENDPOINT(Y)) / LEAST_WIDTH_FM
WRITE (22,*) ' '
WRITE (22,"(A,F6.3,A,F6.3,A,F6.3)") ' (X - 0.5)**2 + 2.75 gives [ ', &
TO_DP(LEFT_ENDPOINT(Y)), ' , ', TO_DP(RIGHT_ENDPOINT(Y)), &
' ]. Magnification = ', TO_DP(C_FM)
WRITE (22,*) ' '
! Example 2. Sum 1 / n^7 from 1 to 100,000.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Sum')
WRITE (K_GRAPH,"(//A//)") 'Example 2. Sum 1 / n^7 from 1 to 100,000.'
N = 10**5
S = 0
DO J = 1, N
TERM = J
S = S + 1 / TERM**7
IF (MOD(J,N/1000) == 1) THEN
WIDTH_FM = MAX( RIGHT_ENDPOINT(S)-LEFT_ENDPOINT(S) , EPSILON(LEFT_ENDPOINT(S)) )
WRITE (K_GRAPH,"(A,I7,A,F8.3,A)") '{', J, ',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ENDIF
ENDDO
WRITE (22,*) ' '
WRITE (22,*) " Example 2. Sum 1 / n^7 from 1 to 100,000 ="
CALL FMPRINT_INTERVAL(S)
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(S)-LEFT_ENDPOINT(S)) / RIGHT_ENDPOINT(S) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints agree to about',J,' decimal digits.'
WRITE (22,*) ' '
CLOSE(K_GRAPH)
! Example 3. Use composite 9-point Gauss quadrature with 1,000 subintervals
! to integrate sin(t) / t from t = 0 to t = 20.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Integration')
WRITE (K_GRAPH,"(//A//)") &
'Example 3. Use composite 9-point Gauss quadrature with 1,000 subintervals'
NF = 1
N = 1000
A = 0
B = 20
CALL GAUSS_9(NF, A, B, N, RESULT)
WRITE (22,*) ' '
WRITE (22,*) " Example 3. Integrate sin(t) / t from t = 0 to t = 20:"
CALL FMPRINT_INTERVAL(RESULT)
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(RESULT)-LEFT_ENDPOINT(RESULT)) / RIGHT_ENDPOINT(RESULT) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints agree to about',J,' decimal digits.'
WRITE (22,*) ' '
CLOSE(K_GRAPH)
! Example 4. Start with ( x, y ) = ( 1, 0 ) and step around the unit circle
! using the recurrence ( x, y ) <-- ( x*c - y*d, y*c + x*d ),
! c = cos(2*pi/n) and s = sin(2*pi/n)
! After 23 trips around the circle with 100 steps per trip,
! x = ( -58.6 , 60.6 ) and y = ( -59.6 , 59.6 )
! After 21 trips around the circle with 1000 steps per trip,
! x = ( 0.49 , 1.51 ) and y = ( -0.51 , 0.51 )
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Circle Recurrence')
WRITE (K_GRAPH,"(//A//)") &
'Example 4. Start with ( x, y ) = ( 1, 0 ) and step around the unit circle'
N = 100
X = 1
Y = 0
PI = ACOS(TO_FM(-1))
C = COS(2*PI/N)
S = SIN(2*PI/N)
WRITE (22,*) ' '
WRITE (22,*) " Example 4. Step around the unit circle multiple times using a recurrence"
WRITE (22,*) ' '
DO K = 1, 23
DO J = 1, N
A = X*C - Y*S
B = Y*C + X*S
X = A
Y = B
WIDTH_FM = MAX( RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X) , EPSILON(LEFT_ENDPOINT(X))/10**7 )
WRITE (K_GRAPH,"(A,I7,A,F8.3,A)") '{', N*(K-1)+J, &
',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ENDDO
ERROR_FM = ABS( (RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X)) / RIGHT_ENDPOINT(X) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(I4,A,I3,A)") K,' trips. The two endpoints for x agree to about', &
J,' decimal digits.'
ENDDO
WRITE (22,*) ' '
WRITE (22,*) ' X ='
CALL FMPRINT_INTERVAL(X)
WRITE (22,*) ' '
WRITE (22,*) ' Y ='
CALL FMPRINT_INTERVAL(Y)
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X)) / RIGHT_ENDPOINT(X) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints for x agree to about',J,' decimal digits.'
