General mathematical functions#

acos#

Name#

acos(3) - [MATHEMATICS:TRIGONOMETRIC] Arccosine (inverse cosine) function

Synopsis#

    result = acos(x)
     elemental TYPE(kind=KIND) function acos(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • TYPE may be real or complex

  • KIND may be any kind supported by the associated type.

  • The returned value will be of the same type and kind as the argument.

Description#

acos(3) computes the arccosine of x (inverse of cos(x)).

Options#

  • x

    The value to compute the arctangent of.

    If the type is real, the value must satisfy |x| <= 1.

Result#

The return value is of the same type and kind as x. The real part of the result is in radians and lies in the range 0 <= acos(x%re) <= PI .

Examples#

Sample program:

program demo_acos
use, intrinsic :: iso_fortran_env, only : real_kinds,real32,real64,real128
implicit none
character(len=*),parameter :: all='(*(g0,1x))'
real(kind=real64) :: x , d2r

   ! basics
    x = 0.866_real64
    print all,'acos(',x,') is ', acos(x)

   ! acos(-1) should be PI
    print all,'for reference &
    &PI ~= 3.14159265358979323846264338327950288419716939937510'
    write(*,*) acos(-1.0_real64)
    d2r=acos(-1.0_real64)/180.0_real64
    print all,'90 degrees is ', d2r*90.0_real64, ' radians'
   ! elemental
    print all,'elemental',acos([-1.0,-0.5,0.0,0.50,1.0])
   ! complex
    print *,'complex',acos( (-1.0,  0.0) )
    print *,'complex',acos( (-1.0, -1.0) )
    print *,'complex',acos( ( 0.0, -0.0) )
    print *,'complex',acos( ( 1.0,  0.0) )

end program demo_acos

Results:

 acos( 0.86599999999999999 ) is  0.52364958093182890
 for reference PI ~= 3.14159265358979323846264338327950288419716939937510
    3.1415926535897931
 90 degrees is  1.5707963267948966  radians
 elemental 3.14159274 2.09439516 1.57079637 1.04719758 0.00000000
  complex            (3.14159274,-0.00000000)
  complex             (2.23703575,1.06127501)
  complex             (1.57079637,0.00000000)
  complex            (0.00000000,-0.00000000)

Standard#

FORTRAN 77 ; for a complex argument - Fortran 2008

See Also#

Inverse function: cos(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

acosh#

Name#

acosh(3) - [MATHEMATICS:TRIGONOMETRIC] Inverse hyperbolic cosine function

Synopsis#

    result = acosh(x)
     elemental TYPE(kind=KIND) function acosh(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • TYPE may be real or complex

  • KIND may be any kind supported by the associated type.

  • The returned value will be of the same type and kind as the argument.

Description#

acosh(3) computes the inverse hyperbolic cosine of x in radians.

Options#

  • x

    The value to compute the hyperbolic cosine of

Result#

The result has a value equal to a processor-dependent approximation to the inverse hyperbolic cosine function of X.

If x is complex, the imaginary part of the result is in radians and lies between

 0 <= aimag(acosh(x)) <= PI

Examples#

Sample program:

program demo_acosh
use,intrinsic :: iso_fortran_env, only : dp=>real64,sp=>real32
implicit none
real(kind=dp), dimension(3) :: x = [ 1.0d0, 2.0d0, 3.0d0 ]
   write (*,*) acosh(x)
end program demo_acosh

Results:

 0.000000000000000E+000   1.31695789692482        1.76274717403909

Standard#

Fortran 2008

See Also#

Inverse function: cosh(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

asin#

Name#

asin(3) - [MATHEMATICS:TRIGONOMETRIC] Arcsine function

Synopsis#

    result = asin(x)
     elemental TYPE(kind=KIND) function asin(x)

      TYPE(kind=KIND) :: x

Characteristics#

  • TYPE may be real or complex

  • KIND may be any kind supported by the associated type.

  • The returned value will be of the same type and kind as the argument.

Description#

asin(3) computes the arcsine of its argument x.

The arcsine is the inverse function of the sine function. It is commonly used in trigonometry when trying to find the angle when the lengths of the hypotenuse and the opposite side of a right triangle are known.

Options#

  • x

    The value to compute the arcsine of

    The type shall be either real and a magnitude that is less than or equal to one; or be complex.

Result#

  • result The result has a value equal to a processor-dependent approximation to arcsin(x).

    If x is real the result is real and it is expressed in radians and lies in the range

        PI/2 <= ASIN (X) <= PI/2.

If the argument (and therefore the result) is imaginary the real part of the result is in radians and lies in the range

    -PI/2 <= real(asin(x)) <= PI/2

Examples#

The arcsine will allow you to find the measure of a right angle when you know the ratio of the side opposite the angle to the hypotenuse.

So if you knew that a train track rose 1.25 vertical miles on a track that was 50 miles long, you could determine the average angle of incline of the track using the arcsine. Given

 sin(theta) = 1.25 miles/50 miles (opposite/hypotenuse)

Sample program:

program demo_asin
use, intrinsic :: iso_fortran_env, only : dp=>real64
implicit none
! value to convert degrees to radians
real(kind=dp),parameter :: D2R=acos(-1.0_dp)/180.0_dp
real(kind=dp)           :: angle, rise, run
character(len=*),parameter :: all='(*(g0,1x))'
  ! given sine(theta) = 1.25 miles/50 miles (opposite/hypotenuse)
  ! then taking the arcsine of both sides of the equality yields
  ! theta = arcsine(1.25 miles/50 miles) ie. arcsine(opposite/hypotenuse)
  rise=1.250_dp
  run=50.00_dp
  angle = asin(rise/run)
  print all, 'angle of incline(radians) = ', angle
  angle = angle/D2R
  print all, 'angle of incline(degrees) = ', angle

  print all, 'percent grade=',rise/run*100.0_dp
end program demo_asin

Results:

    angle of incline(radians) =    2.5002604899361139E-002
    angle of incline(degrees) =    1.4325437375665075
    percent grade=   2.5000000000000000

The percentage grade is the slope, written as a percent. To calculate the slope you divide the rise by the run. In the example the rise is 1.25 mile over a run of 50 miles so the slope is 1.25/50 = 0.025. Written as a percent this is 2.5 %.

For the US, two and 1/2 percent is generally thought of as the upper limit. This means a rise of 2.5 feet when going 100 feet forward. In the US this was the maximum grade on the first major US railroad, the Baltimore and Ohio. Note curves increase the frictional drag on a train reducing the allowable grade.

Standard#

FORTRAN 77 , for a complex argument Fortran 2008

See Also#

Inverse function: sin(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

asinh#

Name#

asinh(3) - [MATHEMATICS:TRIGONOMETRIC] Inverse hyperbolic sine function

Synopsis#

    result = asinh(x)
     elemental TYPE(kind=KIND) function asinh(x)

      TYPE(kind=KIND) :: x

Characteristics#

  • x may be any real or complex type

  • KIND may be any kind supported by the associated type

  • The returned value will be of the same type and kind as the argument x

Description#

asinh(3) computes the inverse hyperbolic sine of x.

Options#

  • x

    The value to compute the inverse hyperbolic sine of

Result#

The result has a value equal to a processor-dependent approximation to the inverse hyperbolic sine function of x.

If x is complex, the imaginary part of the result is in radians and lies between -PI/2 <= aimag(asinh(x)) <= PI/2.

