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fennecs/src/elliptic_mod.f90
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Module elliptic
IMPLICIT NONE
Contains
function elliptic_ea ( a )
!*****************************************************************************80
!
!! ELLIPTIC_EA evaluates the complete elliptic integral E(A).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! E(a) = RF ( 0, 1-sin^2(a), 1 ) - 1/3 sin^2(a) RD ( 0, 1-sin^2(a), 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) A, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_EA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) elliptic_ea
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
k = sin ( a * r8_pi / 180.0D+00 )
x = 0.0D+00
y = ( 1.0D+00 - k ) * ( 1.0D+00 + k )
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
- k * k * rd ( x, y, z, errtol, ierr ) / 3.0D+00
elliptic_ea = value
return
end
subroutine elliptic_ea_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_EA_VALUES returns values of the complete elliptic integral E(A).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the second kind.
!
! The function is defined by the formula:
!
! E(A) = integral ( 0 <= T <= PI/2 )
! sqrt ( 1 - sin ( A )^2 * sin ( T )^2 ) dT
!
! In Mathematica, the function can be evaluated by:
!
! EllipticE[(Sin[Pi*a/180])^2]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 19 August 2004
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function, measured
! in degrees.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 19
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
1.570796326794897D+00, &
1.567809073977622D+00, &
1.558887196601596D+00, &
1.544150496914673D+00, &
1.523799205259774D+00, &
1.498114928422116D+00, &
1.467462209339427D+00, &
1.432290969306756D+00, &
1.393140248523812D+00, &
1.350643881047676D+00, &
1.305539094297794D+00, &
1.258679624779997D+00, &
1.211056027568459D+00, &
1.163827964493139D+00, &
1.118377737969864D+00, &
1.076405113076403D+00, &
1.040114395706010D+00, &
1.012663506234396D+00, &
1.000000000000000D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.0D+00, &
5.0D+00, &
10.0D+00, &
15.0D+00, &
20.0D+00, &
25.0D+00, &
30.0D+00, &
35.0D+00, &
40.0D+00, &
45.0D+00, &
50.0D+00, &
55.0D+00, &
60.0D+00, &
65.0D+00, &
70.0D+00, &
75.0D+00, &
80.0D+00, &
85.0D+00, &
90.0D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_ek ( k )
!*****************************************************************************80
!
!! ELLIPTIC_EK evaluates the complete elliptic integral E(K).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! E(k) = RF ( 0, 1-k^2, 1 ) - 1/3 k^2 RD ( 0, 1-k^2, 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) K, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_EK, the function value.
!
implicit none
real ( kind = 8 ) elliptic_ek
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = ( 1.0D+00 - k ) * ( 1.0D+00 + k )
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
- k * k * rd ( x, y, z, errtol, ierr ) / 3.0D+00
elliptic_ek = value
return
end
subroutine elliptic_ek_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_EK_VALUES returns values of the complete elliptic integral E(K).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the second kind.
!
! The function is defined by the formula:
!
! E(K) = integral ( 0 <= T <= PI/2 )
! sqrt ( 1 - K^2 * sin ( T )^2 ) dT
!
! In Mathematica, the function can be evaluated by:
!
! EllipticE[m]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 29 May 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 21
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
1.570796326794897D+00, &
1.550973351780472D+00, &
1.530757636897763D+00, &
1.510121832092819D+00, &
1.489035058095853D+00, &
1.467462209339427D+00, &
1.445363064412665D+00, &
1.422691133490879D+00, &
1.399392138897432D+00, &
1.375401971871116D+00, &
1.350643881047676D+00, &
1.325024497958230D+00, &
1.298428035046913D+00, &
1.270707479650149D+00, &
1.241670567945823D+00, &
1.211056027568459D+00, &
1.178489924327839D+00, &
1.143395791883166D+00, &
1.104774732704073D+00, &
1.060473727766278D+00, &
1.000000000000000D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.0000000000000000D+00, &
0.2236067977499790D+00, &
0.3162277660168379D+00, &
0.3872983346207417D+00, &
0.4472135954999579D+00, &
0.5000000000000000D+00, &
0.5477225575051661D+00, &
0.5916079783099616D+00, &
0.6324555320336759D+00, &
0.6708203932499369D+00, &
0.7071067811865476D+00, &
0.7416198487095663D+00, &
0.7745966692414834D+00, &
0.8062257748298550D+00, &
0.8366600265340756D+00, &
0.8660254037844386D+00, &
0.8944271909999159D+00, &
0.9219544457292888D+00, &
0.9486832980505138D+00, &
0.9746794344808963D+00, &
1.0000000000000000D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_em ( m )
!*****************************************************************************80
!
!! ELLIPTIC_EM evaluates the complete elliptic integral E(M).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! E(m) = RF ( 0, 1-m, 1 ) - 1/3 m RD ( 0, 1-m, 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) M, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_EM, the function value.
!
implicit none
real ( kind = 8 ) elliptic_em
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = 1.0D+00 - m
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
- m * rd ( x, y, z, errtol, ierr ) / 3.0D+00
elliptic_em = value
return
end
subroutine elliptic_em_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_EM_VALUES returns values of the complete elliptic integral E(M).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the second kind.
!
! The function is defined by the formula:
!
! E(M) = integral ( 0 <= T <= PI/2 )
! sqrt ( 1 - M * sin ( T )^2 ) dT
!
! In Mathematica, the function can be evaluated by:
!
! EllipticE[m]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 14 August 2004
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 21
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
1.570796326794897D+00, &
1.550973351780472D+00, &
1.530757636897763D+00, &
1.510121832092819D+00, &
1.489035058095853D+00, &
1.467462209339427D+00, &
1.445363064412665D+00, &
1.422691133490879D+00, &
1.399392138897432D+00, &
1.375401971871116D+00, &
1.350643881047676D+00, &
1.325024497958230D+00, &
1.298428035046913D+00, &
1.270707479650149D+00, &
1.241670567945823D+00, &
1.211056027568459D+00, &
1.178489924327839D+00, &
1.143395791883166D+00, &
1.104774732704073D+00, &
1.060473727766278D+00, &
1.000000000000000D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.00D+00, &
0.05D+00, &
0.10D+00, &
0.15D+00, &
0.20D+00, &
0.25D+00, &
0.30D+00, &
0.35D+00, &
0.40D+00, &
0.45D+00, &
0.50D+00, &
0.55D+00, &
0.60D+00, &
0.65D+00, &
0.70D+00, &
0.75D+00, &
0.80D+00, &
0.85D+00, &
0.90D+00, &
0.95D+00, &
1.00D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_fa ( a )
!*****************************************************************************80
!
!! ELLIPTIC_FA evaluates the complete elliptic integral F(A).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! F(a) = RF ( 0, 1-sin^2(a), 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 29 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) A, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_FA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) elliptic_fa
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = 1.0D+00 - ( sin ( a * r8_pi / 180.0 ) ) ** 2
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
elliptic_fa = value
return
end
subroutine elliptic_fa_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_FA_VALUES returns values of the complete elliptic integral F(A).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic integral
! of the first kind.
!
! The function is defined by the formula:
!
! F(A) = integral ( 0 <= T <= PI/2 )
! dT / sqrt ( 1 - sin ( A )^2 * sin ( T )^2 )
!
! In Mathematica, the function can be evaluated by:
!
! EllipticK[(Sin[a*Pi/180])^2]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 19 August 2004
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function, measured
! in degrees.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 18
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
0.1570796326794897D+01, &
0.1573792130924768D+01, &
0.1582842804338351D+01, &
0.1598142002112540D+01, &
0.1620025899124204D+01, &
0.1648995218478530D+01, &
0.1685750354812596D+01, &
0.1731245175657058D+01, &
0.1786769134885021D+01, &
0.1854074677301372D+01, &
0.1935581096004722D+01, &
0.2034715312185791D+01, &
0.2156515647499643D+01, &
0.2308786798167196D+01, &
0.2504550079001634D+01, &
0.2768063145368768D+01, &
0.3153385251887839D+01, &
0.3831741999784146D+01 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.0D+00, &
5.0D+00, &
10.0D+00, &
15.0D+00, &
20.0D+00, &
25.0D+00, &
30.0D+00, &
35.0D+00, &
40.0D+00, &
45.0D+00, &
50.0D+00, &
55.0D+00, &
60.0D+00, &
65.0D+00, &
70.0D+00, &
75.0D+00, &
80.0D+00, &
85.0D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_fk ( k )
!*****************************************************************************80
!
!! ELLIPTIC_FK evaluates the complete elliptic integral F(K).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! F(k) = RF ( 0, 1-k^2, 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 29 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) K, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_FK, the function value.
!
implicit none
real ( kind = 8 ) elliptic_fk
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = ( 1.0D+00 - k ) * ( 1.0D+00 + k )
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
elliptic_fk = value
return
end
subroutine elliptic_fk_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_FK_VALUES returns values of the complete elliptic integral F(K).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the first kind.
!
! The function is defined by the formula:
!
! F(K) = integral ( 0 <= T <= PI/2 )
! dT / sqrt ( 1 - K^2 * sin ( T )^2 )
!
! In Mathematica, the function can be evaluated by:
!
! EllipticK[k^2]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 10 August 2004
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
1.570796326794897D+00, &
1.591003453790792D+00, &
1.612441348720219D+00, &
1.635256732264580D+00, &
1.659623598610528D+00, &
1.685750354812596D+00, &
1.713889448178791D+00, &
1.744350597225613D+00, &
1.777519371491253D+00, &
1.813883936816983D+00, &
1.854074677301372D+00, &
1.898924910271554D+00, &
1.949567749806026D+00, &
2.007598398424376D+00, &
2.075363135292469D+00, &
2.156515647499643D+00, &
2.257205326820854D+00, &
2.389016486325580D+00, &
2.578092113348173D+00, &
2.908337248444552D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.0000000000000000D+00, &
0.2236067977499790D+00, &
0.3162277660168379D+00, &
0.3872983346207417D+00, &
0.4472135954999579D+00, &
0.5000000000000000D+00, &
0.5477225575051661D+00, &
0.5916079783099616D+00, &
0.6324555320336759D+00, &
0.6708203932499369D+00, &
0.7071067811865476D+00, &
0.7416198487095663D+00, &
0.7745966692414834D+00, &
0.8062257748298550D+00, &
0.8366600265340756D+00, &
0.8660254037844386D+00, &
0.8944271909999159D+00, &
0.9219544457292888D+00, &
0.9486832980505138D+00, &
0.9746794344808963D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_fm ( m )
!*****************************************************************************80
!
!! ELLIPTIC_FM evaluates the complete elliptic integral F(M).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! F(m) = RF ( 0, 1-m, 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 29 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) M, the argument.
!
! Output, real ( kind = 8 ) ELLIPTIC_FM, the function value.
!
implicit none
real ( kind = 8 ) elliptic_fm
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
!real ( kind = 8 ) rf
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = 1.0D+00 - m
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
elliptic_fm = value
return
end
subroutine elliptic_fm_values ( n_data, x, fx )
!*****************************************************************************80
!