WRITE (22,*) ' '
! Compare the same calculation in ordinary FM arithmetic.
! After 23 trips around the circle with 100 steps per trip,
! x = 1.0 to 60 digits and y = -2.3e-55
! After 23 trips around the circle with 1000 steps per trip,
! x = 1.0 exactly and y = -2.3e-55
WRITE (22,*) ' '
WRITE (22,*) ' Compare the same calculation in ordinary FM arithmetic.'
WRITE (22,*) ' '
N = 100
X_FM = 1
Y_FM = 0
PI_FM = ACOS(TO_FM(-1))
C_FM = COS(2*PI_FM/N)
S_FM = SIN(2*PI_FM/N)
DO K = 1, 23
DO J = 1, N
A_FM = X_FM*C_FM - Y_FM*S_FM
B_FM = Y_FM*C_FM + X_FM*S_FM
X_FM = A_FM
Y_FM = B_FM
ENDDO
ERROR_FM = ABS( X_FM - 1 ) + TO_FM(' 1.0e-99 ')
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(I4,A,I3,A)") K,' trips. x agrees with 1.0 to about', &
J,' decimal digits.'
ENDDO
WRITE (22,*) ' '
WRITE (22,*) ' X ='
CALL FMPRINT(X_FM)
WRITE (22,*) ' '
WRITE (22,*) ' Y ='
CALL FMPRINT(Y_FM)
WRITE (22,*) ' '
CLOSE(K_GRAPH)
! Test to see if it is x and y oscillating between positive and negative values
! that causes the exponential growth of interval width, or if oscillation through
! positive values alone can cause it.
! Use the recurrence it oscillate back and forth between step number 8 and
! step number 18 of the original recurrence.
! Once back and forth takes 20 steps, so 5*23 = 115 trips back and forth will
! take the same number of steps as the 23 trips around the circle above.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Back and Forth Recurrence')
WRITE (K_GRAPH,"(//A/A//)") &
'Example 4b. Start with ( x, y ) = ( cos( 8* 2*pi/n), sin( 8* 2*pi/n) ) ', &
' and step around the unit circle to ( cos(18* 2*pi/n), sin(18* 2*pi/n) ) and back'
WRITE (22,*) ' '
WRITE (22,*) " Example 4b. Step back and forth around the unit circle using a recurrence"
WRITE (22,*) ' '
N = 100
PI = ACOS(TO_FM(-1))
X = COS(8* 2*PI/N)
Y = SIN(8* 2*PI/N)
C = COS(2*PI/N)
S = SIN(2*PI/N)
WRITE (22,*) ' '
WRITE (22,*) ' Starting point for X ='
CALL FMPRINT_INTERVAL(X)
WRITE (22,*) ' '
WRITE (22,*) ' Starting point for Y ='
CALL FMPRINT_INTERVAL(Y)
WRITE (22,*) ' '
! 5 back and forth trips equals 100 steps, the same as one circle trip above.
DO K = 1, 23 * 5
DO J = 1, 20
A = X*C - Y*S
B = Y*C + X*S
X = A
Y = B
WIDTH_FM = MAX( RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X) , EPSILON(LEFT_ENDPOINT(X))/10**7 )
WRITE (K_GRAPH,"(A,I7,A,F8.3,A)") '{', (20*(K-1)+J), &
',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
IF (MOD(J,10) == 0) THEN
S = -S
ENDIF
ENDDO
IF (MOD(K,5) == 0) THEN
ERROR_FM = ABS( (RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X)) / RIGHT_ENDPOINT(X) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(I4,A,I3,A)") K,' trips. The two endpoints for x agree to about', &
J,' decimal digits.'
ENDIF
ENDDO
WRITE (22,*) ' '
WRITE (22,*) ' X ='
CALL FMPRINT_INTERVAL(X)
WRITE (22,*) ' '
WRITE (22,*) ' Y ='
CALL FMPRINT_INTERVAL(Y)
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(X)-LEFT_ENDPOINT(X)) / RIGHT_ENDPOINT(X) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints for x agree to about',J,' decimal digits.'
WRITE (22,*) ' '
CLOSE(K_GRAPH)
! Example 5. 2 * Product 4*n^2 / (4*n^2-1) from 1 to 10,000.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Product')
WRITE (K_GRAPH,"(//A//)") &
'Example 5. 2 * Product 4*n^2 / (4*n^2-1) from 1 to 10,000.'