Examples#

Sample program:

program demo_asinh
use,intrinsic :: iso_fortran_env, only : dp=>real64,sp=>real32
implicit none
real(kind=dp), dimension(3) :: x = [ -1.0d0, 0.0d0, 1.0d0 ]

   ! elemental
    write (*,*) asinh(x)

end program demo_asinh

Results:

  -0.88137358701954305  0.0000000000000000  0.88137358701954305

Standard#

Fortran 2008

See Also#

Inverse function: sinh(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

atan#

Name#

atan(3) - [MATHEMATICS:TRIGONOMETRIC] Arctangent AKA inverse tangent function

Synopsis#

    result = atan([x) | atan(y, x)
     elemental TYPE(kind=KIND) function atan(y,x)

      TYPE(kind=KIND),intent(in) :: x
      TYPE(kind=**),intent(in),optional :: y

Characteristics#

  • If y is present x and y must both be real. Otherwise, x may be complex.

  • KIND can be any kind supported by the associated type.

  • The returned value is of the same type and kind as x.

Description#

atan(3) computes the arctangent of x.

Options#

  • x

    The value to compute the arctangent of. if y is present, x shall be real.

  • y

    is of the same type and kind as x. If x is zero, y must not be zero.

Result#

The returned value is of the same type and kind as x. If y is present, the result is identical to atan2(y,x). Otherwise, it is the arc tangent of x, where the real part of the result is in radians and lies in the range -PI/2 <= atan(x) <= PI/2

Examples#

Sample program:

program demo_atan
use, intrinsic :: iso_fortran_env, only : real_kinds, &
 & real32, real64, real128
implicit none
character(len=*),parameter :: all='(*(g0,1x))'
real(kind=real64),parameter :: &
 Deg_Per_Rad = 57.2957795130823208767981548_real64
real(kind=real64) :: x
    x=2.866_real64
    print all, atan(x)

    print all, atan( 2.0d0, 2.0d0),atan( 2.0d0, 2.0d0)*Deg_Per_Rad
    print all, atan( 2.0d0,-2.0d0),atan( 2.0d0,-2.0d0)*Deg_Per_Rad
    print all, atan(-2.0d0, 2.0d0),atan(-2.0d0, 2.0d0)*Deg_Per_Rad
    print all, atan(-2.0d0,-2.0d0),atan(-2.0d0,-2.0d0)*Deg_Per_Rad

end program demo_atan

Results:

   1.235085437457879
   .7853981633974483 45.00000000000000
   2.356194490192345 135.0000000000000
   -.7853981633974483 -45.00000000000000
   -2.356194490192345 -135.0000000000000

Standard#

FORTRAN 77 for a complex argument; and for two arguments Fortran 2008

See Also#

atan2(3), tan(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

atan2#

Name#

atan2(3) - [MATHEMATICS:TRIGONOMETRIC] Arctangent (inverse tangent) function

Synopsis#

    result = atan2(y, x)
     elemental real(kind=KIND) function atan2(y, x)

      real,kind=KIND) :: atan2
      real,kind=KIND),intent(in) :: y, x

Characteristics#

  • x and y must be reals of the same kind.

  • The return value has the same type and kind as y and x.

Description#

atan2(3) computes in radians a processor-dependent approximation of the arctangent of the complex number ( x, y ) or equivalently the principal value of the arctangent of the value y/x (which determines a unique angle).

If y has the value zero, x shall not have the value zero.

The resulting phase lies in the range -PI <= ATAN2 (Y,X) <= PI and is equal to a processor-dependent approximation to a value of arctan(Y/X).

Options#

  • y

    The imaginary component of the complex value (x,y) or the y component of the point <x,y>.

  • x

    The real component of the complex value (x,y) or the x component of the point <x,y>.

Result#

The value returned is by definition the principal value of the complex number (x, y), or in other terms, the phase of the phasor x+i*y.

The principal value is simply what we get when we adjust a radian value to lie between -PI and PI inclusive,

The classic definition of the arctangent is the angle that is formed in Cartesian coordinates of the line from the origin point <0,0> to the point <x,y> .

Pictured as a vector it is easy to see that if x and y are both zero the angle is indeterminate because it sits directly over the origin, so atan(0.0,0.0) will produce an error.

Range of returned values by quadrant:

>                   +PI/2
>                     |
>                     |
>     PI/2 < z < PI   |   0 > z < PI/2
>                     |
>   +-PI -------------+---------------- +-0
>                     |
>     PI/2 < -z < PI  |   0 < -z < PI/2
>                     |
>                     |
>                   -PI/2
>
     NOTES:

     If the processor distinguishes -0 and +0 then the sign of the
     returned value is that of Y when Y is zero, else when Y is zero
     the returned value is always positive.

Examples#

Sample program:

program demo_atan2
real :: z
complex :: c
 !
 ! basic usage
  ! ATAN2 (1.5574077, 1.0) has the value 1.0 (approximately).
  z=atan2(1.5574077, 1.0)
  write(*,*) 'radians=',z,'degrees=',r2d(z)
 !
 ! elemental arrays
  write(*,*)'elemental',atan2( [10.0, 20.0], [30.0,40.0] )
 !
 ! elemental arrays and scalars
  write(*,*)'elemental',atan2( [10.0, 20.0], 50.0 )
 !
 ! break complex values into real and imaginary components
 ! (note TAN2() can take a complex type value )
  c=(0.0,1.0)
  write(*,*)'complex',c,atan2( x=c%re, y=c%im )
 !
 ! extended sample converting cartesian coordinates to polar
  COMPLEX_VALS: block
  real                :: ang, radius
  complex,allocatable :: vals(:)
 !
  vals=[ &
    ( 1.0, 0.0 ), & ! 0
    ( 1.0, 1.0 ), & ! 45
    ( 0.0, 1.0 ), & ! 90
    (-1.0, 1.0 ), & ! 135
    (-1.0, 0.0 ), & ! 180
    (-1.0,-1.0 ), & ! 225
    ( 0.0,-1.0 )]   ! 270
  do i=1,size(vals)
     call cartesian_to_polar(vals(i)%re, vals(i)%im, radius,ang)
     write(*,101)vals(i),ang,r2d(ang),radius
  enddo
  101 format(             &
  & 'X= ',f5.2,           &
  & ' Y= ',f5.2,          &
  & ' ANGLE= ',g0,        &
  & T38,'DEGREES= ',g0.4, &
  & T54,'DISTANCE=',g0)
 endblock COMPLEX_VALS
!
contains
!
elemental real function r2d(radians)
! input radians to convert to degrees
doubleprecision,parameter :: DEGREE=0.017453292519943d0 ! radians
real,intent(in)           :: radians
   r2d=radians / DEGREE ! do the conversion
end function r2d
!
subroutine cartesian_to_polar(x,y,radius,inclination)
! return angle in radians in range 0 to 2*PI
implicit none
real,intent(in)  :: x,y
real,intent(out) :: radius,inclination
   radius=sqrt(x**2+y**2)
   if(radius.eq.0)then
      inclination=0.0
   else
      inclination=atan2(y,x)
      if(inclination < 0.0)inclination=inclination+2*atan2(0.0d0,-1.0d0)
   endif
end subroutine cartesian_to_polar
!
end program demo_atan2

Results:

 >  radians=   1.000000     degrees=   57.29578
 >  elemental  0.3217506      0.4636476
 >  elemental  0.1973956      0.3805064
 >  complex (0.0000000E+00,1.000000)   1.570796
 > X=  1.00 Y=  0.00 ANGLE= .000000     DEGREES= .000   DISTANCE=1.000000
 > X=  1.00 Y=  1.00 ANGLE= .7853982    DEGREES= 45.00  DISTANCE=1.414214
 > X=  0.00 Y=  1.00 ANGLE= 1.570796    DEGREES= 90.00  DISTANCE=1.000000
 > X= -1.00 Y=  1.00 ANGLE= 2.356194    DEGREES= 135.0  DISTANCE=1.414214
 > X= -1.00 Y=  0.00 ANGLE= 3.141593    DEGREES= 180.0  DISTANCE=1.000000
 > X= -1.00 Y= -1.00 ANGLE= 3.926991    DEGREES= 225.0  DISTANCE=1.414214
 > X=  0.00 Y= -1.00 ANGLE= 4.712389    DEGREES= 270.0  DISTANCE=1.000000

Standard#

FORTRAN 77

See Also#

Resources#

  • arctan:wikipedia fortran-lang intrinsic descriptions (license: MIT) @urbanjost

atanh#

Name#

atanh(3) - [MATHEMATICS:TRIGONOMETRIC] Inverse hyperbolic tangent function

Synopsis#

    result = atanh(x)
     elemental TYPE(kind=KIND) function atanh(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • x may be real or complex of any associated type

  • The returned value will be of the same type and kind as the argument.