!! ELLIPTIC_FM_VALUES returns values of the complete elliptic integral F(M).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the first kind.
!
! The function is defined by the formula:
!
! F(M) = integral ( 0 <= T <= PI/2 )
! dT / sqrt ( 1 - M * sin ( T )^2 )
!
! In Mathematica, the function can be evaluated by:
!
! EllipticK[m]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 10 August 2004
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) X, the argument of the function.
!
! Output, real ( kind = 8 ) FX, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) fx
real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ &
1.570796326794897D+00, &
1.591003453790792D+00, &
1.612441348720219D+00, &
1.635256732264580D+00, &
1.659623598610528D+00, &
1.685750354812596D+00, &
1.713889448178791D+00, &
1.744350597225613D+00, &
1.777519371491253D+00, &
1.813883936816983D+00, &
1.854074677301372D+00, &
1.898924910271554D+00, &
1.949567749806026D+00, &
2.007598398424376D+00, &
2.075363135292469D+00, &
2.156515647499643D+00, &
2.257205326820854D+00, &
2.389016486325580D+00, &
2.578092113348173D+00, &
2.908337248444552D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) x
real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ &
0.00D+00, &
0.05D+00, &
0.10D+00, &
0.15D+00, &
0.20D+00, &
0.25D+00, &
0.30D+00, &
0.35D+00, &
0.40D+00, &
0.45D+00, &
0.50D+00, &
0.55D+00, &
0.60D+00, &
0.65D+00, &
0.70D+00, &
0.75D+00, &
0.80D+00, &
0.85D+00, &
0.90D+00, &
0.95D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
x = 0.0D+00
fx = 0.0D+00
else
x = x_vec(n_data)
fx = fx_vec(n_data)
end if
return
end
function elliptic_inc_ea ( phi, a )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EA evaluates the incomplete elliptic integral E(PHI,A).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! k = sin ( a * pi / 180 )
! E(phi,a) =
! sin ( phi ) RF ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 )
! - 1/3 k^2 sin^3 ( phi ) RD ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, A, the argument.
! 0 <= PHI <= PI/2.
! 0 <= sin^2 ( A * pi / 180 ) * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_EA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_ea
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
k = sin ( a * r8_pi / 180.0D+00 )
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EA - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rd ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EA - Fatal error!'
write ( *, '(a,i2)' ) ' RD returned IERR = ', ierr
stop 1
end if
value = sp * value1 - k ** 2 * sp ** 3 * value2 / 3.0D+00
elliptic_inc_ea = value
return
end
subroutine elliptic_inc_ea_values ( n_data, phi, a, ea )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EA_VALUES: values of the incomplete elliptic integral E(PHI,A).
!
! Discussion:
!
! This is one form of the incomplete elliptic integral of the second kind.
!
! E(PHI,A) = integral ( 0 <= T <= PHI )
! sqrt ( 1 - sin^2 ( A ) * sin^2 ( T ) ) dT
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, A, the arguments of the function.
!
! Output, real ( kind = 8 ) EA, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) a
real ( kind = 8 ) ea
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: a_vec = (/ &
123.0821233267548D+00, &
11.26931745051486D+00, &
-94.88806452075445D+00, &
-99.71407853545323D+00, &
57.05881039324191D+00, &
-19.71363287074183D+00, &
56.31230299738043D+00, &
-91.55605346417718D+00, &
-27.00654574696468D+00, &
-169.2293728595904D+00, &
61.96859564803047D+00, &
-158.7324398933148D+00, &
105.0883958999383D+00, &
-48.95883872360177D+00, &
-42.58568835110901D+00, &
11.65603284687828D+00, &
-8.398113719173338D+00, &
17.69362213019626D+00, &
73.8803420626852D+00, &
-69.82492339645128D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: ea_vec = (/ &
0.3384181367348019D+00, &
1.292924624509506D+00, &
0.6074183768796306D+00, &
0.3939726730783567D+00, &
0.06880814097089803D+00, &
0.0969436473376824D+00, &
0.6025937791452033D+00, &
0.9500549494837583D+00, &
1.342783372140486D+00, &
0.1484915631401388D+00, &
1.085432887050926D+00, &
0.1932136916085597D+00, &
0.3983689593057807D+00, &
0.1780054133336934D+00, &
1.164525270273536D+00, &
1.080167047541845D+00, &
1.346684963830312D+00, &
1.402100272685504D+00, &
0.2928091845544553D+00, &
0.5889342583405707D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
a = 0.0D+00
ea = 0.0D+00
phi = 0.0D+00
else
a = a_vec(n_data)
ea = ea_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_ek ( phi, k )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EK evaluates the incomplete elliptic integral E(PHI,K).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! E(phi,k) =
! sin ( phi ) RF ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 )
! - 1/3 k^2 sin^3 ( phi ) RD ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, K, the argument.
! 0 <= PHI <= PI/2.
! 0 <= K^2 * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_EK, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_ek
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EK - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rd ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EK - Fatal error!'
write ( *, '(a,i2)' ) ' RD returned IERR = ', ierr
stop 1
end if
value = sp * value1 - k ** 2 * sp ** 3 * value2 / 3.0D+00
elliptic_inc_ek = value
return
end
subroutine elliptic_inc_ek_values ( n_data, phi, k, ek )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EK_VALUES: values of the incomplete elliptic integral E(PHI,K).
!
! Discussion:
!
! This is the incomplete elliptic integral of the second kind.
!
! E(PHI,K) = integral ( 0 <= T <= PHI )
! sqrt ( 1 - K^2 * sin ( T )^2 ) dT
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 22 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, K, the arguments.
!
! Output, real ( kind = 8 ) EK, the function value.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) ek
real ( kind = 8 ) k
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: ek_vec = (/ &
0.2852345328295404D+00, &
1.298690225567921D+00, &
0.5508100202571943D+00, &
0.3575401358115371D+00, &
0.06801307805507453D+00, &
0.09679584980231837D+00, &
0.6003112504412838D+00, &
0.8996717721794724D+00, &
1.380715261453875D+00, &
0.1191644625202453D+00, &
1.196994838171557D+00, &
0.1536260979667945D+00, &
0.3546768920544152D+00, &
0.1758756066650882D+00, &
1.229819109410569D+00, &
1.08381066114337D+00, &
1.35023378157378D+00, &
1.419775884709218D+00, &
0.2824895528020034D+00, &
0.5770427720982867D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: k_vec = (/ &
2.712952582080266D+00, &
0.1279518954120547D+00, &
-1.429437513650137D+00, &
-1.981659235625333D+00, &
3.894801879555818D+00, &
-1.042486024983672D+00, &
0.8641142168759754D+00, &
-1.049058412826877D+00, &
-0.3024062128402472D+00, &
-6.574288841527263D+00, &
0.6987397421988888D+00, &
-5.12558591600033D+00, &
2.074947853793764D+00, &
-1.670886158426681D+00, &
-0.4843595000931672D+00, &
0.1393061679635559D+00, &
-0.0946527302537008D+00, &
0.1977207111754007D+00, &
1.788159919089993D+00, &
-1.077780624681256D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
ek = 0.0D+00
k = 0.0D+00
phi = 0.0D+00
else
ek = ek_vec(n_data)
k = k_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_em ( phi, m )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EM evaluates the incomplete elliptic integral E(PHI,M).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! E(phi,m) =
! sin ( phi ) RF ( cos^2 ( phi ), 1-m sin^2 ( phi ), 1 )
! - 1/3 m sin^3 ( phi ) RD ( cos^2 ( phi ), 1-m sin^2 ( phi ), 1 ).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, K, the argument.
! 0 <= PHI <= PI/2.
! 0 <= M * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_EM, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_em
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rd
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = 1.0D+00 - m * sp * sp
z = 1.0D+00
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EM - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rd ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_EM - Fatal error!'
write ( *, '(a,i2)' ) ' RD returned IERR = ', ierr
stop 1
end if
value = sp * value1 - m * sp ** 3 * value2 / 3.0D+00
elliptic_inc_em = value
return
end
subroutine elliptic_inc_em_values ( n_data, phi, m, em )
!*****************************************************************************80
!
!! ELLIPTIC_INC_EM_VALUES: values of the incomplete elliptic integral E(PHI,M).
!
! Discussion:
!
! This is the incomplete elliptic integral of the second kind.
!
! E(PHI,M) = integral ( 0 <= T <= PHI )
! sqrt ( 1 - M * sin ( T )^2 ) dT
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, M, the arguments.
!
! Output, real ( kind = 8 ) EM, the function value.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) em
real ( kind = 8 ) m
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: em_vec = (/ &
0.2732317284159052D+00, &
1.124749725099781D+00, &
0.6446601913679151D+00, &
0.3968902354370061D+00, &
0.06063960799944668D+00, &
0.08909411577948728D+00, &
0.532402014802015D+00, &
1.251888640660265D+00, &
1.28897116191626D+00, &
0.1481718153599732D+00, &
1.038090185639913D+00, &
0.1931275771541276D+00, &
0.3304419611986801D+00, &
0.167394796063963D+00, &
1.214501175324736D+00, &
0.9516560179840655D+00, &
1.203682959526176D+00, &
1.206426326185419D+00, &
0.2522791382096692D+00, &
0.6026499038720986D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
8.450689756874594D+00, &
0.6039878267930615D+00, &
0.1794126658351454D+00, &
0.7095689301026752D+00, &
133.9643389059188D+00, &
47.96621393936416D+00, &
2.172070586163255D+00, &
0.002038130569431913D+00, &
0.3600036705339421D+00, &
0.6219544540067304D+00, &
0.8834215943508453D+00, &
0.2034290670379481D+00, &
5.772526076430922D+00, &
11.14853902343298D+00, &
0.2889238477277305D+00, &
0.7166617182589116D+00, &
0.4760623731559658D+00, &
0.6094948502068943D+00, &
8.902276887883076D+00, &
0.5434439226321253D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
em = 0.0D+00
m = 0.0D+00
phi = 0.0D+00
else
em = em_vec(n_data)
m = m_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_fa ( phi, a )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FA evaluates the incomplete elliptic integral F(PHI,A).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! k = sin ( a * pi / 180 )
! F(phi,k) = sin(phi) * RF ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, A, the argument.
! 0 <= PHI <= PI/2.
! 0 <= sin^2 ( A * pi / 180 ) * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_FA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_fa
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
k = sin ( a * r8_pi / 180.0D+00 )
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_FA - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
elliptic_inc_fa = sp * value
return
end
subroutine elliptic_inc_fa_values ( n_data, phi, a, fa )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FA_VALUES: values of the incomplete elliptic integral F(PHI,A).
!
! Discussion:
!
! This is the incomplete elliptic integral of the first kind.
!
! F(PHI,A) = integral ( 0 <= T <= PHI )
! dT / sqrt ( 1 - sin^2 ( A ) * sin^2 ( T ) )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 22 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, A, the arguments.
!