N = 10**4
S = 2
DO J = 1, N
K = 4 * J * J
TERM = K
TERM = TERM / ( K - 1 )
S = S * TERM
IF (MOD(J,N/1000) == 1) THEN
WIDTH_FM = MAX( RIGHT_ENDPOINT(S)-LEFT_ENDPOINT(S) , EPSILON(LEFT_ENDPOINT(S)) )
WRITE (K_GRAPH,"(A,I7,A,F8.3,A)") '{', J, ',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ENDIF
ENDDO
WRITE (22,*) ' '
WRITE (22,*) " Example 5. 2 * Product 4*n^2 / (4*n^2-1) from 1 to 10,000. ="
CALL FMPRINT_INTERVAL(S)
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(S)-LEFT_ENDPOINT(S)) / RIGHT_ENDPOINT(S) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints of y(30) agree to about',J,' decimal digits.'
WRITE (22,*) ' '
! Example 6. Differential equation. y'' = -y' / 10 - 2 * y / (x+2),
! y(0) = 0, y'(0) = 1
! This solution has 2 roots between 0 and 30.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Diff Eq 1')
WRITE (K_GRAPH,"(//A//)") &
"Example 6. Differential equation. y'' = -y' / 10 - 2 * y / (x+2),"
WRITE (22,*) ' '
WRITE (22,*) " Example 6. Differential equation. y'' = -y' / 10 - 2 * y / (x+2)"
WRITE (22,*) ' '
N_ORDER = 2
A = 0
B = 30
N_STEPS = 10000
V(1) = 0
V(2) = 1
K_FUNCTION = 1
KPRT = N_STEPS / 10
KW = 22
CALL RK4(N_ORDER, A, B, N_STEPS, V, K_FUNCTION, KPRT, KW)
WRITE (22,*) ' '
WRITE (22,*) ' y(30) = '
WRITE (22,*) ' '
CALL FMPRINT_INTERVAL(V(1))
WRITE (22,*) ' '
WRITE (22,*) " y'(30) = "
WRITE (22,*) ' '
CALL FMPRINT_INTERVAL(V(2))
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(V(1))-LEFT_ENDPOINT(V(1))) / RIGHT_ENDPOINT(V(1)) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints of y(30) agree to about',J,' decimal digits.'
WRITE (22,*) ' '
! Example 7. Differential equation. y'' = -y' / 10 - 200 * y / (x+2),
! y(0) = 0, y'(0) = 1
! This solution has 38 roots between 0 and 30.
! The rapid oscillation causes the interval calculation to be unstable
! and lose accuracy in a manner similar to example 4, the stepping
! recurrence making multiple trips around the unit circle.
OPEN(K_GRAPH,FILE='INTERVAL_EXAMPLES Diff Eq 2')
WRITE (K_GRAPH,"(//A//)") &
"Example 7. Differential equation. y'' = -y' / 10 - 200 * y / (x+2),"
WRITE (22,*) ' '
WRITE (22,*) " Example 7. Differential equation. y'' = -y' / 10 - 200 * y / (x+2)"
WRITE (22,*) ' '
N_ORDER = 2
A = 0
B = 30
N_STEPS = 10000
V(1) = 0
V(2) = 1
K_FUNCTION = 2
KPRT = N_STEPS / 10
KW = 22
CALL RK4(N_ORDER, A, B, N_STEPS, V, K_FUNCTION, KPRT, KW)
WRITE (22,*) ' '
WRITE (22,*) ' y(30) = '
WRITE (22,*) ' '
CALL FMPRINT_INTERVAL(V(1))
WRITE (22,*) ' '
WRITE (22,*) " y'(30) = "
WRITE (22,*) ' '
CALL FMPRINT_INTERVAL(V(2))
WRITE (22,*) ' '
ERROR_FM = ABS( (RIGHT_ENDPOINT(V(1))-LEFT_ENDPOINT(V(1))) / RIGHT_ENDPOINT(V(1)) )
J = -NINT(LOG10(ERROR_FM))
WRITE (22,"(A,I3,A)") ' The two endpoints of y(30) agree to about',J,' decimal digits.'
WRITE (22,*) ' '
CLOSE(K_GRAPH)
! Example 8. Solve a "random" NxN linear system.