Description#

atanh(3) computes the inverse hyperbolic tangent of x.

Options#

  • x

    The type shall be real or complex.

Result#

The return value has same type and kind as x. If x is complex, the imaginary part of the result is in radians and lies between

       **-PI/2 <= aimag(atanh(x)) <= PI/2**

Examples#

Sample program:

program demo_atanh
implicit none
real, dimension(3) :: x = [ -1.0, 0.0, 1.0 ]

   write (*,*) atanh(x)

end program demo_atanh

Results:

 >       -Infinity  0.0000000E+00       Infinity

Standard#

Fortran 2008

See Also#

Inverse function: tanh(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

cos#

Name#

cos(3) - [MATHEMATICS:TRIGONOMETRIC] Cosine function

Synopsis#

    result = cos(x)
     elemental TYPE(kind=KIND) function cos(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • x is of type real or complex of any valid kind.

  • KIND may be any kind supported by the associated type of x.

  • The returned value will be of the same type and kind as the argument x.

Description#

cos(3) computes the cosine of an angle x given the size of the angle in radians.

The cosine of a real value is the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

Options#

  • x

    The angle in radians to compute the cosine of.

Result#

The return value is the tangent of x.

If x is of the type real, the return value is in radians and lies in the range -1 <= cos(x) <= 1 .

If x is of type complex, its real part is regarded as a value in radians, often called the phase.

Examples#

Sample program:

program demo_cos
implicit none
character(len=*),parameter :: g2='(a,t20,g0)'
doubleprecision,parameter :: PI=atan(1.0d0)*4.0d0
   write(*,g2)'COS(0.0)=',cos(0.0)
   write(*,g2)'COS(PI)=',cos(PI)
   write(*,g2)'COS(PI/2.0d0)=',cos(PI/2.0d0),'EPSILON=',epsilon(PI)
   write(*,g2)'COS(2*PI)=',cos(2*PI)
   write(*,g2)'COS(-2*PI)=',cos(-2*PI)
   write(*,g2)'COS(-2000*PI)=',cos(-2000*PI)
   write(*,g2)'COS(3000*PI)=',cos(3000*PI)
end program demo_cos

Results:

 > COS(0.0)=          1.000000
 > COS(PI)=           -1.000000000000000
 > COS(PI/2.0d0)=     .6123233995736766E-16
 > EPSILON=           .2220446049250313E-15
 > COS(2*PI)=         1.000000000000000
 > COS(-2*PI)=        1.000000000000000
 > COS(-2000*PI)=     1.000000000000000
 > COS(3000*PI)=      1.000000000000000

Standard#

FORTRAN 77

See Also#

acos(3), sin(3), tan(3)

Resources#

fortran-lang intrinsic descriptions

cosh#

Name#

cosh(3) - [MATHEMATICS:TRIGONOMETRIC] Hyperbolic cosine function

Synopsis#

    result = cosh(x)
     elemental TYPE(kind=KIND) function cosh(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • TYPE may be real or complex of any kind.

  • The returned value will be of the same type and kind as the argument.

Description#

cosh(3) computes the hyperbolic cosine of x.

If x is of type complex its imaginary part is regarded as a value in radians.

Options#

  • x

    the value to compute the hyperbolic cosine of

Result#

If x is complex, the imaginary part of the result is in radians.

If x is real, the return value has a lower bound of one, cosh(x) >= 1.

Examples#

Sample program:

program demo_cosh
use, intrinsic :: iso_fortran_env, only : &
 & real_kinds, real32, real64, real128
implicit none
real(kind=real64) :: x = 1.0_real64
    write(*,*)'X=',x,'COSH(X=)',cosh(x)
end program demo_cosh

Results:

 >  X=   1.00000000000000      COSH(X=)   1.54308063481524

Standard#

FORTRAN 77 , for a complex argument - Fortran 2008

See Also#

Inverse function: acosh(3)

Resources#

fortran-lang intrinsic descriptions

sin#

Name#

sin(3) - [MATHEMATICS:TRIGONOMETRIC] Sine function

Synopsis#

    result = sin(x)
     elemental TYPE(kind=KIND) function sin(x)

      TYPE(kind=KIND) :: x

Characteristics#

  • x may be any real or complex type

  • KIND may be any kind supported by the associated type of x.

  • The returned value will be of the same type and kind as the argument x.

Description#

sin(3) computes the sine of an angle given the size of the angle in radians.

The sine of an angle in a right-angled triangle is the ratio of the length of the side opposite the given angle divided by the length of the hypotenuse.

Options#

  • x

    The angle in radians to compute the sine of.

Result#

  • result The return value contains the processor-dependent approximation of the sine of x

    If X is of type real, it is regarded as a value in radians.

    If X is of type complex, its real part is regarded as a value in radians.

Examples#

Sample program:

program sample_sin
implicit none
real :: x = 0.0
   x = sin(x)
   write(*,*)'X=',x
end program sample_sin

Results:

 >  X=  0.0000000E+00

Extended Example#

Haversine Formula#

From the article on “Haversine formula” in Wikipedia:

    The haversine formula is an equation important in navigation,
    giving great-circle distances between two points on a sphere from
    their longitudes and latitudes.

So to show the great-circle distance between the Nashville International Airport (BNA) in TN, USA, and the Los Angeles International Airport (LAX) in CA, USA you would start with their latitude and longitude, commonly given as

  BNA: N 36 degrees 7.2',   W 86 degrees 40.2'
  LAX: N 33 degrees 56.4',  W 118 degrees 24.0'

which converted to floating-point values in degrees is:

       Latitude Longitude

     - BNA
       36.12, -86.67

     - LAX
       33.94, -118.40

And then use the haversine formula to roughly calculate the distance along the surface of the Earth between the locations:

Sample program:

program demo_sin
implicit none
real :: d
    d = haversine(36.12,-86.67, 33.94,-118.40) ! BNA to LAX
    print '(A,F9.4,A)', 'distance: ',d,' km'
contains
function haversine(latA,lonA,latB,lonB) result (dist)
!
! calculate great circle distance in kilometers
! given latitude and longitude in degrees
!
real,intent(in) :: latA,lonA,latB,lonB
real :: a,c,dist,delta_lat,delta_lon,lat1,lat2
real,parameter :: radius = 6371 ! mean earth radius in kilometers,
! recommended by the International Union of Geodesy and Geophysics

! generate constant pi/180
real, parameter :: deg_to_rad = atan(1.0)/45.0
   delta_lat = deg_to_rad*(latB-latA)
   delta_lon = deg_to_rad*(lonB-lonA)
   lat1 = deg_to_rad*(latA)
   lat2 = deg_to_rad*(latB)
   a = (sin(delta_lat/2))**2 + &
          & cos(lat1)*cos(lat2)*(sin(delta_lon/2))**2
   c = 2*asin(sqrt(a))
   dist = radius*c
end function haversine
end program demo_sin

Results:

 > distance: 2886.4446 km

Standard#

FORTRAN 77

See Also#

asin(3), cos(3), tan(3), acosh(3), acos(3), asinh(3), atan2(3), atanh(3), acosh(3), asinh(3), atanh(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

sinh#

Name#

sinh(3) - [MATHEMATICS:TRIGONOMETRIC] Hyperbolic sine function

Synopsis#

    result = sinh(x)
     elemental TYPE(kind=KIND) function sinh(x)

      TYPE(kind=KIND) :: x

Characteristics#

  • TYPE may be real or complex

  • KIND may be any kind supported by the associated type.