! Output, real ( kind = 8 ) FA, the function value.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) a
real ( kind = 8 ) fa
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: a_vec = (/ &
123.0821233267548D+00, &
11.26931745051486D+00, &
-94.88806452075445D+00, &
-99.71407853545323D+00, &
57.05881039324191D+00, &
-19.71363287074183D+00, &
56.31230299738043D+00, &
-91.55605346417718D+00, &
-27.00654574696468D+00, &
-169.2293728595904D+00, &
61.96859564803047D+00, &
-158.7324398933148D+00, &
105.0883958999383D+00, &
-48.95883872360177D+00, &
-42.58568835110901D+00, &
11.65603284687828D+00, &
-8.398113719173338D+00, &
17.69362213019626D+00, &
73.8803420626852D+00, &
-69.82492339645128D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: fa_vec = (/ &
0.3478806460316299D+00, &
1.313180577009584D+00, &
0.7037956689264326D+00, &
0.4157626844675118D+00, &
0.06888475483285136D+00, &
0.09697816754845832D+00, &
0.6605394722518515D+00, &
1.82758346036751D+00, &
1.482258783392487D+00, &
0.1485295339221232D+00, &
1.753800062701494D+00, &
0.193528896465351D+00, &
0.4199100508706138D+00, &
0.1790836490491233D+00, &
1.446048832279763D+00, &
1.094097652100984D+00, &
1.358947908427035D+00, &
1.46400078231538D+00, &
0.3009092014525799D+00, &
0.6621341112075102D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
a = 0.0D+00
fa = 0.0D+00
phi = 0.0D+00
else
a = a_vec(n_data)
fa = fa_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_fk ( phi, k )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FK evaluates the incomplete elliptic integral F(PHI,K).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! F(phi,k) = sin(phi) * RF ( cos^2 ( phi ), 1-k^2 sin^2 ( phi ), 1 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, K, the argument.
! 0 <= PHI <= PI/2.
! 0 <= K^2 * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_FK, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_fk
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_FK - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
elliptic_inc_fk = sp * value
return
end
subroutine elliptic_inc_fk_values ( n_data, phi, k, fk )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FK_VALUES: values of the incomplete elliptic integral F(PHI,K).
!
! Discussion:
!
! This is the incomplete elliptic integral of the first kind.
!
! F(PHI,K) = integral ( 0 <= T <= PHI )
! dT / sqrt ( 1 - K^2 * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 22 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, K, the arguments.
!
! Output, real ( kind = 8 ) FK, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) fk
real ( kind = 8 ) k
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: fk_vec = (/ &
0.4340870330108736D+00, &
1.307312511398114D+00, &
0.8005154258533936D+00, &
0.4656721451084328D+00, &
0.06969849613441773D+00, &
0.09712646708750489D+00, &
0.6632598061016007D+00, &
2.2308677858579D+00, &
1.439846282888019D+00, &
0.2043389243773096D+00, &
1.537183574881771D+00, &
0.2749229901565622D+00, &
0.4828388342828284D+00, &
0.1812848567886627D+00, &
1.360729522341841D+00, &
1.09039680912027D+00, &
1.355363051581808D+00, &
1.445462819732441D+00, &
0.3125355489354676D+00, &
0.6775731623807174D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: k_vec = (/ &
2.712952582080266D+00, &
0.1279518954120547D+00, &
-1.429437513650137D+00, &
-1.981659235625333D+00, &
3.894801879555818D+00, &
-1.042486024983672D+00, &
0.8641142168759754D+00, &
-1.049058412826877D+00, &
-0.3024062128402472D+00, &
-6.574288841527263D+00, &
0.6987397421988888D+00, &
-5.12558591600033D+00, &
2.074947853793764D+00, &
-1.670886158426681D+00, &
-0.4843595000931672D+00, &
0.1393061679635559D+00, &
-0.0946527302537008D+00, &
0.1977207111754007D+00, &
1.788159919089993D+00, &
-1.077780624681256D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
fk = 0.0D+00
k = 0.0D+00
phi = 0.0D+00
else
fk = fk_vec(n_data)
k = k_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_fm ( phi, m )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FM evaluates the incomplete elliptic integral F(PHI,M).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! F(phi,m) = sin(phi) * RF ( cos^2 ( phi ), 1-m sin^2 ( phi ), 1 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, M, the argument.
! 0 <= PHI <= PI/2.
! 0 <= M * sin^2(PHI) <= 1.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_FM, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_fm
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = 1.0D+00 - m * sp ** 2
z = 1.0D+00
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_FM - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
elliptic_inc_fm = sp * value
return
end
subroutine elliptic_inc_fm_values ( n_data, phi, m, fm )
!*****************************************************************************80
!
!! ELLIPTIC_INC_FM_VALUES: values of the incomplete elliptic integral F(PHI,M).
!
! Discussion:
!
! This is the incomplete elliptic integral of the first kind.
!
! F(PHI,M) = integral ( 0 <= T <= PHI )
! dT / sqrt ( 1 - M * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 22 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, M, the arguments.
!
! Output, real ( kind = 8 ) FM, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) fm
real ( kind = 8 ) m
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ), save, dimension ( n_max ) :: fm_vec = (/ &
0.4804314075855023D+00, &
1.535634981092025D+00, &
0.6602285297476601D+00, &
0.4125884303785135D+00, &
0.07964566007155376D+00, &
0.1062834070535258D+00, &
0.7733990864393913D+00, &
1.252862499892228D+00, &
1.549988686611532D+00, &
0.1488506735822822D+00, &
1.892229900799662D+00, &
0.1936153327753556D+00, &
0.5481932935424454D+00, &
0.1911795073571756D+00, &
1.379225069349756D+00, &
1.261282453331402D+00, &
1.535239838525378D+00, &
1.739782418156071D+00, &
0.3616930047198503D+00, &
0.6458627645916422D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
8.450689756874594D+00, &
0.6039878267930615D+00, &
0.1794126658351454D+00, &
0.7095689301026752D+00, &
133.9643389059188D+00, &
47.96621393936416D+00, &
2.172070586163255D+00, &
0.002038130569431913D+00, &
0.3600036705339421D+00, &
0.6219544540067304D+00, &
0.8834215943508453D+00, &
0.2034290670379481D+00, &
5.772526076430922D+00, &
11.14853902343298D+00, &
0.2889238477277305D+00, &
0.7166617182589116D+00, &
0.4760623731559658D+00, &
0.6094948502068943D+00, &
8.902276887883076D+00, &
0.5434439226321253D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
1.302990057703935D+00, &
0.6523628380743488D+00, &
0.4046022501376546D+00, &
0.06884642871852312D+00, &
0.0969609046794745D+00, &
0.630370432896175D+00, &
1.252375418911598D+00, &
1.409796082144801D+00, &
0.1485105463502483D+00, &
1.349466184634646D+00, &
0.1933711786970301D+00, &
0.4088829927466769D+00, &
0.1785430666405224D+00, &
1.292588374416351D+00, &
1.087095515757691D+00, &
1.352794600489329D+00, &
1.432530166308616D+00, &
0.2968093345769761D+00, &
0.6235880396594726D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
fm = 0.0D+00
m = 0.0D+00
phi = 0.0D+00
else
fm = fm_vec(n_data)
m = m_vec(n_data)
phi = phi_vec(n_data)
end if
return
end
function elliptic_inc_pia ( phi, n, a )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIA evaluates the incomplete elliptic integral Pi(PHI,N,A).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! Pi(PHI,N,A) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - sin^2(A*pi/180) * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, N, A, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_PIA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_pia
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) n
real ( kind = 8 ) p
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
k = sin ( a * r8_pi / 180.0D+00 )
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
p = 1.0D+00 - n * sp ** 2
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIA - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rj ( x, y, z, p, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIA - Fatal error!'
write ( *, '(a,i2)' ) ' RJ returned IERR = ', ierr
stop 1
end if
value = sp * value1 + n * sp ** 3 * value2 / 3.0D+00
elliptic_inc_pia = value
return
end
subroutine elliptic_inc_pia_values ( n_data, phi, n, a, pia )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIA_VALUES: values of incomplete elliptic integral Pi(PHI,N,A).
!
! Discussion:
!
! This is the incomplete elliptic integral of the third kind.
!
! Pi(PHI,N,A) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - sin^2(A) * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 22 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, N, A, the arguments of the function.
!
! Output, real ( kind = 8 ) PIA, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) a
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ) pia
real ( kind = 8 ), save, dimension ( n_max ) :: a_vec = (/ &
88.87822485052908D+00, &
-86.55208740039521D+00, &
-116.6195703112117D+00, &
-9.742878017582015D+00, &
65.73480919446207D+00, &
-115.0387719677141D+00, &
124.9421177735846D+00, &
-89.78704401263703D+00, &
-98.42673771271734D+00, &
-53.74936192418378D+00, &
68.28047574440727D+00, &
20.82174673810708D+00, &
-29.1042364797769D+00, &
-37.80176710944693D+00, &
-55.81173355852393D+00, &
-37.66594589748672D+00, &
-80.09408170610219D+00, &
52.23806528467412D+00, &
74.30945212430545D+00, &
-17.22920703094039D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
8.064681366127422D+00, &
-0.2840588974558835D+00, &
-5.034023488967104D+00, &
-1.244606253942751D+00, &
1.465981775919188D+00, &
95338.12857321106D+00, &
-44.43130633436311D+00, &
-0.8029374966926196D+00, &
5.218883222649502D+00, &
2.345821782626782D+00, &
0.157358332363011D+00, &
1.926593468907062D+00, &
6.113982855261652D+00, &
1.805710621498681D+00, &
-0.4072847419780592D+00, &
-0.9416404038595624D+00, &
0.7009655305226739D+00, &
-1.019830985340273D+00, &
-0.4510798219577842D+00, &
0.6028821390092596D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
0.8823091382756705D+00, &
0.4046022501376546D+00, &
0.9958310121985398D+00, &
0.630370432896175D+00, &
0.002887706662908567D+00, &
0.1485105463502483D+00, &
1.320800086884777D+00, &
0.4088829927466769D+00, &
0.552337007372852D+00, &
1.087095515757691D+00, &
0.7128175949111615D+00, &
0.2968093345769761D+00, &
0.2910907344062498D+00, &
0.9695030752034163D+00, &
1.122288759723523D+00, &
1.295911610809573D+00, &
1.116491437736542D+00, &
1.170719322533712D+00, &
1.199360682338851D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: pia_vec = (/ &
0.7099335174334724D+00, &
0.9601963779142505D+00, &
0.3362852532098376D+00, &
0.7785343427543768D+00, &
0.857889755214478D+00, &
0.004630772344931844D+00, &
0.1173842687902911D+00, &
1.505788070660267D+00, &
0.7213264194624553D+00, &
0.8073261799642218D+00, &
1.402853811110838D+00, &
1.259245331474513D+00, &
0.3779079263971614D+00, &
0.3088493910496766D+00, &
0.9782829177005183D+00, &
0.9430491574504173D+00, &
3.320796277384155D+00, &
0.9730988737054799D+00, &
1.301988094953789D+00, &
1.64558360445259D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
a = 0.0D+00
n = 0.0D+00
phi = 0.0D+00
pia = 0.0D+00
else
a = a_vec(n_data)
n = n_vec(n_data)
phi = phi_vec(n_data)
pia = pia_vec(n_data)
end if
return
end
function elliptic_inc_pik ( phi, n, k )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIK evaluates the incomplete elliptic integral Pi(PHI,N,K).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! Pi(PHI,N,K) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - k^2 * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, N, K, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_PIK, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_pik
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) n
real ( kind = 8 ) p
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = ( 1.0D+00 - k * sp ) * ( 1.0D+00 + k * sp )
z = 1.0D+00
p = 1.0D+00 - n * sp ** 2
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIK - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rj ( x, y, z, p, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIK - Fatal error!'
write ( *, '(a,i2)' ) ' RJ returned IERR = ', ierr
stop 1
end if
value = sp * value1 + n * sp ** 3 * value2 / 3.0D+00
elliptic_inc_pik = value
return
end
subroutine elliptic_inc_pik_values ( n_data, phi, n, k, pik )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIK_VALUES: values of incomplete elliptic integral Pi(PHI,N,K).