WRITE (22,*) ' '
WRITE (22,*) ' Example 8. Solve a "random" NxN linear system.'
WRITE (22,*) ' '
DO N = 10, 100, 10
ALLOCATE( MATRIX(N,N), RHS(N), SOLN(N), KSWAP(N) )
DO I = 1, N
DO J = 1, N
CALL FM_RANDOM_NUMBER(RAND)
MATRIX(I,J) = RAND
ENDDO
RHS(I) = I
ENDDO
CALL FM_INTERVAL_FACTOR_LU(MATRIX, N, DET, KSWAP)
CALL FM_INTERVAL_SOLVE_LU (MATRIX, N, RHS, SOLN, KSWAP)
WRITE (22,*) ' '
J_MIN = 99
J_MAX = 0
DO I = 1, N
ERROR_FM = ABS( (RIGHT_ENDPOINT(SOLN(I))-LEFT_ENDPOINT(SOLN(I))) / &
RIGHT_ENDPOINT(SOLN(I)) )
J = -NINT(LOG10(ERROR_FM))
J_MIN = MIN( J_MIN, J )
J_MAX = MAX( J_MAX, J )
ENDDO
WRITE (22,"(A,I3,A,I3,A,I3,A)") ' For N = ', N, &
' the solution elements agree to between ', &
J_MIN, ' and ', J_MAX, ' significant digits.'
DEALLOCATE( MATRIX, RHS, SOLN, KSWAP )
ENDDO
! Example 9. Newton's method starting with a single point. Solve x*exp(x) - 2 = 0.
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) " Example 9. Newton's method starting with a single point."
WRITE (22,*) ' '
X = 1
DO J = 1, 10
A = X*EXP(X) - 2
B = (X+1)*EXP(X)
X = X - A/B
WRITE (22,*) ' '
WRITE (22,"(A,I2,A)") " Iteration ",J,'. X ='
CALL FMPRINT_INTERVAL(X)
ENDDO
! Example 10. Newton's method starting with an interval containing a root.
! Solve x*exp(x) - 2 = 0.
! The next interval is xt - f(xt)/f'(x) intersected with x.
! xt is a single point from the interval x (the midpoint is used here).
! The idea is to have a contracting sequence of intervals that
! each contain a root.
! Refs: Hansen-Greenberg 83, Baker Kearfott 95-97,
! Mayer 95, van Hentenryck et al. 97
WRITE (22,*) ' '
WRITE (22,*) ' '
WRITE (22,*) " Example 10. Newton's method starting with an interval containing a root"
WRITE (22,*) ' '
X = TO_FM_INTERVAL( " 0.5 ", " 1.0 " )
DO J = 1, 9
X_FM = LEFT_ENDPOINT(X)/2 + RIGHT_ENDPOINT(X)/2
Y_FM = X_FM*EXP(X_FM) - 2
B = (X+1)*EXP(X)
Y = X_FM - Y_FM/B
X = TO_FM_INTERVAL( MAX( LEFT_ENDPOINT(X),LEFT_ENDPOINT(Y) ) , &
MIN( RIGHT_ENDPOINT(X),RIGHT_ENDPOINT(Y) ) )
WRITE (22,*) ' '
WRITE (22,"(A,I2,A)") " Iteration ",J,'. X ='
CALL FMPRINT_INTERVAL(X)
ENDDO
WRITE (22,*) ' '
WRITE (22,*) ' '
END PROGRAM INTERVAL_EXAMPLES
FUNCTION F(NF, X)
USE FMZM
USE FM_INTERVAL_ARITHMETIC
IMPLICIT NONE
INTEGER :: NF
TYPE (FM_INTERVAL) :: F, X
CALL FM_ENTER_USER_FUNCTION(F)
IF (NF == 1) THEN
F = SIN(X) / X
ENDIF
CALL FM_EXIT_USER_FUNCTION(F)
END FUNCTION F
SUBROUTINE GAUSS_9(NF, A, B, N, RESULT)
! Sample subroutine usage for FM.
! Integrate F(NF, X) from A to B using N subintervals, and return the answer in RESULT.
! This does numerical integration using a 9-point Gauss quadrature rule.
! It is not a very good way to do high-precision integration, but it is a short routine
! and can often get 50 digits if f(x) is well-behaved and the interval of integration
! is not too big.
! Note that subroutines using FM variables need a call to FM_ENTER_USER_ROUTINE upon entry
! to the routine and one to FM_EXIT_USER_ROUTINE upon exit.