  • The returned value will be of the same type and kind as the argument.

Description#

sinh(3) computes the hyperbolic sine of x.

The hyperbolic sine of x is defined mathematically as:

     sinh(x) = (exp(x) - exp(-x)) / 2.0

Options#

  • x

    The value to calculate the hyperbolic sine of

Result#

The result has a value equal to a processor-dependent approximation to sinh(X). If X is of type complex its imaginary part is regarded as a value in radians.

Examples#

Sample program:

program demo_sinh
use, intrinsic :: iso_fortran_env, only : &
& real_kinds, real32, real64, real128
implicit none
real(kind=real64) :: x = - 1.0_real64
real(kind=real64) :: nan, inf
character(len=20) :: line

  ! basics
   print *, sinh(x)
   print *, (exp(x)-exp(-x))/2.0

  ! sinh(3) is elemental and can handle an array
   print *, sinh([x,2.0*x,x/3.0])

   ! a NaN input returns NaN
   line='NAN'
   read(line,*) nan
   print *, sinh(nan)

   ! a Inf input returns Inf
   line='Infinity'
   read(line,*) inf
   print *, sinh(inf)

   ! an overflow returns Inf
   x=huge(0.0d0)
   print *, sinh(x)

end program demo_sinh

Results:

  -1.1752011936438014
  -1.1752011936438014
  -1.1752011936438014       -3.6268604078470190      -0.33954055725615012
                       NaN
                  Infinity
                  Infinity

Standard#

Fortran 95 , for a complex argument Fortran 2008

See Also#

asinh(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

tan#

Name#

tan(3) - [MATHEMATICS:TRIGONOMETRIC] Tangent function

Synopsis#

result = tan(x)
 elemental TYPE(kind=KIND) function tan(x)

  TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • the TYPE of x may be real or complex of any supported kind

  • The returned value will be of the same type and kind as the argument x.

Description#

tan(3) computes the tangent of x.

Options#

  • x

    The angle in radians to compute the tangent of for real input. If x is of type complex, its real part is regarded as a value in radians.

Result#

The return value is the tangent of the value x.

Examples#

Sample program:

program demo_tan
use, intrinsic :: iso_fortran_env, only : real_kinds, &
& real32, real64, real128
implicit none
real(kind=real64) :: x = 0.165_real64
     write(*,*)x, tan(x)
end program demo_tan

Results:

     0.16500000000000001       0.16651386310913616

Standard#

FORTRAN 77 . For a complex argument, Fortran 2008 .

See Also#

atan(3), atan2(3), cos(3), sin(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

tanh#

Name#

tanh(3) - [MATHEMATICS:TRIGONOMETRIC] Hyperbolic tangent function

Synopsis#

    result = tanh(x)
     elemental TYPE(kind=KIND) function tanh(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • x may be real or complex and any associated kind supported by the processor.

  • The returned value will be of the same type and kind as the argument.

Description#

tanh(3) computes the hyperbolic tangent of x.

Options#

  • x

    The value to compute the Hyperbolic tangent of.

Result#

Returns the hyperbolic tangent of x.

If x is complex, the imaginary part of the result is regarded as a radian value.

If x is real, the return value lies in the range

      -1 <= tanh(x) <= 1.

Examples#

Sample program:

program demo_tanh
use, intrinsic :: iso_fortran_env, only : &
& real_kinds, real32, real64, real128
implicit none
real(kind=real64) :: x = 2.1_real64
   write(*,*)x, tanh(x)
end program demo_tanh

Results:

      2.1000000000000001       0.97045193661345386

Standard#

FORTRAN 77 , for a complex argument Fortran 2008

See Also#

atanh(3)

Resources#

fortran-lang intrinsic descriptions

random_number#

Name#

random_number(3) - [MATHEMATICS:RANDOM] Pseudo-random number

Synopsis#

    call random_number(harvest)
     subroutine random_number(harvest)

      real,intent(out) :: harvest(..)

Characteristics#

  • harvest and the result are default real variables

Description#

random_number(3) returns a single pseudorandom number or an array of pseudorandom numbers from the uniform distribution over the range 0 <= x < 1.

Options#

  • harvest

    Shall be a scalar or an array of type real.

Examples#

Sample program:

program demo_random_number
use, intrinsic :: iso_fortran_env, only : dp=>real64
implicit none
integer, allocatable :: seed(:)
integer              :: n
integer              :: first,last
integer              :: i
integer              :: rand_int
integer,allocatable  :: count(:)
real(kind=dp)        :: rand_val
   call random_seed(size = n)
   allocate(seed(n))
   call random_seed(get=seed)
   first=1
   last=10
   allocate(count(last-first+1))
   ! To have a discrete uniform distribution on the integers
   ! [first, first+1, ..., last-1, last] carve the continuous
   ! distribution up into last+1-first equal sized chunks,
   ! mapping each chunk to an integer.
   !
   ! One way is:
   !   call random_number(rand_val)
   ! choose one from last-first+1 integers
   !   rand_int = first + FLOOR((last+1-first)*rand_val)
      count=0
      ! generate a lot of random integers from 1 to 10 and count them.
      ! with a large number of values you should get about the same
      ! number of each value
      do i=1,100000000
         call random_number(rand_val)
         rand_int=first+floor((last+1-first)*rand_val)
         if(rand_int.ge.first.and.rand_int.le.last)then
            count(rand_int)=count(rand_int)+1
         else
            write(*,*)rand_int,' is out of range'
         endif
      enddo
      write(*,'(i0,1x,i0)')(i,count(i),i=1,size(count))
end program demo_random_number

Results:

   1 10003588
   2 10000104
   3 10000169
   4 9997996
   5 9995349
   6 10001304
   7 10001909
   8 9999133
   9 10000252
   10 10000196

Standard#

Fortran 95

See Also#

random_seed(3)

fortran-lang intrinsic descriptions

random_seed#

Name#

random_seed(3) - [MATHEMATICS:RANDOM] Initialize a pseudo-random number sequence

Synopsis#

    call random_seed( [size] [,put] [,get] )
     subroutine random_seed( size, put, get )

      integer,intent(out),optional :: size
      integer,intent(in),optional :: put(*)
      integer,intent(out),optional :: get(*)

Characteristics#

  • size a scalar default integer

  • put a rank-one default integer array

  • get a rank-one default integer array

  • the result

Description#

random_seed(3) restarts or queries the state of the pseudorandom number generator used by random_number.

If random_seed is called without arguments, it is seeded with random data retrieved from the operating system.

Options#

  • size

    specifies the minimum size of the arrays used with the put and get arguments.

  • put

    the size of the array must be larger than or equal to the number returned by the size argument.

  • get

    It is intent(out) and the size of the array must be larger than or equal to the number returned by the size argument.

Examples#

Sample program:

program demo_random_seed
implicit none
integer, allocatable :: seed(:)
integer :: n

   call random_seed(size = n)
   allocate(seed(n))
   call random_seed(get=seed)
   write (*, *) seed

end program demo_random_seed

Results:

     -674862499 -1750483360  -183136071  -317862567   682500039
     349459   344020729 -1725483289

Standard#

Fortran 95

See Also#

random_number(3)

fortran-lang intrinsic descriptions

exp#

Name#

exp(3) - [MATHEMATICS] Base-e exponential function

Synopsis#

    result = exp(x)
     elemental TYPE(kind=KIND) function exp(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • x may be real or complex of any kind.