!
! Discussion:
!
! This is the incomplete elliptic integral of the third kind.
!
! Pi(PHI,N,K) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - K^2 * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 23 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, N, K, the arguments of the function.
!
! Output, real ( kind = 8 ) PIK, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) k
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ) pik
real ( kind = 8 ), save, dimension ( n_max ) :: k_vec = (/ &
1.959036804709882D+00, &
-1.123741823223131D+00, &
-2.317629084640271D+00, &
-0.1202582658444815D+00, &
1.008702896970963D+00, &
-103.3677494756118D+00, &
4.853800240677973D+00, &
-1.016577251056124D+00, &
-1.94341484065839D+00, &
-0.8876593284500023D+00, &
0.8160487832898813D+00, &
0.2994546721661018D+00, &
-0.7044232294525243D+00, &
-0.9266523277404759D+00, &
-0.6962608926846425D+00, &
-0.4453932031991797D+00, &
-0.9104582513322106D+00, &
0.6187501419936026D+00, &
0.8672305032589989D+00, &
-0.1996772638241632D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
8.064681366127422D+00, &
-0.2840588974558835D+00, &
-5.034023488967104D+00, &
-1.244606253942751D+00, &
1.465981775919188D+00, &
95338.12857321106D+00, &
-44.43130633436311D+00, &
-0.8029374966926196D+00, &
5.218883222649502D+00, &
2.345821782626782D+00, &
0.157358332363011D+00, &
1.926593468907062D+00, &
6.113982855261652D+00, &
1.805710621498681D+00, &
-0.4072847419780592D+00, &
-0.9416404038595624D+00, &
0.7009655305226739D+00, &
-1.019830985340273D+00, &
-0.4510798219577842D+00, &
0.6028821390092596D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
0.8823091382756705D+00, &
0.4046022501376546D+00, &
0.9958310121985398D+00, &
0.630370432896175D+00, &
0.002887706662908567D+00, &
0.1485105463502483D+00, &
1.320800086884777D+00, &
0.4088829927466769D+00, &
0.552337007372852D+00, &
1.087095515757691D+00, &
0.7128175949111615D+00, &
0.2968093345769761D+00, &
0.2910907344062498D+00, &
0.9695030752034163D+00, &
1.122288759723523D+00, &
1.295911610809573D+00, &
1.116491437736542D+00, &
1.170719322533712D+00, &
1.199360682338851D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: pik_vec = (/ &
0.7982975462595892D+00, &
1.024022134726036D+00, &
0.40158120852642D+00, &
0.7772649487439858D+00, &
0.8737159913132074D+00, &
0.004733334297691273D+00, &
0.1280656893638068D+00, &
1.594376037512564D+00, &
0.8521145133671923D+00, &
0.8154325229803082D+00, &
1.31594514075427D+00, &
1.25394623148424D+00, &
0.3796503567258643D+00, &
0.3111034454739552D+00, &
0.9442477901112342D+00, &
0.9153111661980959D+00, &
2.842080644328393D+00, &
0.9263253777034376D+00, &
1.212396018757624D+00, &
1.628083572710471D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
k = 0.0D+00
n = 0.0D+00
phi = 0.0D+00
pik = 0.0D+00
else
k = k_vec(n_data)
n = n_vec(n_data)
phi = phi_vec(n_data)
pik = pik_vec(n_data)
end if
return
end
function elliptic_inc_pim ( phi, n, m )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIM evaluates the incomplete elliptic integral Pi(PHI,N,M).
!
! Discussion:
!
! The value is computed using Carlson elliptic integrals:
!
! Pi(PHI,N,M) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - m * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) PHI, N, M, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_INC_PIM, the function value.
!
implicit none
real ( kind = 8 ) cp
real ( kind = 8 ) elliptic_inc_pim
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
real ( kind = 8 ) n
real ( kind = 8 ) p
real ( kind = 8 ) phi
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) sp
real ( kind = 8 ) value
real ( kind = 8 ) value1
real ( kind = 8 ) value2
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
cp = cos ( phi )
sp = sin ( phi )
x = cp * cp
y = 1.0D+00 - m * sp ** 2
z = 1.0D+00
p = 1.0D+00 - n * sp ** 2
errtol = 1.0D-03
value1 = rf ( x, y, z, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIM - Fatal error!'
write ( *, '(a,i2)' ) ' RF returned IERR = ', ierr
stop 1
end if
value2 = rj ( x, y, z, p, errtol, ierr )
if ( ierr /= 0 ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'ELLIPTIC_INC_PIM - Fatal error!'
write ( *, '(a,i2)' ) ' RJ returned IERR = ', ierr
stop 1
end if
value = sp * value1 + n * sp ** 3 * value2 / 3.0D+00
elliptic_inc_pim = value
return
end
subroutine elliptic_inc_pim_values ( n_data, phi, n, m, pim )
!*****************************************************************************80
!
!! ELLIPTIC_INC_PIM_VALUES: values of incomplete elliptic integral Pi(PHI,N,M).
!
! Discussion:
!
! This is the incomplete elliptic integral of the third kind.
!
! Pi(PHI,N,M) = integral ( 0 <= T <= PHI )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - M * sin ( T )^2 )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! US Department of Commerce, 1964.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Wolfram Media / Cambridge University Press, 1999.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) PHI, N, M, the arguments of the function.
!
! Output, real ( kind = 8 ) PIM, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) m
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ) phi
real ( kind = 8 ) pim
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
7.330122710928245D+00, &
0.1108806690614566D+00, &
0.2828355944410993D+00, &
0.6382999794812498D+00, &
2.294718938593894D+00, &
42062.55329826538D+00, &
39.2394337789563D+00, &
0.008002151065098688D+00, &
0.7190579590867517D+00, &
0.9703767630929055D+00, &
1.098881295982823D+00, &
1.398066725917478D+00, &
4.641021931654496D+00, &
4.455969064311461D+00, &
0.3131448239736511D+00, &
0.3686443684703166D+00, &
0.06678210908100803D+00, &
0.9635538974026796D+00, &
1.060208762696207D+00, &
0.4687160847955397D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
8.064681366127422D+00, &
-0.2840588974558835D+00, &
-5.034023488967104D+00, &
-1.244606253942751D+00, &
1.465981775919188D+00, &
95338.12857321106D+00, &
-44.43130633436311D+00, &
-0.8029374966926196D+00, &
5.218883222649502D+00, &
2.345821782626782D+00, &
0.157358332363011D+00, &
1.926593468907062D+00, &
6.113982855261652D+00, &
1.805710621498681D+00, &
-0.4072847419780592D+00, &
-0.9416404038595624D+00, &
0.7009655305226739D+00, &
-1.019830985340273D+00, &
-0.4510798219577842D+00, &
0.6028821390092596D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: phi_vec = (/ &
0.3430906586047127D+00, &
0.8823091382756705D+00, &
0.4046022501376546D+00, &
0.9958310121985398D+00, &
0.630370432896175D+00, &
0.002887706662908567D+00, &
0.1485105463502483D+00, &
1.320800086884777D+00, &
0.4088829927466769D+00, &
0.552337007372852D+00, &
1.087095515757691D+00, &
0.7128175949111615D+00, &
0.2968093345769761D+00, &
0.2910907344062498D+00, &
0.9695030752034163D+00, &
1.122288759723523D+00, &
1.295911610809573D+00, &
1.116491437736542D+00, &
1.170719322533712D+00, &
1.199360682338851D+00 /)
real ( kind = 8 ), save, dimension ( n_max ) :: pim_vec = (/ &
1.0469349800785D+00, &
0.842114448140669D+00, &
0.3321642201520043D+00, &
0.8483033529960849D+00, &
1.055753817656772D+00, &
0.005108896144265593D+00, &
0.1426848042785896D+00, &
1.031350958206424D+00, &
0.7131013701418496D+00, &
0.8268044665355507D+00, &
1.57632867896015D+00, &
1.542817120857211D+00, &
0.4144629799126912D+00, &
0.3313231611366746D+00, &
0.9195822851915201D+00, &
0.9422320754002217D+00, &
2.036599002815859D+00, &
1.076799231499882D+00, &
1.416084462957852D+00, &
1.824124922310891D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
m = 0.0D+00
n = 0.0D+00
phi = 0.0D+00
pim = 0.0D+00
else
m = m_vec(n_data)
n = n_vec(n_data)
phi = phi_vec(n_data)
pim = pim_vec(n_data)
end if
return
end
function elliptic_pia ( n, a )
!*****************************************************************************80
!
!! ELLIPTIC_PIA evaluates the complete elliptic integral Pi(N,A).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,A) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - sin^2(A) * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,(sin(a*pi/180)^2)
!
! The value is computed using Carlson elliptic integrals:
!
! k = sin ( a * pi / 180 )
! Pi(n,k) = RF ( 0, 1 - k^2, 1 ) + 1/3 n RJ ( 0, 1 - k^2, 1, 1 - n )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) N, A, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_PIA, the function value.
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) elliptic_pia
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) n
real ( kind = 8 ) p
real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
k = sin ( a * r8_pi / 180.0D+00 )
x = 0.0D+00
y = ( 1.0D+00 - k ) * ( 1.0D+00 + k )
z = 1.0D+00
p = 1.0D+00 - n
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
+ n * rj ( x, y, z, p, errtol, ierr ) / 3.0D+00
elliptic_pia = value
return
end
subroutine elliptic_pia_values ( n_data, n, a, pia )
!*****************************************************************************80
!
!! ELLIPTIC_PIA_VALUES returns values of the complete elliptic integral Pi(N,A).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,A) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - sin^2(A) * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,(sin(A*pi/180))^2)
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) N, A, the arguments of the function.
!