! To keep from wasting memory, local variables like XJ should have the SAVE attribute.
USE FMVALS
USE FMZM
USE FM_INTERVAL_ARITHMETIC
IMPLICIT NONE
TYPE (FM_INTERVAL) :: A, B, RESULT
TYPE (FM_INTERVAL), SAVE :: AJ, H2, XJ, XI(9), WI(9)
TYPE (FM_INTERVAL), EXTERNAL :: F
TYPE (FM), SAVE :: WIDTH_FM
INTEGER :: NF, N, J, K
INTEGER, SAVE :: COEFF_BASE = 0, COEFF_PRECISION = 0
INTENT (IN) :: N, A, B
INTENT (INOUT) :: RESULT
CALL FM_ENTER_USER_ROUTINE
IF (COEFF_BASE /= MBASE .OR. COEFF_PRECISION < NDIG) THEN
COEFF_BASE = MBASE
COEFF_PRECISION = NDIG
XI(9) = TO_FM(' 0.968160239507626089835576202903672870049404800491925329550023 ')
XI(1) = -XI(9)
XI(8) = TO_FM(' 0.836031107326635794299429788069734876544106718124675996104372 ')
XI(2) = -XI(8)
XI(7) = TO_FM(' 0.613371432700590397308702039341474184785720604940564692872813 ')
XI(3) = -XI(7)
XI(6) = TO_FM(' 0.324253423403808929038538014643336608571956260736973088827047 ')
XI(4) = -XI(6)
XI(5) = 0
WI(9) = TO_FM(' 0.081274388361574411971892158110523650675661720782410750711108 ')
WI(1) = WI(9)
WI(8) = TO_FM(' 0.180648160694857404058472031242912809514337821732040484498336 ')
WI(2) = WI(8)
WI(7) = TO_FM(' 0.260610696402935462318742869418632849771840204437299951939997 ')
WI(3) = WI(7)
WI(6) = TO_FM(' 0.312347077040002840068630406584443665598754861261904645554011 ')
WI(4) = WI(6)
WI(5) = TO_FM(' 32768 ') / 99225
ENDIF
RESULT = 0
H2 = ( B - A ) / ( 2 * N )
DO J = 1, N
AJ = A + (J-1) * 2 * H2
DO K = 1, 9
XJ = AJ + H2 + H2 * XI(K)
RESULT = RESULT + F( NF, XJ ) * WI(K)
ENDDO
IF (MOD(J,N/1000) == 0) THEN
WIDTH_FM = MAX( RIGHT_ENDPOINT(RESULT)-LEFT_ENDPOINT(RESULT) , &
EPSILON(LEFT_ENDPOINT(RESULT)) )
WRITE (31,"(A,I7,A,F8.3,A)") '{', J, ',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ENDIF
ENDDO
RESULT = RESULT * H2
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE GAUSS_9
MODULE RK_FUNCT
! For a function to return an array as its function value, there must be an explicit interface.
! F is the generic name of the function, F_RK is the specific version for 1-dimensional arrays.
INTERFACE F
MODULE PROCEDURE F_RK
END INTERFACE
CONTAINS
FUNCTION F_RK( X, V, K_FUNCTION )
USE FMZM
USE FM_INTERVAL_ARITHMETIC
IMPLICIT NONE
INTEGER :: K_FUNCTION
TYPE (FM_INTERVAL) :: X, V(5), F_RK(5)
CALL FM_ENTER_USER_FUNCTION(F_RK)
F_RK = 0
IF (K_FUNCTION == 1) THEN
! Function 1. y'' = -y' / 10 - 2 * y / (x+2),
! y(0) = 0, y'(0) = 1
! v' = { y' , y'' } = { u , -u / 10 - 2 y / (x+2) } = F(x,v)
F_RK(1) = V(2)
F_RK(2) = -V(2) / 10 - 2 * V(1) / ( X + 2 )
ELSE IF (K_FUNCTION == 2) THEN
! Function 2. y'' = -y' / 10 - 200 * y / (x+2),
! y(0) = 0, y'(0) = 1
! v' = { y' , y'' } = { u , -u / 10 - 2 y / (x+2) } = F(x,v)
F_RK(1) = V(2)
F_RK(2) = -V(2) / 10 - 200 * V(1) / ( X + 2 )
ELSE
F_RK(1) = X*V(1)
ENDIF
CALL FM_EXIT_USER_FUNCTION(F_RK)
END FUNCTION F_RK
END MODULE RK_FUNCT
SUBROUTINE RK4(N_ORDER, A, B, N_STEPS, V, K_FUNCTION, KPRT, KW)
USE FMZM
USE FM_INTERVAL_ARITHMETIC
USE RK_FUNCT
! N_ORDER is the order of the DE
! A is the initial x-value
! B is the final x-value
! N_STEPS is the number of steps done.