  • The return value has the same type and kind as x.

Description#

exp(3) returns the value of e (the base of natural logarithms) raised to the power of x.

e” is also known as Euler’s constant.

If x is of type complex, its imaginary part is regarded as a value in radians such that if (see Euler’s formula):

    cx=(re,im)

then

    exp(cx) = exp(re) * cmplx(cos(im),sin(im),kind=kind(cx))

Since exp(3) is the inverse function of log(3) the maximum valid magnitude of the real component of x is log(huge(x)).

Options#

  • x

    The type shall be real or complex.

Result#

The value of the result is e**x where e is Euler’s constant.

If x is of type complex, its imaginary part is regarded as a value in radians.

Examples#

Sample program:

program demo_exp
implicit none
real :: x, re, im
complex :: cx

   x = 1.0
   write(*,*)"Euler's constant is approximately",exp(x)

   !! complex values
   ! given
   re=3.0
   im=4.0
   cx=cmplx(re,im)

   ! complex results from complex arguments are Related to Euler's formula
   write(*,*)'given the complex value ',cx
   write(*,*)'exp(x) is',exp(cx)
   write(*,*)'is the same as',exp(re)*cmplx(cos(im),sin(im),kind=kind(cx))

   ! exp(3) is the inverse function of log(3) so
   ! the real component of the input must be less than or equal to
   write(*,*)'maximum real component',log(huge(0.0))
   ! or for double precision
   write(*,*)'maximum doubleprecision component',log(huge(0.0d0))

   ! but since the imaginary component is passed to the cos(3) and sin(3)
   ! functions the imaginary component can be any real value

end program demo_exp

Results:

 Euler's constant is approximately   2.718282
 given the complex value  (3.000000,4.000000)
 exp(x) is (-13.12878,-15.20078)
 is the same as (-13.12878,-15.20078)
 maximum real component   88.72284
 maximum doubleprecision component   709.782712893384

Standard#

FORTRAN 77

See Also#

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

log#

Name#

log(3) - [MATHEMATICS] Natural logarithm

Synopsis#

  result = log(x)
   elemental TYPE(kind=KIND) function log(x)

    TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • x may be any real or complex kind.

  • the result is the same type and characteristics as x.

Description#

log(3) computes the natural logarithm of x, i.e. the logarithm to the base “e”.

Options#

  • x

    The value to compute the natural log of. If x is real, its value shall be greater than zero. If x is complex, its value shall not be zero.

Result#

The natural logarithm of x. If x is the complex value (r,i) , the imaginary part “i” is in the range

    -PI < i <= PI

If the real part of x is less than zero and the imaginary part of x is zero, then the imaginary part of the result is approximately PI if the imaginary part of PI is positive real zero or the processor does not distinguish between positive and negative real zero, and approximately -PI if the imaginary part of x is negative real zero.

Examples#

Sample program:

program demo_log
implicit none
  real(kind(0.0d0)) :: x = 2.71828182845904518d0
  complex :: z = (1.0, 2.0)
  write(*,*)x, log(x)    ! will yield (approximately) 1
  write(*,*)z, log(z)
end program demo_log

Results:

      2.7182818284590451        1.0000000000000000
   (1.00000000,2.00000000) (0.804718971,1.10714877)

Standard#

FORTRAN 77

See also#

****(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

log10#

Name#

log10(3) - [MATHEMATICS] Base 10 or common logarithm

Synopsis#

    result = log10(x)
     elemental real(kind=KIND) function log10(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x may be any kind of real value

  • the result is the same type and characteristics as x.

Description#

log10(3) computes the base 10 logarithm of x. This is generally called the “common logarithm”.

Options#

  • x

    A real value > 0 to take the log of.

Result#

The logarithm to base 10 of x

Examples#

Sample program:

program demo_log10
use, intrinsic :: iso_fortran_env, only : real_kinds, &
 & real32, real64, real128
implicit none
real(kind=real64) :: x = 10.0_real64

   x = log10(x)
   write(*,'(*(g0))')'log10(',x,') is ',log10(x)

   ! elemental
   write(*, *)log10([1.0, 10.0, 100.0, 1000.0, 10000.0, &
                     & 100000.0, 1000000.0, 10000000.0])

end program demo_log10

Results:

 > log10(1.000000000000000) is .000000000000000
 >   0.0000000E+00   1.000000       2.000000       3.000000       4.000000
 >    5.000000       6.000000       7.000000

Standard#

FORTRAN 77

See also#

****(3)

fortran-lang intrinsic descriptions

sqrt#

Name#

sqrt(3) - [MATHEMATICS] Square-root function

Synopsis#

    result = sqrt(x)
     elemental TYPE(kind=KIND) function sqrt(x)

      TYPE(kind=KIND),intent(in) :: x

Characteristics#

  • TYPE may be real or complex.

  • KIND may be any kind valid for the declared type.

  • the result has the same characteristics as x.

Description#

sqrt(3) computes the principal square root of x.

The number whose square root is being considered is known as the radicand.

In mathematics, a square root of a radicand x is a number y such that y*y = x.

Every nonnegative radicand x has two square roots of the same unique magnitude, one positive and one negative. The nonnegative square root is called the principal square root.

The principal square root of 9 is 3, for example, even though (-3)*(-3) is also 9.

Square roots of negative numbers are a special case of complex numbers, where with complex input the components of the radicand need not be positive in order to have a valid square root.

Options#

  • x

    The radicand to find the principal square root of. If x is real its value must be greater than or equal to zero.

Result#

The principal square root of x is returned.

For a complex result the real part is greater than or equal to zero.

When the real part of the result is zero, the imaginary part has the same sign as the imaginary part of x.

Examples#

Sample program:

program demo_sqrt
use, intrinsic :: iso_fortran_env, only : real_kinds, &
 & real32, real64, real128
implicit none
real(kind=real64) :: x, x2
complex :: z, z2

  ! basics
   x = 2.0_real64
   ! complex
   z = (1.0, 2.0)
   write(*,*)'input values ',x,z

   x2 = sqrt(x)
   z2 = sqrt(z)
   write(*,*)'output values ',x2,z2

  ! elemental
  write(*,*)'elemental',sqrt([64.0,121.0,30.0])

  ! alternatives
   x2 = x**0.5
   z2 = z**0.5
   write(*,*)'alternatively',x2,z2

end program demo_sqrt

Results:

    input values    2.00000000000000      (1.000000,2.000000)
    output values    1.41421356237310      (1.272020,0.7861513)
    elemental   8.000000       11.00000       5.477226
    alternatively   1.41421356237310      (1.272020,0.7861513)

Standard#

FORTRAN 77

See also#

exp(3), log(3), log10(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

hypot#

Name#

hypot(3) - [MATHEMATICS] Returns the Euclidean distance - the distance between a point and the origin.

Synopsis#

    result = hypot(x, y)
     elemental real(kind=KIND) function hypot(x,y)

      real(kind=KIND),intent(in) :: x
      real(kind=KIND),intent(in) :: y

Characteristics#

  • x,y and the result shall all be real and of the same kind.

Description#

hypot(3) is referred to as the Euclidean distance function. It is equal to

sqrt(x**2+y**2)

without undue underflow or overflow.

In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between two points.

hypot(x,y) returns the distance between the point <x,y> and the origin.

Options#

  • x

    The type shall be real.

  • y

    The type and kind type parameter shall be the same as x.

Result#

The return value has the same type and kind type parameter as x.

