stdlib_math modulestdlib_math module provides general purpose mathematical functions.
clip functionReturns a value which lies in the given interval [xmin, xmax] (interval is xmin and xmax inclusive) and is closest to the input value x.
res = clip (x, xmin, xmax)
Experimental
Elemental function.
x: scalar of either integer or real type. This argument is intent(in).
xmin: scalar of either integer or real type. This argument is intent(in).
xmax: scalar of either integer or real type, which must be greater than or equal to xmin. This argument is intent(in).
Note: All arguments must have same type and same kind.
The output is a scalar of type and kind same as to that of the arguments.
Here inputs are of type integer and kind int32
program example_clip_integer
use stdlib_math, only: clip
use stdlib_kinds, only: int32
implicit none
integer(int32) :: x
integer(int32) :: xmin
integer(int32) :: xmax
integer(int32) :: clipped_value
xmin = -5_int32
xmax = 5_int32
x = 12_int32
clipped_value = clip(x, xmin, xmax)
! clipped_value <- 5
end program example_clip_integer
Here inputs are of type real and kind sp
program example_clip_real
use stdlib_math, only: clip
use stdlib_kinds, only: sp
implicit none
real(sp) :: x
real(sp) :: xmin
real(sp) :: xmax
real(sp) :: clipped_value
xmin = -5.769_sp
xmax = 3.025_sp
x = 3.025_sp
clipped_value = clip(x, xmin, xmax)
! clipped_value <- 3.02500010
end program example_clip_real
gcd functionReturns the greatest common divisor of two integers.
res = gcd (a, b)
Experimental
Elemental function.
a: One integer with intent(in) to get the divisor for.
b: Another integer with intent(in) to get the divisor for.
Note: All arguments must be integers of the same kind.
Returns an integer of the same kind as that of the arguments.
program example_gcd
use stdlib_math, only: gcd
implicit none
integer :: a, b, c
a = 48
b = 18
c = gcd(a, b) ! returns 6
end program example_gcd
linspace - Create a linearly spaced rank one arrayReturns a linearly spaced rank 1 array from [start, end]. Optionally, you can specify the length of the returned array by passing n.
res = linspace (start, end [, n])
Experimental
Pure function.
start: Shall be scalar of any numeric type or kind. This argument is intent(in).
end: Shall be the same type and kind as start. This argument is intent(in).
n: Shall be an integer specifying the length of the output. This argument is optional and intent(in).
The output is a rank 1 array whose length is either 100 (default value) or n.
If n == 1, return a rank 1 array whose only element is end.
If n <= 0, return a rank 1 array with length 0.
If start/end are real or complex types, the result will be of the same type and kind as start/end.
If start/end are integer types, the result will default to a real(dp) array.
Here inputs are of type complex and kind dp
program example_linspace_complex
use stdlib_math, only: linspace
use stdlib_kinds, only: dp
implicit none
complex(dp) :: start = cmplx(10.0_dp, 5.0_dp, kind=dp)
complex(dp) :: end = cmplx(-10.0_dp, 15.0_dp, kind=dp)
complex(dp) :: z(11)
z = linspace(start, end, 11)
end program example_linspace_complex
Here inputs are of type integer and kind int16, with the result defaulting to real(dp).
program example_linspace_int16
use stdlib_math, only: linspace
use stdlib_kinds, only: int16, dp
implicit none
integer(int16) :: start = 10_int16
integer(int16) :: end = 23_int16
real(dp) :: r(15)
r = linspace(start, end, 15)
end program example_linspace_int16
logspace - Create a logarithmically spaced rank one arrayReturns a logarithmically spaced rank 1 array from [base^start, base^end]. The default size of the array is 50. Optionally, you can specify the length of the returned array by passing n. You can also specify the base used to compute the range (default 10).
res = logspace (start, end [, n [, base]])
Experimental
Pure function.
start: Shall be a scalar of any numeric type. All kinds are supported for real and complex arguments. For integers, only the default kind is currently implemented. This argument is intent(in).
end: Shall be the same type and kind as start. This argument is intent(in).
n: Shall be an integer specifying the length of the output. This argument is optional and intent(in).
base : Shall be a scalar of any numeric type. All kinds are supported for real and complex arguments. For integers, only the default kind is currently implemented. This argument is optional and intent(in).
The output is a rank 1 array whose length is either 50 (default value) or n.
If n == 1, return a rank 1 array whose only element is base^end.
If n <= 0, return a rank 1 array with length 0
The type and kind of the output is dependent on the type and kind of the passed parameters.
For function calls where the base is not specified: logspace(start, end)/logspace(start, end, n), the type and kind of
the output follows the same scheme as above for linspace.
If
start/endarerealorcomplextypes, theresultwill be the same type and kind asstart/end. Ifstart/endare integer types, theresultwill default to areal(dp)array.