! V contains the initial values at x = a on input, and is returned with the solution
! values at x = b.
! K_FUNCTION is the function number for the RHS.
! KPRT > 0 causes the solution to be printed on unit KW after each KPRT steps.
IMPLICIT NONE
INTEGER :: J, K, K_FUNCTION, KPRT, KW, N_ORDER, N_STEPS
TYPE (FM_INTERVAL) :: A, B, H, X, V(5), K1(5), K2(5), K3(5), K4(5)
TYPE (FM) :: ERROR_FM, WIDTH_FM
SAVE :: H, X, K1, K2, K3, K4
CALL FM_ENTER_USER_ROUTINE
X = A
H = (B-A)/N_STEPS
IF (N_ORDER < 5) V(N_ORDER+1:5) = 0
DO J = 1, N_STEPS
K1 = H * F( X , V , K_FUNCTION )
K2 = H * F( X+H/2 , V+K1/2 , K_FUNCTION )
K3 = H * F( X+H/2 , V+K2/2 , K_FUNCTION )
K4 = H * F( X+H , V+K3 , K_FUNCTION )
X = A + J*H
V = V + ( K1 + 2*K2 + 2*K3 + K4 ) / 6
IF (KPRT > 0) THEN
IF (MOD(J,KPRT) == 0) THEN
ERROR_FM = ABS( (RIGHT_ENDPOINT(V(1))-LEFT_ENDPOINT(V(1))) / RIGHT_ENDPOINT(V(1)) )
K = -NINT(LOG10(ERROR_FM))
WRITE (KW,"(A,F15.10,A,2F20.15,A,I3,A)") ' X = ',TO_DP(X),' V = ', &
TO_DP(V(1:N_ORDER)),' Endpoints of V(1) agree to ', K, ' digits.'
ENDIF
ENDIF
IF (MOD(J,N_STEPS/1000) == 1) THEN
WIDTH_FM = MAX( RIGHT_ENDPOINT(V(1))-LEFT_ENDPOINT(V(1)) , &
EPSILON(LEFT_ENDPOINT(V(1))) )
IF (K_FUNCTION == 1) THEN
WRITE (31,"(A,I7,A,F8.3,A)") '{', J, ',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ELSE
WRITE (31,"(A,I7,A,F8.3,A)") '{', J, ',', TO_DP(LOG(WIDTH_FM)) / LOG(10.0D0), '},'
ENDIF
ENDIF
ENDDO
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE RK4
SUBROUTINE FM_INTERVAL_FACTOR_LU(A,N,DET,KSWAP)
USE FMZM
USE FM_INTERVAL_ARITHMETIC
IMPLICIT NONE
! Gauss elimination to factor the NxN matrix A (LU decomposition).
! The time is proportional to N**3.
! Once this factorization has been done, a linear system A x = b
! with the same coefficient matrix A and Nx1 vector b can be solved
! for x using routine FM_INTERVAL_SOLVE_LU in time proportional to N**2.
! DET is returned as the determinant of A.
! Nonzero DET means a solution can be found.
! DET = 0 is returned if the system is singular.
! KSWAP is a list of row interchanges made by the partial pivoting strategy during the
! elimination phase.
! After returning, the values in matrix A have been replaced by the multipliers
! used during elimination. This is equivalent to factoring the A matrix into
! a lower triangular matrix L times an upper triangular matrix U.
INTEGER :: N
INTEGER :: JCOL, JDIAG, JMAX, JROW, KSWAP(N)
TYPE (FM_INTERVAL) :: A(N,N), DET
TYPE (FM_INTERVAL), SAVE :: AMAX, AMULT, TEMP
CALL FM_ENTER_USER_ROUTINE
DET = 1
KSWAP(1:N) = 1
IF (N <= 0) THEN
DET = 0
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
IF (N == 1) THEN
KSWAP(1) = 1
DET = A(1,1)
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
! Do the elimination phase.