The result is the positive magnitude of the distance of the point <x,y> from the origin <0.0,0.0> .

Examples#

Sample program:

program demo_hypot
use, intrinsic :: iso_fortran_env, only : &
 & real_kinds, real32, real64, real128
implicit none
real(kind=real32) :: x, y
real(kind=real32),allocatable :: xs(:), ys(:)
integer :: i
character(len=*),parameter :: f='(a,/,SP,*(3x,g0,1x,g0:,/))'

   x = 1.e0_real32
   y = 0.5e0_real32

   write(*,*)
   write(*,'(*(g0))')'point <',x,',',y,'> is ',hypot(x,y)
   write(*,'(*(g0))')'units away from the origin'
   write(*,*)

   ! elemental
   xs=[  x,  x**2,  x*10.0,  x*15.0, -x**2  ]
   ys=[  y,  y**2, -y*20.0,  y**2,   -y**2  ]

   write(*,f)"the points",(xs(i),ys(i),i=1,size(xs))
   write(*,f)"have distances from the origin of ",hypot(xs,ys)
   write(*,f)"the closest is",minval(hypot(xs,ys))

end program demo_hypot

Results:

   point <1.00000000,0.500000000> is 1.11803401
   units away from the origin

   the points
      +1.00000000 +0.500000000
      +1.00000000 +0.250000000
      +10.0000000 -10.0000000
      +15.0000000 +0.250000000
      -1.00000000 -0.250000000
   have distances from the origin of
      +1.11803401 +1.03077638
      +14.1421356 +15.0020828
      +1.03077638
   the closest is
      +1.03077638

Standard#

Fortran 2008

See also#

****(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

bessel_j0#

Name#

bessel_j0(3) - [MATHEMATICS] Bessel function of the first kind of order 0

Synopsis#

    result = bessel_j0(x)
     elemental real(kind=KIND) function bessel_j0(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • KIND may be any KIND supported by the real type.

  • The result is the same type and kind as x.

Description#

bessel_j0(3) computes the Bessel function of the first kind of order 0 of x.

Options#

  • x

    The value to operate on.

Result#

the Bessel function of the first kind of order 0 of x. The result lies in the range -0.4027 <= bessel(0,x) <= 1.

Examples#

Sample program:

program demo_bessel_j0
use, intrinsic :: iso_fortran_env, only : real_kinds, &
& real32, real64, real128
   implicit none
   real(kind=real64) :: x
   x = 0.0_real64
   x = bessel_j0(x)
   write(*,*)x
end program demo_bessel_j0

Results:

      1.0000000000000000

Standard#

Fortran 2008

See Also#

bessel_j1(3), bessel_jn(3), bessel_y0(3), bessel_y1(3), bessel_yn(3)

fortran-lang intrinsic descriptions

bessel_j1#

Name#

bessel_j1(3) - [MATHEMATICS] Bessel function of the first kind of order 1

Synopsis#

    result = bessel_j1(x)
     elemental real(kind=KIND) function bessel_j1(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • KIND may be any supported real KIND.

  • the result is of the same type and kind as x

Description#

bessel_j1(3) computes the Bessel function of the first kind of order 1 of x.

Options#

  • x

    The type shall be real.

Result#

The return value is of type real and lies in the range -0.5818 <= bessel(0,x) <= 0.5818 . It has the same kind as x.

Examples#

Sample program:

program demo_bessel_j1
use, intrinsic :: iso_fortran_env, only : real_kinds, &
 & real32, real64, real128
implicit none
real(kind=real64) :: x = 1.0_real64
   x = bessel_j1(x)
   write(*,*)x
end program demo_bessel_j1

Results:

     0.44005058574493350

Standard#

Fortran 2008

See Also#

bessel_j0(3), bessel_jn(3), bessel_y0(3), bessel_y1(3), bessel_yn(3)

fortran-lang intrinsic descriptions

bessel_jn#

Name#

bessel_jn(3) - [MATHEMATICS] Bessel function of the first kind

Synopsis#

    result = bessel_jn(n, x)
     elemental real(kind=KIND) function bessel_jn(n,x)

      integer(kind=**),intent(in) :: n
      real(kind=KIND),intent(in) :: x
  • KIND may be any valid value for type real

  • x is real

  • The return value has the same type and kind as x.

    result = bessel_jn(n1, n2, x)
     real(kind=KIND) function bessel_jn(n1, n2, ,x)

     integer(kind=**),intent(in) :: n1
     integer(kind=**),intent(in) :: n2
     real(kind=KIND),intent(in) :: x
  • n1 is integer

  • n2 is integer

  • x is real

  • The return value has the same type and kind as x.

Description#

bessel_jn( n, x ) computes the Bessel function of the first kind of order n of x.

bessel_jn(n1, n2, x) returns an array with the Bessel function|Bessel functions of the first kind of the orders n1 to n2.

Options#

  • n

    a non-negative scalar integer..

  • n1

    a non-negative scalar integer.

  • n2

    a non-negative scalar integer.

  • x

    Shall be a scalar for bessel_jn(n,x) or an array For bessel_jn(n1, n2, x).

Result#

The result value of BESSEL_JN (N, X) is a processor-dependent approximation to the Bessel function of the first kind and order N of X.

The result of BESSEL_JN (N1, N2, X) is a rank-one array with extent MAX (N2-N1+1, 0). Element i of the result value of BESSEL_JN (N1, N2, X) is a processor-dependent approximation to the Bessel function of the first kind and order N1+i-1 of X.

Examples#

Sample program:

program demo_bessel_jn
use, intrinsic :: iso_fortran_env, only : real_kinds, &
   & real32, real64, real128
implicit none
real(kind=real64) :: x = 1.0_real64
    x = bessel_jn(5,x)
    write(*,*)x
end program demo_bessel_jn

Results:

      2.4975773021123450E-004

Standard#

Fortran 2008

See Also#

bessel_j0(3), bessel_j1(3), bessel_y0(3), bessel_y1(3), bessel_yn(3)

fortran-lang intrinsic descriptions

bessel_y0#

Name#

bessel_y0(3) - [MATHEMATICS] Bessel function of the second kind of order 0

Synopsis#

    result = bessel_y0(x)
     elemental real(kind=KIND) function bessel_y0(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • KIND may be any supported real KIND.

  • the result characteristics (type, kind) are the same as x

Description#

bessel_y0(3) computes the Bessel function of the second kind of order 0 of x.

Options#

  • x

    The type shall be real. Its value shall be greater than zero.

Result#

The return value is of type real. It has the same kind as x.

Examples#

Sample program:

program demo_bessel_y0
use, intrinsic :: iso_fortran_env, only : real_kinds, &
& real32, real64, real128
implicit none
  real(kind=real64) :: x = 0.0_real64
  x = bessel_y0(x)
  write(*,*)x
end program demo_bessel_y0

Results:

                    -Infinity

Standard#

Fortran 2008

See Also#

bessel_j0(3), bessel_j1(3), bessel_jn(3), bessel_y1(3), bessel_yn(3)

fortran-lang intrinsic descriptions

bessel_y1#

Name#

bessel_y1(3) - [MATHEMATICS] Bessel function of the second kind of order 1

Synopsis#

    result = bessel_y1(x)
     elemental real(kind=KIND) function bessel_y1(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • KIND may be any supported real KIND.

  • the characteristics (type, kind) of the result are the same as x

Description#

bessel_y1(3) computes the Bessel function of the second kind of order 1 of x.

Options#

  • x

    The type shall be real. Its value shall be greater than zero.

Result#

The return value is real. It has the same kind as x.