For function calls where the base is specified, the type and kind of the result is in accordance with the following table:
start/end |
n |
base |
output |
|---|---|---|---|
real(KIND) |
Integer |
real(KIND) |
real(KIND) |
| " " | " " | complex(KIND) |
complex(KIND) |
| " " | " " | Integer |
real(KIND) |
complex(KIND) |
" " | real(KIND) |
complex(KIND) |
| " " | " " | complex(KIND) |
complex(KIND) |
| " " | " " | Integer |
complex(KIND) |
Integer |
" " | real(KIND) |
real(KIND) |
| " " | " " | complex(KIND) |
complex(KIND) |
| " " | " " | Integer |
Integer |
Here inputs are of type complex and kind dp. n and base is not specified and thus default to 50 and 10, respectively.
program example_logspace_complex
use stdlib_math, only: logspace
use stdlib_kinds, only: dp
implicit none
complex(dp) :: start = (10.0_dp, 5.0_dp)
complex(dp) :: end = (-10.0_dp, 15.0_dp)
complex(dp) :: z(11) ! Complex values raised to complex powers results in complex values
z = logspace(start, end, 11)
end program example_logspace_complex
Here inputs are of type integer and default kind. base is not specified and thus defaults to 10.
program example_logspace_int
use stdlib_math, only: logspace
use stdlib_kinds, only: dp
implicit none
integer, parameter :: start = 10
integer, parameter :: end = 23
integer, parameter :: n = 15
real(dp) :: r(n) ! Integer values raised to real powers results in real values
r = logspace(start, end, n)
end program example_logspace_int
Here start/end are of type real and double precision. base is type complex and also double precision.
program example_logspace_rstart_cbase
use stdlib_math, only: logspace
use stdlib_kinds, only: dp
implicit none
real(dp) :: start = 0.0_dp
real(dp) :: end = 3.0_dp
integer, parameter :: n = 4
complex(dp) :: base = (0.0_dp, 1.0_dp)
complex(dp) :: z(n) ! complex values raised to real powers result in complex values
z = logspace(start, end, n, base)
end program example_logspace_rstart_cbase
arange functionExperimental
Pure function.
Creates a rank-1 array of the integer/real type with fixed-spaced values of given spacing, within a given interval.
result = arange (start [, end, step])
All arguments should be the same type and kind.
start: Shall be an integer/real scalar.
This is an intent(in) argument.
The default start value is 1.
end: Shall be an integer/real scalar.
This is an intent(in) and optional argument.
The default end value is the inputted start value.
step: Shall be an integer/real scalar and large than 0.
This is an intent(in) and optional argument.
The default step value is 1.
If step = 0, the step argument will be corrected to 1/1.0 by the internal process of the arange function.
If step < 0, the step argument will be corrected to abs(step) by the internal process of the arange function.
Returns a rank-1 array of fixed-spaced values.
For integer type arguments, the length of the result vector is (end - start)/step + 1.
For real type arguments, the length of the result vector is floor((end - start)/step) + 1.
program example_math_arange
use stdlib_math, only: arange
implicit none
print *, arange(3) ! [1,2,3]
print *, arange(-1) ! [1,0,-1]
print *, arange(0, 2) ! [0,1,2]
print *, arange(1, -1) ! [1,0,-1]
print *, arange(0, 2, 2) ! [0,2]
print *, arange(3.0) ! [1.0,2.0,3.0]
print *, arange(0.0, 5.0) ! [0.0,1.0,2.0,3.0,4.0,5.0]
print *, arange(0.0, 6.0, 2.5) ! [0.0,2.5,5.0]
print *, (1.0, 1.0)*arange(3) ! [(1.0,1.0),(2.0,2.0),[3.0,3.0]]
print *, arange(0.0, 2.0, -2.0) ! [0.0,2.0]. Not recommended: `step` argument is negative!
print *, arange(0.0, 2.0, 0.0) ! [0.0,1.0,2.0]. Not recommended: `step` argument is zero!
end program example_math_arange
arg functionExperimental
Elemental function.
arg computes the phase angle (radian version) of complex scalar in the interval (-π,π].
The angles in θ are such that z = abs(z)*exp((0.0, θ)).
result = arg (z)
z: Shall be a complex scalar/array.
This is an intent(in) argument.
Returns the real type phase angle (radian version) of the complex argument z.
Notes: Although the angle of the complex number 0 is undefined, arg((0,0)) returns the value 0.
program example_math_arg
use stdlib_math, only: arg
implicit none
print *, arg((0.0, 0.0)) ! 0.0
print *, arg((3.0, 4.0)) ! 0.927
print *, arg(2.0*exp((0.0, 0.5))) ! 0.5
print *, arg([(0.0, 1.0), (1.0, 0.0), (0.0, -1.0), (-1.0, 0.0)]) ! [π/2, 0.0, -π/2, π]
end program example_math_arg
argd functionExperimental
Elemental function.
argd computes the phase angle (degree version) of complex scalar in the interval (-180.0,180.0].