! Output, real ( kind = 8 ) PIA, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) a
real ( kind = 8 ), save, dimension ( n_max ) :: a_vec = (/ &
30.00000000000000D+00, &
45.00000000000000D+00, &
60.00000000000000D+00, &
77.07903361841643D+00, &
30.00000000000000D+00, &
45.00000000000000D+00, &
60.00000000000000D+00, &
77.07903361841643D+00, &
30.00000000000000D+00, &
45.00000000000000D+00, &
60.00000000000000D+00, &
77.07903361841643D+00, &
30.00000000000000D+00, &
45.00000000000000D+00, &
60.00000000000000D+00, &
77.07903361841643D+00, &
30.00000000000000D+00, &
45.00000000000000D+00, &
60.00000000000000D+00, &
77.07903361841643D+00 /)
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00 /)
real ( kind = 8 ) pia
real ( kind = 8 ), save, dimension ( n_max ) :: pia_vec = (/ &
0.4892245275965397D+00, &
0.5106765677902629D+00, &
0.5460409271920561D+00, &
0.6237325893535237D+00, &
0.823045542660675D+00, &
0.8760028274011437D+00, &
0.9660073560143946D+00, &
1.171952391481798D+00, &
1.177446843000566D+00, &
1.273127366749682D+00, &
1.440034318657551D+00, &
1.836472172302591D+00, &
1.685750354812596D+00, &
1.854074677301372D+00, &
2.156515647499643D+00, &
2.908337248444552D+00, &
2.413671504201195D+00, &
2.701287762095351D+00, &
3.234773471249465D+00, &
4.633308147279891D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
a = 0.0D+00
n = 0.0D+00
pia = 0.0D+00
else
a = a_vec(n_data)
n = n_vec(n_data)
pia = pia_vec(n_data)
end if
return
end
function elliptic_pik ( n, k )
!*****************************************************************************80
!
!! ELLIPTIC_PIK evaluates the complete elliptic integral Pi(N,K).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,K) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - K^2 * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,k^2)
!
! The value is computed using Carlson elliptic integrals:
!
! Pi(n,k) = RF ( 0, 1 - k^2, 1 ) + 1/3 n RJ ( 0, 1 - k^2, 1, 1 - n )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) N, K, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_PIK, the function value.
!
implicit none
real ( kind = 8 ) elliptic_pik
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) k
real ( kind = 8 ) n
real ( kind = 8 ) p
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = ( 1.0D+00 - k ) * ( 1.0D+00 + k )
z = 1.0D+00
p = 1.0D+00 - n
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
+ n * rj ( x, y, z, p, errtol, ierr ) / 3.0D+00
elliptic_pik = value
return
end
subroutine elliptic_pik_values ( n_data, n, k, pik )
!*****************************************************************************80
!
!! ELLIPTIC_PIK_VALUES returns values of the complete elliptic integral Pi(N,K).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,K) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - K^2 * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,k^2)
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) N, K, the arguments of the function.
!
! Output, real ( kind = 8 ) PIK, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) k
real ( kind = 8 ), save, dimension ( n_max ) :: k_vec = (/ &
0.5000000000000000D+00, &
0.7071067811865476D+00, &
0.8660254037844386D+00, &
0.9746794344808963D+00, &
0.5000000000000000D+00, &
0.7071067811865476D+00, &
0.8660254037844386D+00, &
0.9746794344808963D+00, &
0.5000000000000000D+00, &
0.7071067811865476D+00, &
0.8660254037844386D+00, &
0.9746794344808963D+00, &
0.5000000000000000D+00, &
0.7071067811865476D+00, &
0.8660254037844386D+00, &
0.9746794344808963D+00, &
0.5000000000000000D+00, &
0.7071067811865476D+00, &
0.8660254037844386D+00, &
0.9746794344808963D+00 /)
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00 /)
real ( kind = 8 ) pik
real ( kind = 8 ), save, dimension ( n_max ) :: pik_vec = (/ &
0.4892245275965397D+00, &
0.5106765677902629D+00, &
0.5460409271920561D+00, &
0.6237325893535237D+00, &
0.823045542660675D+00, &
0.8760028274011437D+00, &
0.9660073560143946D+00, &
1.171952391481798D+00, &
1.177446843000566D+00, &
1.273127366749682D+00, &
1.440034318657551D+00, &
1.836472172302591D+00, &
1.685750354812596D+00, &
1.854074677301372D+00, &
2.156515647499643D+00, &
2.908337248444552D+00, &
2.413671504201195D+00, &
2.701287762095351D+00, &
3.234773471249465D+00, &
4.633308147279891D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
k = 0.0D+00
n = 0.0D+00
pik = 0.0D+00
else
k = k_vec(n_data)
n = n_vec(n_data)
pik = pik_vec(n_data)
end if
return
end
function elliptic_pim ( n, m )
!*****************************************************************************80
!
!! ELLIPTIC_PIM evaluates the complete elliptic integral Pi(N,M).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,M) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - M * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,m)
!
! The value is computed using Carlson elliptic integrals:
!
! Pi(n,m) = RF ( 0, 1 - m, 1 ) + 1/3 n RJ ( 0, 1 - m, 1, 1 - n )
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! Input, real ( kind = 8 ) N, M, the arguments.
!
! Output, real ( kind = 8 ) ELLIPTIC_PIM, the function value.
!
implicit none
real ( kind = 8 ) elliptic_pim
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) m
real ( kind = 8 ) n
real ( kind = 8 ) p
!real ( kind = 8 ) rf
!real ( kind = 8 ) rj
real ( kind = 8 ) value
real ( kind = 8 ) x
real ( kind = 8 ) y
real ( kind = 8 ) z
x = 0.0D+00
y = 1.0D+00 - m
z = 1.0D+00
p = 1.0D+00 - n
errtol = 1.0D-03
value = rf ( x, y, z, errtol, ierr ) &
+ n * rj ( x, y, z, p, errtol, ierr ) / 3.0D+00
elliptic_pim = value
return
end
subroutine elliptic_pim_values ( n_data, n, m, pim )
!*****************************************************************************80
!
!! ELLIPTIC_PIM_VALUES returns values of the complete elliptic integral Pi(N,M).
!
! Discussion:
!
! This is one form of what is sometimes called the complete elliptic
! integral of the third kind.
!
! The function is defined by the formula:
!
! Pi(N,M) = integral ( 0 <= T <= PI/2 )
! dT / (1 - N sin^2(T) ) sqrt ( 1 - M * sin ( T )^2 )
!
! In MATLAB, the function can be evaluated by:
!
! ellipticPi(n,m)
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) N, M, the arguments of the function.
!
! Output, real ( kind = 8 ) PIM, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) m
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
0.25D+00, &
0.50D+00, &
0.75D+00, &
0.95D+00, &
0.25D+00, &
0.50D+00, &
0.75D+00, &
0.95D+00, &
0.25D+00, &
0.50D+00, &
0.75D+00, &
0.95D+00, &
0.25D+00, &
0.50D+00, &
0.75D+00, &
0.95D+00, &
0.25D+00, &
0.50D+00, &
0.75D+00, &
0.95D+00 /)
real ( kind = 8 ) n
integer ( kind = 4 ) n_data
real ( kind = 8 ), save, dimension ( n_max ) :: n_vec = (/ &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-10.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-3.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
-1.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00 /)
real ( kind = 8 ) pim
real ( kind = 8 ), save, dimension ( n_max ) :: pim_vec = (/ &
0.4892245275965397D+00, &
0.5106765677902629D+00, &
0.5460409271920561D+00, &
0.6237325893535237D+00, &
0.823045542660675D+00, &
0.8760028274011437D+00, &
0.9660073560143946D+00, &
1.171952391481798D+00, &
1.177446843000566D+00, &
1.273127366749682D+00, &
1.440034318657551D+00, &
1.836472172302591D+00, &
1.685750354812596D+00, &
1.854074677301372D+00, &
2.156515647499643D+00, &
2.908337248444552D+00, &
2.413671504201195D+00, &
2.701287762095351D+00, &
3.234773471249465D+00, &
4.633308147279891D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
m = 0.0D+00
n = 0.0D+00
pim = 0.0D+00
else
m = m_vec(n_data)
n = n_vec(n_data)
pim = pim_vec(n_data)
end if
return
end
function jacobi_cn ( u, m )
!*****************************************************************************80
!
!! JACOBI_CN evaluates the Jacobi elliptic function CN(U,M).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! Original ALGOL version by Roland Bulirsch.
! FORTRAN90 version by John Burkardt
!
! Reference:
!
! Roland Bulirsch,
! Numerical calculation of elliptic integrals and elliptic functions,
! Numerische Mathematik,
! Volume 7, Number 1, 1965, pages 78-90.
!
! Parameters:
!
! Input, real ( kind = 8 ) U, M, the arguments.
!
! Output, real ( kind = 8 ) JACOBI_CN, the function value.
!
implicit none
real ( kind = 8 ) cn
real ( kind = 8 ) dn
real ( kind = 8 ) jacobi_cn
real ( kind = 8 ) m
real ( kind = 8 ) sn
real ( kind = 8 ) u
call sncndn ( u, m, sn, cn, dn )
jacobi_cn = cn
return
end
subroutine jacobi_cn_values ( n_data, u, m, cn )
!*****************************************************************************80
!
!! JACOBI_CN_VALUES returns some values of the Jacobi elliptic function CN(U,M).
!
! Discussion:
!
! In Mathematica, the function can be evaluated by:
!
! JacobiCN[ u, m ]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) U, the argument of the function.
!
! Output, real ( kind = 8 ) M, the parameter of the function.
!
! Output, real ( kind = 8 ) CN, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) m
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00 /)
real ( kind = 8 ) cn
real ( kind = 8 ), save, dimension ( n_max ) :: cn_vec = (/ &
0.9950041652780258D+00, &
0.9800665778412416D+00, &
0.8775825618903727D+00, &
0.5403023058681397D+00, &
-0.4161468365471424D+00, &
0.9950124626090582D+00, &
0.9801976276784098D+00, &
0.8822663948904403D+00, &
0.5959765676721407D+00, &
-0.1031836155277618D+00, &
0.9950207489532265D+00, &
0.9803279976447253D+00, &
0.8868188839700739D+00, &
0.6480542736638854D+00, &
0.2658022288340797D+00, &
0.3661899347368653D-01, &
0.9803279976447253D+00, &
0.8868188839700739D+00, &
0.6480542736638854D+00, &
0.2658022288340797D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) u
real ( kind = 8 ), save, dimension ( n_max ) :: u_vec = (/ &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
4.0D+00, &
-0.2D+00, &
-0.5D+00, &
-1.0D+00, &
-2.0D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
m = 0.0D+00
u = 0.0D+00
cn = 0.0D+00
else
m = m_vec(n_data)
u = u_vec(n_data)
cn = cn_vec(n_data)
end if
return
end
function jacobi_dn ( u, m )
!*****************************************************************************80
!
!! JACOBI_DN evaluates the Jacobi elliptic function DN(U,M).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! Original ALGOL version by Roland Bulirsch.
! FORTRAN90 version by John Burkardt
!
! Reference:
!
! Roland Bulirsch,
! Numerical calculation of elliptic integrals and elliptic functions,
! Numerische Mathematik,
! Volume 7, Number 1, 1965, pages 78-90.
!
! Parameters:
!
! Input, real ( kind = 8 ) U, M, the arguments.
!
! Output, real ( kind = 8 ) JACOBI_DN, the function value.