Examples#

Sample program:

program demo_bessel_y1
use, intrinsic :: iso_fortran_env, only : real_kinds, &
& real32, real64, real128
implicit none
  real(kind=real64) :: x = 1.0_real64
  write(*,*)x, bessel_y1(x)
end program demo_bessel_y1

Results:

 >    1.00000000000000      -0.781212821300289

Standard#

Fortran 2008

See Also#

bessel_j0(3), bessel_j1(3), bessel_jn(3), bessel_y0(3), bessel_yn(3)

fortran-lang intrinsic descriptions

bessel_yn#

Name#

bessel_yn(3) - [MATHEMATICS] Bessel function of the second kind

Synopsis#

    result = bessel_yn(n, x)
     elemental real(kind=KIND) function bessel_yn(n,x)

      integer(kind=**),intent(in) :: n
      real(kind=KIND),intent(in) :: x

Characteristics#

  • n is integer

  • x is real

  • The return value has the same type and kind as x.

    result = bessel_yn(n1, n2, x)
     real(kind=KIND) function bessel_yn(n1, n2, ,x)

      integer(kind=**),intent(in) :: n1
      integer(kind=**),intent(in) :: n2
      real(kind=KIND),intent(in) :: x
  • n1 is integer

  • n2 is integer

  • x is real

  • The return value has the same type and kind as x.

Description#

bessel_yn(n, x) computes the Bessel function of the second kind of order n of x.

bessel_yn(n1, n2, x) returns an array with the Bessel function|Bessel functions of the first kind of the orders n1 to n2.

Options#

  • n

    Shall be a scalar or an array of type integer and non-negative.

  • n1

    Shall be a non-negative scalar of type integer and non-negative.

  • n2

    Shall be a non-negative scalar of type integer and non-negative.

  • x

    A real non-negative value. Note bessel_yn(n1, n2, x) is not elemental, in which case it must be a scalar.

Result#

The result value of BESSEL_YN (N, X) is a processor-dependent approximation to the Bessel function of the second kind and order N of X.

The result of BESSEL_YN (N1, N2, X) is a rank-one array with extent MAX (N2-N1+1, 0). Element i of the result value of BESSEL_YN (N1, N2, X) is a processor-dependent approximation to the Bessel function of the second kind and order N1+i-1 of X.

Examples#

Sample program:

program demo_bessel_yn
use, intrinsic :: iso_fortran_env, only : real_kinds, &
& real32, real64, real128
implicit none
real(kind=real64) :: x = 1.0_real64
  write(*,*) x,bessel_yn(5,x)
end program demo_bessel_yn

Results:

      1.0000000000000000       -260.40586662581222

Standard#

Fortran 2008

See Also#

bessel_j0(3), bessel_j1(3), bessel_jn(3), bessel_y0(3), bessel_y1(3)

fortran-lang intrinsic descriptions

erf#

Name#

erf(3) - [MATHEMATICS] Error function

Synopsis#

    result = erf(x)
     elemental real(kind=KIND) function erf(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x is of type real

  • The result is of the same type and kind as x.

Description#

erf(3) computes the error function of x, defined as

\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt. \]

Options#

  • x

    The type shall be real.

Result#

The return value is of type real, of the same kind as x and lies in the range -1 <= erf(x) <= 1 .

Examples#

Sample program:

program demo_erf
use, intrinsic :: iso_fortran_env, only : real_kinds, &
 & real32, real64, real128
implicit none
real(kind=real64) :: x = 0.17_real64
    write(*,*)x, erf(x)
end program demo_erf

Results:

     0.17000000000000001       0.18999246120180879

Standard#

Fortran 2008

See also#

erfc(3), erf_scaled(3)

Resources#

fortran-lang intrinsic descriptions

erfc#

Name#

erfc(3) - [MATHEMATICS] Complementary error function

Synopsis#

    result = erfc(x)
     elemental real(kind=KIND) function erfc(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x is of type real and any valid kind

  • KIND is any value valid for type real

  • the result has the same characteristics as x

Description#

erfc(3) computes the complementary error function of x. Simply put this is equivalent to 1 - erf(x), but erfc is provided because of the extreme loss of relative accuracy if erf(x) is called for large x and the result is subtracted from 1.

erfc(x) is defined as

\[ \text{erfc}(x) = 1 - \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt. \]

Options#

  • x

    The type shall be real.

Result#

The return value is of type real and of the same kind as x. It lies in the range

     0 \<= **erfc**(x) \<= 2.

and is a processor-dependent approximation to the complementary error function of x ( **1-erf(x) ).

Examples#

Sample program:

program demo_erfc
use, intrinsic :: iso_fortran_env, only : &
 & real_kinds, real32, real64, real128
implicit none
real(kind=real64) :: x = 0.17_real64
   write(*,'(*(g0))')'X=',x, ' ERFC(X)=',erfc(x)
   write(*,'(*(g0))')'equivalently 1-ERF(X)=',1-erf(x)
end program demo_erfc

Results:

 > X=.1700000000000000 ERFC(X)=.8100075387981912
 > equivalently 1-ERF(X)=.8100075387981912

Standard#

Fortran 2008

See also#

erf(3) erf_scaled(3)

Resources#

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

erfc_scaled#

Name#

erfc_scaled(3) - [MATHEMATICS] Scaled complementary error function

Synopsis#

    result = erfc_scaled(x)
     elemental real(kind=KIND) function erfc_scaled(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x is of type real of any valid kind

  • KIND is any kind valid for a real type

  • the result has the same characteristics as x

Description#

erfc_scaled(3) computes the exponentially-scaled complementary error function of x:

\[ e^{x^2} \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt. \]

erfc_scaled(x)=exp(x*x)erfc(x)

NOTE1#

The complementary error function is asymptotic to exp(-X2)/(X/PI). As such it underflows at approximately X >= 9 when using ISO/IEC/IEEE 60559:2011 single precision arithmetic. The exponentially-scaled complementary error function is asymptotic to 1/(X PI). As such it does not underflow until X > HUGE (X)/PI.

Options#

  • x the value to apply the erfc function to

Result#

The approximation to the exponentially-scaled complementary error function of x

Examples#

Sample program:

program demo_erfc_scaled
implicit none
real(kind(0.0d0)) :: x = 0.17d0
   x = erfc_scaled(x)
   print *, x
end program demo_erfc_scaled

Results:

 >   0.833758302149981

Standard#

Fortran 2008

See also#

erf(3), exp(3), erfc(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost

gamma#

Name#

gamma(3) - [MATHEMATICS] Gamma function, which yields factorials for positive whole numbers

Synopsis#

    result = gamma(x)
     elemental real(kind=KIND) function gamma( x)

      type(real,kind=KIND),intent(in) :: x

Characteristics#

  • x is a real value

  • returns a real value with the kind as x.

Description#

gamma(x) computes Gamma of x. For positive whole number values of n the Gamma function can be used to calculate factorials, as (n-1)! == gamma(real(n)). That is

n! == gamma(real(n+1))
\[\begin{split} \\__Gamma__(x) = \\int\_0\*\*\\infty t\*\*{x-1}{\\mathrm{e}}\*\*{__-t__}\\,{\\mathrm{d}}t \end{split}\]

Options#

  • x

    Shall be of type real and neither zero nor a negative integer.

Result#

The return value is of type real of the same kind as x. The result has a value equal to a processor-dependent approximation to the gamma function of x.