! JDIAG is the current diagonal element below which the elimination proceeds.
DO JDIAG = 1, N-1
! Pivot to put the element with the largest absolute value on the diagonal.
AMAX = ABS(A(JDIAG,JDIAG))
JMAX = JDIAG
DO JROW = JDIAG+1, N
IF (ABS(A(JROW,JDIAG)) > AMAX) THEN
AMAX = ABS(A(JROW,JDIAG))
JMAX = JROW
ENDIF
ENDDO
! If AMAX is zero here then the system is singular.
IF (AMAX == 0.0) THEN
DET = 0
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
! Swap rows JDIAG and JMAX unless they are the same row.
KSWAP(JDIAG) = JMAX
IF (JMAX /= JDIAG) THEN
DET = -DET
DO JCOL = JDIAG, N
TEMP = A(JDIAG,JCOL)
A(JDIAG,JCOL) = A(JMAX,JCOL)
A(JMAX,JCOL) = TEMP
ENDDO
ENDIF
DET = DET * A(JDIAG,JDIAG)
! For JROW = JDIAG+1, ..., N, eliminate A(JROW,JDIAG) by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG / A(JDIAG,JDIAG)
DO JROW = JDIAG+1, N
IF (A(JROW,JDIAG) == 0) CYCLE
AMULT = A(JROW,JDIAG)/A(JDIAG,JDIAG)
! Save the multiplier for use later by FM_INTERVAL_SOLVE_LU.
A(JROW,JDIAG) = AMULT
A(JROW,JDIAG+1:N) = A(JROW,JDIAG+1:N) - AMULT*A(JDIAG,JDIAG+1:N)
ENDDO
ENDDO
DET = DET * A(N,N)
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE FM_INTERVAL_FACTOR_LU
SUBROUTINE FM_INTERVAL_SOLVE_LU(A,N,B,X,KSWAP)
USE FMZM
USE FM_INTERVAL_ARITHMETIC
IMPLICIT NONE
! Solve a linear system A x = b.
! A is the NxN coefficient matrix, after having been factored by FM_INTERVAL_FACTOR_LU.
! B is the Nx1 right-hand-side vector.
! X is returned with the solution of the linear system.
! KSWAP is a list of row interchanges made by the partial pivoting strategy during the
! elimination phase in FM_INTERVAL_FACTOR_LU.
! Time for this call is proportional to N**2.
INTEGER :: N, KSWAP(N)
TYPE (FM_INTERVAL) :: A(N,N), B(N), X(N)
TYPE (FM_INTERVAL), SAVE :: TEMP
INTEGER :: JDIAG, JMAX
CALL FM_ENTER_USER_ROUTINE
IF (N <= 0) THEN
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
IF (N == 1) THEN
X(1) = B(1) / A(1,1)
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
X(1:N) = B(1:N)
! Do the elimination phase operations only on X.
! JDIAG is the current diagonal element below which the elimination proceeds.
DO JDIAG = 1, N-1
! Pivot to put the element with the largest absolute value on the diagonal.
JMAX = KSWAP(JDIAG)
! Swap rows JDIAG and JMAX unless they are the same row.
IF (JMAX /= JDIAG) THEN
TEMP = X(JDIAG)
X(JDIAG) = X(JMAX)
X(JMAX) = TEMP
ENDIF
! For JROW = JDIAG+1, ..., N, eliminate A(JROW,JDIAG) by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG / A(JDIAG,JDIAG)
! After factoring, A(JROW,JDIAG) is the original A(JROW,JDIAG) / A(JDIAG,JDIAG).
X(JDIAG+1:N) = X(JDIAG+1:N) - A(JDIAG+1:N,JDIAG)*X(JDIAG)
ENDDO
! Do the back substitution.
DO JDIAG = N, 1, -1
! Divide row JDIAG by the diagonal element.
X(JDIAG) = X(JDIAG) / A(JDIAG,JDIAG)
! Zero above the diagonal in column JDIAG by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG
! For JROW = 1, ..., JDIAG-1.
IF (JDIAG == 1) EXIT
X(1:JDIAG-1) = X(1:JDIAG-1) - A(1:JDIAG-1,JDIAG)*X(JDIAG)
ENDDO
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE FM_INTERVAL_SOLVE_LU
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