!
implicit none
real ( kind = 8 ) cn
real ( kind = 8 ) dn
real ( kind = 8 ) jacobi_dn
real ( kind = 8 ) m
real ( kind = 8 ) sn
real ( kind = 8 ) u
call sncndn ( u, m, sn, cn, dn )
jacobi_dn = dn
return
end
subroutine jacobi_dn_values ( n_data, u, m, dn )
!*****************************************************************************80
!
!! JACOBI_DN_VALUES returns some values of the Jacobi elliptic function DN(U,M).
!
! Discussion:
!
! In Mathematica, the function can be evaluated by:
!
! JacobiDN[ u, m ]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) U, the argument of the function.
!
! Output, real ( kind = 8 ) M, the parameter of the function.
!
! Output, real ( kind = 8 ) DN, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) m
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00 /)
real ( kind = 8 ) dn
real ( kind = 8 ), save, dimension ( n_max ) :: dn_vec = (/ &
0.1000000000000000D+01, &
0.1000000000000000D+01, &
0.1000000000000000D+01, &
0.1000000000000000D+01, &
0.1000000000000000D+01, &
0.9975093485144243D+00, &
0.9901483195224800D+00, &
0.9429724257773857D+00, &
0.8231610016315963D+00, &
0.7108610477840873D+00, &
0.9950207489532265D+00, &
0.9803279976447253D+00, &
0.8868188839700739D+00, &
0.6480542736638854D+00, &
0.2658022288340797D+00, &
0.3661899347368653D-01, &
0.9803279976447253D+00, &
0.8868188839700739D+00, &
0.6480542736638854D+00, &
0.2658022288340797D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) u
real ( kind = 8 ), save, dimension ( n_max ) :: u_vec = (/ &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
4.0D+00, &
-0.2D+00, &
-0.5D+00, &
-1.0D+00, &
-2.0D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
m = 0.0D+00
u = 0.0D+00
dn = 0.0D+00
else
m = m_vec(n_data)
u = u_vec(n_data)
dn = dn_vec(n_data)
end if
return
end
function jacobi_sn ( u, m )
!*****************************************************************************80
!
!! JACOBI_SN evaluates the Jacobi elliptic function SN(U,M).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! Original ALGOL version by Roland Bulirsch.
! FORTRAN90 version by John Burkardt
!
! Reference:
!
! Roland Bulirsch,
! Numerical calculation of elliptic integrals and elliptic functions,
! Numerische Mathematik,
! Volume 7, Number 1, 1965, pages 78-90.
!
! Parameters:
!
! Input, real ( kind = 8 ) U, M, the arguments.
!
! Output, real ( kind = 8 ) JACOBI_SN, the function value.
!
implicit none
real ( kind = 8 ) cn
real ( kind = 8 ) dn
real ( kind = 8 ) jacobi_sn
real ( kind = 8 ) m
real ( kind = 8 ) sn
real ( kind = 8 ) u
call sncndn ( u, m, sn, cn, dn )
jacobi_sn = sn
return
end
subroutine jacobi_sn_values ( n_data, u, m, sn )
!*****************************************************************************80
!
!! JACOBI_SN_VALUES returns some values of the Jacobi elliptic function SN(U,M).
!
! Discussion:
!
! In Mathematica, the function can be evaluated by:
!
! JacobiSN[ u, m ]
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 25 June 2018
!
! Author:
!
! John Burkardt
!
! Reference:
!
! Milton Abramowitz, Irene Stegun,
! Handbook of Mathematical Functions,
! National Bureau of Standards, 1964,
! ISBN: 0-486-61272-4,
! LC: QA47.A34.
!
! Stephen Wolfram,
! The Mathematica Book,
! Fourth Edition,
! Cambridge University Press, 1999,
! ISBN: 0-521-64314-7,
! LC: QA76.95.W65.
!
! Parameters:
!
! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0
! before the first call. On each call, the routine increments N_DATA by 1,
! and returns the corresponding data; when there is no more data, the
! output value of N_DATA will be 0 again.
!
! Output, real ( kind = 8 ) U, the argument of the function.
!
! Output, real ( kind = 8 ) M, the parameter of the function.
!
! Output, real ( kind = 8 ) SN, the value of the function.
!
implicit none
integer ( kind = 4 ), parameter :: n_max = 20
real ( kind = 8 ) m
real ( kind = 8 ), save, dimension ( n_max ) :: m_vec = (/ &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.0D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
0.5D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00, &
1.0D+00 /)
real ( kind = 8 ) sn
real ( kind = 8 ), save, dimension ( n_max ) :: sn_vec = (/ &
0.9983341664682815D-01, &
0.1986693307950612D+00, &
0.4794255386042030D+00, &
0.8414709848078965D+00, &
0.9092974268256817D+00, &
0.9975068547462484D-01, &
0.1980217429819704D+00, &
0.4707504736556573D+00, &
0.8030018248956439D+00, &
0.9946623253580177D+00, &
0.9966799462495582D-01, &
0.1973753202249040D+00, &
0.4621171572600098D+00, &
0.7615941559557649D+00, &
0.9640275800758169D+00, &
0.9993292997390670D+00, &
-0.1973753202249040D+00, &
-0.4621171572600098D+00, &
-0.7615941559557649D+00, &
-0.9640275800758169D+00 /)
integer ( kind = 4 ) n_data
real ( kind = 8 ) u
real ( kind = 8 ), save, dimension ( n_max ) :: u_vec = (/ &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
0.1D+00, &
0.2D+00, &
0.5D+00, &
1.0D+00, &
2.0D+00, &
4.0D+00, &
-0.2D+00, &
-0.5D+00, &
-1.0D+00, &
-2.0D+00 /)
if ( n_data < 0 ) then
n_data = 0
end if
n_data = n_data + 1
if ( n_max < n_data ) then
n_data = 0
m = 0.0D+00
u = 0.0D+00
sn = 0.0D+00
else
m = m_vec(n_data)
u = u_vec(n_data)
sn = sn_vec(n_data)
end if
return
end
function rc ( x, y, errtol, ierr )
!*****************************************************************************80
!
!! RC computes the elementary integral RC(X,Y).
!
! Discussion:
!
! This function computes the elementary integral
!
! RC(X,Y) = Integral ( 0 <= T < oo )
!
! -1/2 -1
! (1/2)(T+X) (T+Y) DT,
!
! where X is nonnegative and Y is positive. The duplication
! theorem is iterated until the variables are nearly equal,
! and the function is then expanded in Taylor series to fifth
! order.
!
! Logarithmic, inverse circular, and inverse hyperbolic
! functions can be expressed in terms of RC.
!
! Check by addition theorem:
!
! RC(X,X+Z) + RC(Y,Y+Z) = RC(0,Z),
! where X, Y, and Z are positive and X * Y = Z * Z.
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! Original FORTRAN77 version by Bille Carlson, Elaine Notis.
! This FORTRAN90 version by John Burkardt.
!
! Reference:
!
! Bille Carlson,
! Computing Elliptic Integrals by Duplication,
! Numerische Mathematik,
! Volume 33, 1979, pages 1-16.
!
! Bille Carlson, Elaine Notis,
! Algorithm 577, Algorithms for Incomplete Elliptic Integrals,
! ACM Transactions on Mathematical Software,
! Volume 7, Number 3, pages 398-403, September 1981.
!
! Parameters:
!
! Input, real ( kind = 8 ) X, Y, the arguments in the integral.
!
! Input, real ( kind = 8 ) ERRTOL, the error tolerance.
! Relative error due to truncation is less than
! 16 * ERRTOL ^ 6 / (1 - 2 * ERRTOL).
! Sample choices:
! ERRTOL Relative truncation error less than
! 1.D-3 2.D-17
! 3.D-3 2.D-14
! 1.D-2 2.D-11
! 3.D-2 2.D-8
! 1.D-1 2.D-5
!
! Output, integer ( kind = 4 ) IERR, the error flag.
! 0, no error occurred.
! 1, abnormal termination.
!
implicit none
real ( kind = 8 ) c1
real ( kind = 8 ) c2
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) lamda
real ( kind = 8 ) lolim
real ( kind = 8 ) mu
real ( kind = 8 ) rc
real ( kind = 8 ) s
real ( kind = 8 ) sn
real ( kind = 8 ) uplim
real ( kind = 8 ) x
real ( kind = 8 ) xn
real ( kind = 8 ) y
real ( kind = 8 ) yn
!
! LOLIM AND UPLIM DETERMINE THE RANGE OF VALID ARGUMENTS.
! LOLIM IS NOT LESS THAN THE MACHINE MINIMUM MULTIPLIED BY 5.
! UPLIM IS NOT GREATER THAN THE MACHINE MAXIMUM DIVIDED BY 5.
!
save lolim
save uplim
data lolim /3.D-78/
data uplim /1.D+75/
if ( &
x < 0.0d0 .or. &
y <= 0.0d0 .or. &
( x + y ) < lolim .or. &
uplim < x .or. &
uplim < y ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'RC - Error!'
write ( *, '(a)' ) ' Invalid input arguments.'
write ( *, '(a,d23.16)' ) ' X = ', x
write ( *, '(a,d23.16)' ) ' Y = ', y
write ( *, '(a)' ) ''
ierr = 1
rc = 0.0D+00
return
end if
ierr = 0
xn = x
yn = y
do
mu = ( xn + yn + yn ) / 3.0d0
sn = ( yn + mu ) / mu - 2.0d0
if ( abs ( sn ) < errtol ) then
c1 = 1.0d0 / 7.0d0
c2 = 9.0d0 / 22.0d0
s = sn * sn * ( 0.3d0 &
+ sn * ( c1 + sn * ( 0.375d0 + sn * c2 ) ) )
rc = ( 1.0d0 + s ) / sqrt ( mu )
return
end if
lamda = 2.0d0 * sqrt ( xn ) * sqrt ( yn ) + yn
xn = ( xn + lamda ) * 0.25d0
yn = ( yn + lamda ) * 0.25d0
end do
end
function rd ( x, y, z, errtol, ierr )
!*****************************************************************************80
!
!! RD computes an incomplete elliptic integral of the second kind, RD(X,Y,Z).
!
! Discussion:
!
! This function computes an incomplete elliptic integral of the second kind.
!
! RD(X,Y,Z) = Integral ( 0 <= T < oo )
!
! -1/2 -1/2 -3/2
! (3/2)(T+X) (T+Y) (T+Z) DT,
!
! where X and Y are nonnegative, X + Y is positive, and Z is positive.
!
! If X or Y is zero, the integral is complete.
!
! The duplication theorem is iterated until the variables are
! nearly equal, and the function is then expanded in Taylor
! series to fifth order.
!
! Check:
!
! RD(X,Y,Z) + RD(Y,Z,X) + RD(Z,X,Y) = 3 / sqrt ( X * Y * Z ),
! where X, Y, and Z are positive.
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! Original FORTRAN77 version by Bille Carlson, Elaine Notis.
! This FORTRAN90 version by John Burkardt.
!
! Reference:
!
! Bille Carlson,
! Computing Elliptic Integrals by Duplication,
! Numerische Mathematik,
! Volume 33, 1979, pages 1-16.
!
! Bille Carlson, Elaine Notis,
! Algorithm 577, Algorithms for Incomplete Elliptic Integrals,
! ACM Transactions on Mathematical Software,
! Volume 7, Number 3, pages 398-403, September 1981.