Examples#

Sample program:

program demo_gamma
use, intrinsic :: iso_fortran_env, only : wp=>real64
implicit none
real :: x, xa(4)
integer :: i

   x = gamma(1.0)
   write(*,*)'gamma(1.0)=',x

   ! elemental
   xa=gamma([1.0,2.0,3.0,4.0])
   write(*,*)xa
   write(*,*)

   ! gamma(3) is related to the factorial function
   do i=1,20
      ! check value is not too big for default integer type
      if(factorial(i).gt.huge(0))then
         write(*,*)i,factorial(i)
      else
         write(*,*)i,factorial(i),int(factorial(i))
      endif
   enddo
   ! more factorials
   FAC: block
   integer,parameter :: n(*)=[0,1,5,11,170]
   integer :: j
      do j=1,size(n)
         write(*,'(*(g0,1x))')'factorial of', n(j),' is ', &
          & product([(real(i,kind=wp),i=1,n(j))]),  &
          & gamma(real(n(j)+1,kind=wp))
      enddo
   endblock FAC

   contains
   function factorial(i) result(f)
   integer,parameter :: dp=kind(0d0)
   integer,intent(in) :: i
   real :: f
      if(i.le.0)then
         write(*,'(*(g0))')'<ERROR> gamma(3) function value ',i,' <= 0'
         stop      '<STOP> bad value in gamma function'
      endif
      f=gamma(real(i+1))
   end function factorial
end program demo_gamma

Results:

    gamma(1.0)=   1.000000
      1.000000       1.000000       2.000000       6.000000

              1   1.000000               1
              2   2.000000               2
              3   6.000000               6
              4   24.00000              24
              5   120.0000             120
              6   720.0000             720
              7   5040.000            5040
              8   40320.00           40320
              9   362880.0          362880
             10   3628800.         3628800
             11  3.9916800E+07    39916800
             12  4.7900160E+08   479001600
             13  6.2270208E+09
             14  8.7178289E+10
             15  1.3076744E+12
             16  2.0922791E+13
             17  3.5568741E+14
             18  6.4023735E+15
             19  1.2164510E+17
             20  2.4329020E+18
   factorial of 0  is  1.000000000000000 1.000000000000000
   factorial of 1  is  1.000000000000000 1.000000000000000
   factorial of 5  is  120.0000000000000 120.0000000000000
   factorial of 11  is  39916800.00000000 39916800.00000000
   factorial of 170  is  .7257415615307994E+307 .7257415615307999E+307

Standard#

Fortran 2008

See Also#

Logarithm of the Gamma function: log_gamma(3)

Resources#

Wikipedia: Gamma_function

fortran-lang intrinsic descriptions

log_gamma#

Name#

log_gamma(3) - [MATHEMATICS] Logarithm of the absolute value of the Gamma function

Synopsis#

    result = log_gamma(x)
     elemental real(kind=KIND) function log_gamma(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x may be any real type

  • the return value is of same type and kind as x.

Description#

log_gamma(3) computes the natural logarithm of the absolute value of the Gamma function.

Options#

  • x

    neither negative nor zero value to render the result for.

Result#

The result has a value equal to a processor-dependent approximation to the natural logarithm of the absolute value of the gamma function of x.

Examples#

Sample program:

program demo_log_gamma
implicit none
real :: x = 1.0
   write(*,*)x,log_gamma(x) ! returns 0.0
   write(*,*)x,log_gamma(3.0) ! returns 0.693 (approximately)
end program demo_log_gamma

Results:

 >    1.000000      0.0000000E+00
 >    1.000000      0.6931472

Standard#

Fortran 2008

See Also#

Gamma function: gamma(3)

fortran-lang intrinsic descriptions

log_gamma#

Name#

log_gamma(3) - [MATHEMATICS] Logarithm of the absolute value of the Gamma function

Synopsis#

    result = log_gamma(x)
     elemental real(kind=KIND) function log_gamma(x)

      real(kind=KIND),intent(in) :: x

Characteristics#

  • x may be any real type

  • the return value is of same type and kind as x.

Description#

log_gamma(3) computes the natural logarithm of the absolute value of the Gamma function.

Options#

  • x

    neither negative nor zero value to render the result for.

Result#

The result has a value equal to a processor-dependent approximation to the natural logarithm of the absolute value of the gamma function of x.

Examples#

Sample program:

program demo_log_gamma
implicit none
real :: x = 1.0
   write(*,*)x,log_gamma(x) ! returns 0.0
   write(*,*)x,log_gamma(3.0) ! returns 0.693 (approximately)
end program demo_log_gamma

Results:

 >    1.000000      0.0000000E+00
 >    1.000000      0.6931472

Standard#

Fortran 2008

See Also#

Gamma function: gamma(3)

fortran-lang intrinsic descriptions

norm2#

Name#

norm2(3) - [MATHEMATICS] Euclidean vector norm

Synopsis#

    result = norm2(array, [dim])
     real(kind=KIND) function norm2(array, dim)

      real(kind=KIND),intent(in) :: array(..)
      integer(kind=**),intent(in),optional :: dim

Characteristics#

  • array shall be an array of type real.

  • dim shall be a scalar of type integer

  • The result is of the same type as array.

Description#

norm2(3) calculates the Euclidean vector norm (L_2 norm or generalized L norm) of array along dimension dim.

Options#

  • array

    the array of input values for the L_2 norm computations

  • dim

    a value in the range from 1 to rank(array).

Result#

If dim is absent, a scalar with the square root of the sum of squares of the elements of array is returned.

Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension DIM dropped is returned.

  Case (i):     The result of NORM2 (X) has a value equal to a
                processor-dependent approximation to the generalized
                L norm of X, which is the square root of the sum of
                the squares of the elements of X. If X has size zero,
                the result has the value zero.

  Case (ii):    The result of NORM2 (X, DIM=DIM) has a value equal
                to that of NORM2 (X) if X has rank one. Otherwise,
                the resulting array is reduced in rank with dimension
                **dim** removed, and each remaining elment is the
                result of NORM2(X) for the values along dimension
                **dim**.

It is recommended that the processor compute the result without undue overflow or underflow.

Examples#

Sample program:

program demo_norm2
implicit none
integer :: i
real :: x(2,3) = reshape([ &
   1, 2, 3, &
   4, 5, 6  &
   ],shape(x),order=[2,1])

  write(*,*) 'input in row-column order'
  write(*,*) 'x='
  write(*,'(4x,3f4.0)')transpose(x)
  write(*,*)
  write(*,*) 'norm2(x)=',norm2(x)
  write(*,*) 'which is equivalent to'
  write(*,*) 'sqrt(sum(x**2))=',sqrt(sum(x**2))
  write(*,*)
  write(*,*) 'for reference the array squared is'
  write(*,*) 'x**2='
  write(*,'(4x,3f4.0)')transpose(x**2)
  write(*,*)
  write(*,*) 'norm2(x,dim=1)=',norm2(x,dim=1)
  write(*,*) 'norm2(x,dim=2)=',norm2(x,dim=2)
  write(*,*) '(sqrt(sum(x(:,i)**2)),i=1,3)=',(sqrt(sum(x(:,i)**2)),i=1,3)
  write(*,*) '(sqrt(sum(x(i,:)**2)),i=1,2)=',(sqrt(sum(x(i,:)**2)),i=1,2)

end program demo_norm2

Results:

 >  input in row-column order
 >  x=
 >       1.  2.  3.
 >       4.  5.  6.
 >
 >  norm2(x)=   9.539392
 >  which is equivalent to
 >  sqrt(sum(x**2))=   9.539392
 >
 >  for reference the array squared is
 >  x**2=
 >       1.  4.  9.
 >      16. 25. 36.
 >
 >  norm2(x,dim=1)=   4.123106       5.385165       6.708204
 >  norm2(x,dim=2)=   3.741657       8.774964
 >  (sqrt(sum(x(:,i)**2)),i=1,3)=   4.123106       5.385165       6.708204
 >  (sqrt(sum(x(i,:)**2)),i=1,2)=   3.741657       8.774964

Standard#

Fortran 2008

See Also#

product(3), sum(3), hypot(3)

fortran-lang intrinsic descriptions (license: MIT) @urbanjost