The angles in θ are such that z = abs(z)*exp((0.0, θ*π/180.0)).
result = argd (z)
z: Shall be a complex scalar/array.
This is an intent(in) argument.
Returns the real type phase angle (degree version) of the complex argument z.
Notes: Although the angle of the complex number 0 is undefined, argd((0,0)) returns the value 0.
program example_math_argd
use stdlib_math, only: argd
implicit none
print *, argd((0.0, 0.0)) ! 0.0°
print *, argd((3.0, 4.0)) ! 53.1°
print *, argd(2.0*exp((0.0, 0.5))) ! 28.64°
print *, argd([(0.0, 1.0), (1.0, 0.0), (0.0, -1.0), (-1.0, 0.0)]) ! [90°, 0°, -90°, 180°]
end program example_math_argd
argpi functionExperimental
Elemental function.
argpi computes the phase angle (IEEE circular version) of complex scalar in the interval (-1.0,1.0].
The angles in θ are such that z = abs(z)*exp((0.0, θ*π)).
result = argpi (z)
z: Shall be a complex scalar/array.
This is an intent(in) argument.
Returns the real type phase angle (circular version) of the complex argument z.
Notes: Although the angle of the complex number 0 is undefined, argpi((0,0)) returns the value 0.
program example_math_argpi
use stdlib_math, only: argpi
implicit none
print *, argpi((0.0, 0.0)) ! 0.0
print *, argpi((3.0, 4.0)) ! 0.295
print *, argpi(2.0*exp((0.0, 0.5))) ! 0.159
print *, argpi([(0.0, 1.0), (1.0, 0.0), (0.0, -1.0), (-1.0, 0.0)]) ! [0.5, 0.0, -0.5, 1.0]
end program example_math_argpi
is_close functionReturns a boolean scalar/array where two scalars/arrays are element-wise equal within a tolerance.
!> For `real` type
is_close(a, b, rel_tol, abs_tol) = abs(a - b) <= max(rel_tol*(abs(a), abs(b)), abs_tol)
!> and for `complex` type
is_close(a, b, rel_tol, abs_tol) = is_close(a%re, b%re, rel_tol, abs_tol) .and. &
is_close(a%im, b%im, rel_tol, abs_tol)
bool = is_close (a, b [, rel_tol, abs_tol, equal_nan])
Experimental.
Elemental function.
Note: All real/complex arguments must have same kind.
If the value of rel_tol/abs_tol is negative (not recommended),
it will be corrected to abs(rel_tol/abs_tol) by the internal process of is_close.
a: Shall be a real/complex scalar/array.
This argument is intent(in).
b: Shall be a real/complex scalar/array.
This argument is intent(in).
rel_tol: Shall be a real scalar/array.
This argument is intent(in) and optional, which is sqrt(epsilon(..)) by default.
abs_tol: Shall be a real scalar/array.
This argument is intent(in) and optional, which is 0.0 by default.
equal_nan: Shall be a logical scalar/array.
This argument is intent(in) and optional, which is .false. by default.
Whether to compare NaN values as equal. If .true.,
NaN values in a will be considered equal to NaN values in b.
Returns a logical scalar/array.
program example_math_is_close
use stdlib_math, only: is_close
implicit none
real :: x(2) = [1, 2], y, NAN
y = -3
NAN = sqrt(y)
print *, is_close(x, [real :: 1, 2.1]) ! [T, F]
print *, is_close(2.0, 2.1, abs_tol=0.1) ! T
print *, NAN, is_close(2.0, NAN), is_close(2.0, NAN, equal_nan=.true.) ! NAN, F, F
print *, is_close(NAN, NAN), is_close(NAN, NAN, equal_nan=.true.) ! F, T
end program example_math_is_close
all_close functionReturns a boolean scalar where two arrays are element-wise equal within a tolerance.
bool = all_close (a, b [, rel_tol, abs_tol, equal_nan])
Experimental.
Pure function.
Note: All real/complex arguments must have same kind.
If the value of rel_tol/abs_tol is negative (not recommended),
it will be corrected to abs(rel_tol/abs_tol) by the internal process of all_close.
a: Shall be a real/complex array.
This argument is intent(in).
b: Shall be a real/complex array.
This argument is intent(in).
rel_tol: Shall be a real scalar.
This argument is intent(in) and optional, which is sqrt(epsilon(..)) by default.
abs_tol: Shall be a real scalar.
This argument is intent(in) and optional, which is 0.0 by default.
equal_nan: Shall be a logical scalar.