!
! Parameters:
!
! Input, real ( kind = 8 ) X, Y, Z, the arguments in the integral.
!
! Input, real ( kind = 8 ) ERRTOL, the error tolerance.
! The relative error due to truncation is less than
! 3 * ERRTOL ^ 6 / (1-ERRTOL) ^ 3/2.
! Sample choices:
! ERRTOL Relative truncation error less than
! 1.D-3 4.D-18
! 3.D-3 3.D-15
! 1.D-2 4.D-12
! 3.D-2 3.D-9
! 1.D-1 4.D-6
!
! Output, integer ( kind = 4 ) IERR, the error flag.
! 0, no error occurred.
! 1, abnormal termination.
!
implicit none
real ( kind = 8 ) c1
real ( kind = 8 ) c2
real ( kind = 8 ) c3
real ( kind = 8 ) c4
real ( kind = 8 ) ea
real ( kind = 8 ) eb
real ( kind = 8 ) ec
real ( kind = 8 ) ed
real ( kind = 8 ) ef
real ( kind = 8 ) epslon
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) lamda
real ( kind = 8 ) lolim
real ( kind = 8 ) mu
real ( kind = 8 ) power4
real ( kind = 8 ) rd
real ( kind = 8 ) sigma
real ( kind = 8 ) s1
real ( kind = 8 ) s2
real ( kind = 8 ) uplim
real ( kind = 8 ) x
real ( kind = 8 ) xn
real ( kind = 8 ) xndev
real ( kind = 8 ) xnroot
real ( kind = 8 ) y
real ( kind = 8 ) yn
real ( kind = 8 ) yndev
real ( kind = 8 ) ynroot
real ( kind = 8 ) z
real ( kind = 8 ) zn
real ( kind = 8 ) zndev
real ( kind = 8 ) znroot
!
! LOLIM AND UPLIM DETERMINE THE RANGE OF VALID ARGUMENTS.
! LOLIM IS NOT LESS THAN 2 / (MACHINE MAXIMUM) ^ (2/3).
! UPLIM IS NOT GREATER THAN (0.1 * ERRTOL / MACHINE
! MINIMUM) ^ (2/3), WHERE ERRTOL IS DESCRIBED BELOW.
! IN THE FOLLOWING TABLE IT IS ASSUMED THAT ERRTOL WILL
! NEVER BE CHOSEN SMALLER THAN 1.D-5.
!
save lolim
save uplim
data lolim /6.D-51/
data uplim /1.D+48/
if ( &
x < 0.0D+00 .or. &
y < 0.0D+00 .or. &
x + y < lolim .or. &
z < lolim .or. &
uplim < x .or. &
uplim < y .or. &
uplim < z ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'RD - Error!'
write ( *, '(a)' ) ' Invalid input arguments.'
write ( *, '(a,d23.16)' ) ' X = ', x
write ( *, '(a,d23.16)' ) ' Y = ', y
write ( *, '(a,d23.16)' ) ' Z = ', z
write ( *, '(a)' ) ''
ierr = 1
rd = 0.0D+00
return
end if
ierr = 0
xn = x
yn = y
zn = z
sigma = 0.0d0
power4 = 1.0d0
do
mu = ( xn + yn + 3.0d0 * zn ) * 0.2d0
xndev = ( mu - xn ) / mu
yndev = ( mu - yn ) / mu
zndev = ( mu - zn ) / mu
epslon = max ( abs ( xndev ), abs ( yndev ), abs ( zndev ) )
if ( epslon < errtol ) then
c1 = 3.0d0 / 14.0d0
c2 = 1.0d0 / 6.0d0
c3 = 9.0d0 / 22.0d0
c4 = 3.0d0 / 26.0d0
ea = xndev * yndev
eb = zndev * zndev
ec = ea - eb
ed = ea - 6.0d0 * eb
ef = ed + ec + ec
s1 = ed * ( - c1 + 0.25d0 * c3 * ed - 1.5d0 * c4 * zndev * ef )
s2 = zndev * ( c2 * ef + zndev * ( - c3 * ec + zndev * c4 * ea ) )
rd = 3.0d0 * sigma + power4 * ( 1.0d0 + s1 + s2 ) / ( mu * sqrt ( mu ) )
return
end if
xnroot = sqrt ( xn )
ynroot = sqrt ( yn )
znroot = sqrt ( zn )
lamda = xnroot * ( ynroot + znroot ) + ynroot * znroot
sigma = sigma + power4 / ( znroot * ( zn + lamda ) )
power4 = power4 * 0.25d0
xn = ( xn + lamda ) * 0.25d0
yn = ( yn + lamda ) * 0.25d0
zn = ( zn + lamda ) * 0.25d0
end do
end
function rf ( x, y, z, errtol, ierr )
!*****************************************************************************80
!
!! RF computes an incomplete elliptic integral of the first kind, RF(X,Y,Z).
!
! Discussion:
!
! This function computes the incomplete elliptic integral of the first kind.
!
! RF(X,Y,Z) = Integral ( 0 <= T < oo )
!
! -1/2 -1/2 -1/2
! (1/2)(T+X) (T+Y) (T+Z) DT,
!
! where X, Y, and Z are nonnegative and at most one of them is zero.
!
! If X or Y or Z is zero, the integral is complete.
!
! The duplication theorem is iterated until the variables are
! nearly equal, and the function is then expanded in Taylor
! series to fifth order.
!
! Check by addition theorem:
!
! RF(X,X+Z,X+W) + RF(Y,Y+Z,Y+W) = RF(0,Z,W),
! where X, Y, Z, W are positive and X * Y = Z * W.
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! Original FORTRAN77 version by Bille Carlson, Elaine Notis.
! This FORTRAN90 version by John Burkardt.
!
! Reference:
!
! Bille Carlson,
! Computing Elliptic Integrals by Duplication,
! Numerische Mathematik,
! Volume 33, 1979, pages 1-16.
!
! Bille Carlson, Elaine Notis,
! Algorithm 577, Algorithms for Incomplete Elliptic Integrals,
! ACM Transactions on Mathematical Software,
! Volume 7, Number 3, pages 398-403, September 1981.
!
! Parameters:
!
! Input, real ( kind = 8 ) X, Y, Z, the arguments in the integral.
!
! Input, real ( kind = 8 ) ERRTOL, the error tolerance.
! Relative error due to truncation is less than
! ERRTOL ^ 6 / (4 * (1 - ERRTOL)).
! Sample choices:
! ERRTOL Relative truncation error less than
! 1.D-3 3.D-19
! 3.D-3 2.D-16
! 1.D-2 3.D-13
! 3.D-2 2.D-10
! 1.D-1 3.D-7
!
! Output, integer ( kind = 4 ) IERR, the error flag.
! 0, no error occurred.
! 1, abnormal termination.
!
implicit none
real ( kind = 8 ) c1
real ( kind = 8 ) c2
real ( kind = 8 ) c3
real ( kind = 8 ) e2
real ( kind = 8 ) e3
real ( kind = 8 ) epslon
real ( kind = 8 ) errtol
integer ( kind = 4 ) ierr
real ( kind = 8 ) lamda
real ( kind = 8 ) lolim
real ( kind = 8 ) mu
real ( kind = 8 ) rf
real ( kind = 8 ) s
real ( kind = 8 ) uplim
real ( kind = 8 ) x
real ( kind = 8 ) xn
real ( kind = 8 ) xndev
real ( kind = 8 ) xnroot
real ( kind = 8 ) y
real ( kind = 8 ) yn
real ( kind = 8 ) yndev
real ( kind = 8 ) ynroot
real ( kind = 8 ) z
real ( kind = 8 ) zn
real ( kind = 8 ) zndev
real ( kind = 8 ) znroot
!
! LOLIM AND UPLIM DETERMINE THE RANGE OF VALID ARGUMENTS.
! LOLIM IS NOT LESS THAN THE MACHINE MINIMUM MULTIPLIED BY 5.
! UPLIM IS NOT GREATER THAN THE MACHINE MAXIMUM DIVIDED BY 5.
!
save lolim
save uplim
data lolim /3.D-78/
data uplim /1.D+75/
if ( &
x < 0.0D+00 .or. &
y < 0.0D+00 .or. &
z < 0.0D+00 .or. &
x + y < lolim .or. &
x + z < lolim .or. &
y + z < lolim .or. &
uplim <= x .or. &
uplim <= y .or. &
uplim <= z ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'RF - Error!'
write ( *, '(a)' ) ' Invalid input arguments.'
write ( *, '(a,d23.16)' ) ' X = ', x
write ( *, '(a,d23.16)' ) ' Y = ', y
write ( *, '(a,d23.16)' ) ' Z = ', z
write ( *, '(a)' ) ''
ierr = 1
rf = 0.0D+00
return
end if
ierr = 0
xn = x
yn = y
zn = z
do
mu = ( xn + yn + zn ) / 3.0d0
xndev = 2.0d0 - ( mu + xn ) / mu
yndev = 2.0d0 - ( mu + yn ) / mu
zndev = 2.0d0 - ( mu + zn ) / mu
epslon = max ( abs ( xndev ), abs ( yndev ), abs ( zndev ) )
if ( epslon < errtol ) then
c1 = 1.0d0 / 24.0d0
c2 = 3.0d0 / 44.0d0
c3 = 1.0d0 / 14.0d0
e2 = xndev * yndev - zndev * zndev
e3 = xndev * yndev * zndev
s = 1.0d0 + ( c1 * e2 - 0.1d0 - c2 * e3 ) * e2 + c3 * e3
rf = s / sqrt ( mu )
return
end if
xnroot = sqrt ( xn )
ynroot = sqrt ( yn )
znroot = sqrt ( zn )
lamda = xnroot * ( ynroot + znroot ) + ynroot * znroot
xn = ( xn + lamda ) * 0.25d0
yn = ( yn + lamda ) * 0.25d0
zn = ( zn + lamda ) * 0.25d0
end do
end
function rj ( x, y, z, p, errtol, ierr )
!*****************************************************************************80
!
!! RJ computes an incomplete elliptic integral of the third kind, RJ(X,Y,Z,P).
!
! Discussion:
!
! This function computes an incomplete elliptic integral of the third kind.
!
! RJ(X,Y,Z,P) = Integral ( 0 <= T < oo )
!
! -1/2 -1/2 -1/2 -1
! (3/2)(T+X) (T+Y) (T+Z) (T+P) DT,
!
! where X, Y, and Z are nonnegative, at most one of them is
! zero, and P is positive.
!
! If X or Y or Z is zero, then the integral is complete.
!
! The duplication theorem is iterated until the variables are nearly equal,
! and the function is then expanded in Taylor series to fifth order.
!
! Check by addition theorem:
!
! RJ(X,X+Z,X+W,X+P)
! + RJ(Y,Y+Z,Y+W,Y+P) + (A-B) * RJ(A,B,B,A) + 3 / sqrt ( A)
! = RJ(0,Z,W,P), where X,Y,Z,W,P are positive and X * Y
! = Z * W, A = P * P * (X+Y+Z+W), B = P * (P+X) * (P+Y),
! and B - A = P * (P-Z) * (P-W).
!
! The sum of the third and fourth terms on the left side is 3 * RC(A,B).
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 30 May 2018
!
! Author:
!
! Original FORTRAN77 version by Bille Carlson, Elaine Notis.
! This FORTRAN90 version by John Burkardt.
!
! Reference:
!
! Bille Carlson,
! Computing Elliptic Integrals by Duplication,
! Numerische Mathematik,
! Volume 33, 1979, pages 1-16.
!
! Bille Carlson, Elaine Notis,
! Algorithm 577, Algorithms for Incomplete Elliptic Integrals,
! ACM Transactions on Mathematical Software,
! Volume 7, Number 3, pages 398-403, September 1981.
!
! Parameters:
!
! Input, real ( kind = 8 ) X, Y, Z, P, the arguments in the integral.
!
! Input, real ( kind = 8 ) ERRTOL, the error tolerance.
! Relative error due to truncation of the series for rj
! is less than 3 * ERRTOL ^ 6 / (1 - ERRTOL) ^ 3/2.
! An error tolerance (ETOLRC) will be passed to the subroutine
! for RC to make the truncation error for RC less than for RJ.
! Sample choices:
! ERRTOL Relative truncation error less than
! 1.D-3 4.D-18
! 3.D-3 3.D-15
! 1.D-2 4.D-12
! 3.D-2 3.D-9
! 1.D-1 4.D-6
!
! Output, integer ( kind = 4 ) IERR, the error flag.
! 0, no error occurred.
! 1, abnormal termination.
!
implicit none
real ( kind = 8 ) alfa
real ( kind = 8 ) beta
real ( kind = 8 ) c1
real ( kind = 8 ) c2
real ( kind = 8 ) c3
real ( kind = 8 ) c4
real ( kind = 8 ) ea
real ( kind = 8 ) eb
real ( kind = 8 ) ec
real ( kind = 8 ) e2
real ( kind = 8 ) e3
real ( kind = 8 ) epslon
real ( kind = 8 ) errtol
real ( kind = 8 ) etolrc
integer ( kind = 4 ) ierr
real ( kind = 8 ) lamda
real ( kind = 8 ) lolim
real ( kind = 8 ) mu
real ( kind = 8 ) p
real ( kind = 8 ) pn
real ( kind = 8 ) pndev
real ( kind = 8 ) power4
!real ( kind = 8 ) rc
real ( kind = 8 ) rj
real ( kind = 8 ) sigma
real ( kind = 8 ) s1
real ( kind = 8 ) s2
real ( kind = 8 ) s3
real ( kind = 8 ) uplim
real ( kind = 8 ) x
real ( kind = 8 ) xn
real ( kind = 8 ) xndev
real ( kind = 8 ) xnroot
real ( kind = 8 ) y
real ( kind = 8 ) yn
real ( kind = 8 ) yndev
real ( kind = 8 ) ynroot
real ( kind = 8 ) z
real ( kind = 8 ) zn
real ( kind = 8 ) zndev
real ( kind = 8 ) znroot
!
! LOLIM AND UPLIM DETERMINE THE RANGE OF VALID ARGUMENTS.
! LOLIM IS NOT LESS THAN THE CUBE ROOT OF THE VALUE
! OF LOLIM USED IN THE SUBROUTINE FOR RC.
! UPLIM IS NOT GREATER THAN 0.3 TIMES THE CUBE ROOT OF
! THE VALUE OF UPLIM USED IN THE SUBROUTINE FOR RC.
!
save lolim
save uplim
data lolim /2.D-26/
data uplim /3.D+24/
if ( &
x < 0.0D+00 .or. &
y < 0.0D+00 .or. &
z < 0.0D+00 .or. &
x + y < lolim .or. &
x + z < lolim .or. &
y + z < lolim .or. &
p < lolim .or. &
uplim < x .or. &
uplim < y .or. &
uplim < z .or. &
uplim < p ) then
write ( *, '(a)' ) ''
write ( *, '(a)' ) 'RJ - Error!'
write ( *, '(a)' ) ' Invalid input arguments.'
write ( *, '(a,d23.16)' ) ' X = ', x
write ( *, '(a,d23.16)' ) ' Y = ', y
write ( *, '(a,d23.16)' ) ' Z = ', z
write ( *, '(a,d23.16)' ) ' P = ', p
write ( *, '(a)' ) ''
ierr = 1
rj = 0.0D+00
return
end if
ierr = 0
xn = x
yn = y
zn = z
pn = p
sigma = 0.0d0
power4 = 1.0d0
etolrc = 0.5d0 * errtol
do
mu = ( xn + yn + zn + pn + pn ) * 0.2d0
xndev = ( mu - xn ) / mu
yndev = ( mu - yn ) / mu
zndev = ( mu - zn ) / mu
pndev = ( mu - pn ) / mu
epslon = max ( abs ( xndev ), abs ( yndev ), abs ( zndev ), abs ( pndev ) )
if ( epslon < errtol ) then
c1 = 3.0d0 / 14.0d0
c2 = 1.0d0 / 3.0d0
c3 = 3.0d0 / 22.0d0
c4 = 3.0d0 / 26.0d0
ea = xndev * ( yndev + zndev ) + yndev * zndev
eb = xndev * yndev * zndev
ec = pndev * pndev
e2 = ea - 3.0d0 * ec
e3 = eb + 2.0d0 * pndev * ( ea - ec )
s1 = 1.0d0 + e2 * ( - c1 + 0.75d0 * c3 * e2 - 1.5d0 * c4 * e3 )
s2 = eb * ( 0.5d0 * c2 + pndev * ( - c3 - c3 + pndev * c4 ) )
s3 = pndev * ea * ( c2 - pndev * c3 ) - c2 * pndev * ec
rj = 3.0d0 * sigma + power4 * ( s1 + s2 + s3 ) / ( mu * sqrt ( mu ) )
return
end if
xnroot = sqrt ( xn )
ynroot = sqrt ( yn )
znroot = sqrt ( zn )
lamda = xnroot * ( ynroot + znroot ) + ynroot * znroot
alfa = pn * ( xnroot + ynroot + znroot ) &
+ xnroot * ynroot * znroot
alfa = alfa * alfa
beta = pn * ( pn + lamda ) * ( pn + lamda )
sigma = sigma + power4 * rc ( alfa, beta, etolrc, ierr )
if ( ierr /= 0 ) then
rj = 0.0D+00
return
end if
power4 = power4 * 0.25d0
xn = ( xn + lamda ) * 0.25d0
yn = ( yn + lamda ) * 0.25d0
zn = ( zn + lamda ) * 0.25d0
pn = ( pn + lamda ) * 0.25d0
end do
end
subroutine sncndn ( u, m, sn, cn, dn )
!*****************************************************************************80
!
!! SNCNDN evaluates Jacobi elliptic functions.
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 24 June 2018
!
! Author:
!
! Original ALGOL version by Roland Bulirsch.
! FORTRAN90 version by John Burkardt
!
! Reference:
!
! Roland Bulirsch,
! Numerical calculation of elliptic integrals and elliptic functions,
! Numerische Mathematik,
! Volume 7, Number 1, 1965, pages 78-90.
!
! Parameters:
!
! Input, real ( kind = 8 ) U, M, the arguments.
!
! Output, real ( kind = 8 ) SN, CN, DN, the value of the Jacobi
! elliptic functions sn(u,m), cn(u,m), and dn(u,m).
!
implicit none
real ( kind = 8 ) a
real ( kind = 8 ) b
real ( kind = 8 ) c
real ( kind = 8 ) ca
real ( kind = 8 ) cn
real ( kind = 8 ) d
real ( kind = 8 ) dn
real ( kind = 8 ) m_array(25)
real ( kind = 8 ) n_array(25)
integer ( kind = 4 ) i
integer ( kind = 4 ) l
real ( kind = 8 ) m
real ( kind = 8 ) m_comp
real ( kind = 8 ) sn
real ( kind = 8 ) u
real ( kind = 8 ) u_copy
m_comp = 1.0D+00 - m
u_copy = u
if ( m_comp == 0.0D+00 ) then
cn = 1.0D+00 / cosh ( u_copy )
dn = cn
sn = tanh ( u_copy )
return
end if
if ( 1.0D+00 < m ) then
d = 1.0D+00 - m_comp
m_comp = - m_comp / d
d = sqrt ( d )
u_copy = d * u_copy
end if
ca = sqrt ( epsilon ( ca ) )
a = 1.0D+00
dn = 1.0D+00
l = 25
do i = 1, 25
m_array(i) = a
m_comp = sqrt ( m_comp )
n_array(i) = m_comp
c = 0.5D+00 * ( a + m_comp )
if ( abs ( a - m_comp ) <= ca * a ) then
l = i
exit
end if
m_comp = a * m_comp
a = c
end do
u_copy = c * u_copy
sn = sin ( u_copy )
cn = cos ( u_copy )
if ( sn /= 0.0D+00 ) then
a = cn / sn
c = a * c
do i = l, 1, -1
b = m_array(i)
a = c * a
c = dn * c
dn = ( n_array(i) + a ) / ( b + a )
a = c / b
end do
a = 1.0D+00 / sqrt ( c * c + 1.0D+00 )
if ( sn < 0.0D+00 ) then
sn = - a
else
sn = a
end if
cn = c * sn
end if
if ( 1.0D+00 < m ) then
a = dn
dn = cn
cn = a
sn = sn / d
end if
return
end
subroutine timestamp ( )
!*****************************************************************************80
!
!! TIMESTAMP prints the current YMDHMS date as a time stamp.
!
! Example:
!
! 31 May 2001 9:45:54.872 AM
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 18 May 2013
!
! Author:
!
! John Burkardt
!
! Parameters:
!
! None
!
implicit none
character ( len = 8 ) ampm
integer ( kind = 4 ) d
integer ( kind = 4 ) h
integer ( kind = 4 ) m
integer ( kind = 4 ) mm
character ( len = 9 ), parameter, dimension(12) :: month = (/ &
'January ', 'February ', 'March ', 'April ', &
'May ', 'June ', 'July ', 'August ', &
'September', 'October ', 'November ', 'December ' /)
integer ( kind = 4 ) n
integer ( kind = 4 ) s
integer ( kind = 4 ) values(8)
integer ( kind = 4 ) y
call date_and_time ( values = values )
y = values(1)
m = values(2)
d = values(3)
h = values(5)
n = values(6)
s = values(7)
mm = values(8)
if ( h < 12 ) then
ampm = 'AM'
else if ( h == 12 ) then
if ( n == 0 .and. s == 0 ) then
ampm = 'Noon'
else
ampm = 'PM'
end if
else
h = h - 12
if ( h < 12 ) then
ampm = 'PM'
else if ( h == 12 ) then
if ( n == 0 .and. s == 0 ) then
ampm = 'Midnight'
else
ampm = 'AM'
end if
end if
end if
write ( *, '(i2.2,1x,a,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)' ) &
d, trim ( month(m) ), y, h, ':', n, ':', s, '.', mm, trim ( ampm )
return
end
end module
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