This argument is intent(in) and optional, which is .false. by default.
Whether to compare NaN values as equal. If .true.,
NaN values in a will be considered equal to NaN values in b.
Returns a logical scalar.
program example_math_all_close
use stdlib_math, only: all_close
implicit none
real :: y, NAN
complex :: z(4, 4)
y = -3
NAN = sqrt(y)
z = (1.0, 1.0)
print *, all_close(z + cmplx(1.0e-11, 1.0e-11), z) ! T
print *, NAN, all_close([NAN], [NAN]), all_close([NAN], [NAN], equal_nan=.true.)
! NAN, F, T
end program example_math_all_close
diff functionComputes differences between adjacent elements of an array.
For a rank-1 array:
y = diff (x [, n, prepend, append])
and for a rank-2 array:
y = diff (x [, n, dim, prepend, append])
Experimental.
Pure function.
x: The array to take a difference of.
Shall be a real/integer and rank-1/rank-2 array.
This argument is intent(in).
n: How many times to iteratively calculate the difference.
Shall be an integer scalar.
This argument is intent(in) and optional, and has value of 1 by default.
dim: The dimension of the input array along which to calculate the difference.
Its value must be between 1 and rank(x).
Shall be an integer scalar.
This argument is intent(in) and optional and has a value of 1 by default.
prepend, append: Arrays to prepend or append to a along axis prior to performing the difference.
The dimension and shape must match a except along axis.
Shall be a real/integer and rank-1/rank-2 array.
This argument is intent(in) and optional, which is no value by default.
Note:
x, prepend and append arguments must have the same type, kind and rank. n is less than or equal to 0 (which is not recommended), the return value of diff is x. dim is not equal to 1 or 2 (which is not recommended),
1 will be used by the internal process of diff.Returns the finite difference of the input array.
Shall be a real/integer and rank-1/rank-2 array.
When both prepend and append are not present, the result y has one fewer element than x alongside the dimension dim.
program example_diff
use stdlib_math, only: diff
implicit none
integer :: i(7) = [1, 1, 2, 3, 5, 8, 13]
real :: x(6) = [0, 5, 15, 30, 50, 75]
integer :: A(3, 3) = reshape([1, 7, 17, 3, 11, 19, 5, 13, 23], [3, 3])
integer :: Y(3, 2)
print *, diff(i) ! [0, 1, 1, 2, 3, 5]
print *, diff(x, 2) ! [5.0, 5.0, 5.0, 5.0]
Y = diff(A, n=1, dim=2)
print *, Y(1, :) ! [2, 2]
print *, Y(2, :) ! [4, 2]
print *, Y(3, :) ! [2, 4]
print *, diff(i, prepend=[0]) ! [1, 0, 1, 1, 2, 3, 5]
print *, diff(i, append=[21]) ! [0, 1, 1, 2, 3, 5, 8]
end program example_diff
meshgrid subroutineComputes a list of coordinate matrices from coordinate vectors.
For $n \geq 1$ coordinate vectors $(x_1, x_2, ..., x_n)$ of sizes $(s_1, s_2, ..., s_n)$, meshgrid computes $n$ coordinate matrices $(X_1, X_2, ..., X_n)$ with identical shape corresponding to the selected indexing:
- Cartesian indexing (default behavior): the shape of the coordinate matrices is $(s_2, s_1, s_3, s_4, ... s_n)$.
- matrix indexing: the shape of the coordinate matrices is $(s_1, s_2, s_3, s_4, ... s_n)$.
For a 2D problem in Cartesian indexing:
call meshgrid (x, y, xm, ym)
For a 3D problem in Cartesian indexing:
call meshgrid (x, y, z, xm, ym, zm)
For a 3D problem in matrix indexing:
call meshgrid (x, y, z, xm, ym, zm, indexing="ij")
The subroutine can be called in n-dimensional situations, as long as n is inferior to the maximum allowed array rank.
Experimental.
Subroutine.
For a n-dimensional problem, with n >= 1:
x1, x2, ..., xn: The coordinate vectors.
Shall be real/integer and rank-1 arrays.
These arguments are intent(in).
xm1, xm2, ..., xmn: The coordinate matrices.
Shall be arrays of type real or integer of adequate shape:
- for Cartesian indexing, the shape of the coordinate matrices must be [size(x2), size(x1), size(x3), ..., size(xn)].
- for matrix indexing, the shape of the coordinate matrices must be [size(x1), size(x2), size(x3), ..., size(xn)].
These argument are intent(out).
indexing: the selected indexing.
Shall be an integer equal to stdlib_meshgrid_xy for Cartesian indexing (default), or stdlib_meshgrid_ij for matrix indexing. stdlib_meshgrid_xy and stdlib_meshgrid_ij are public constants defined in the module.
This argument is intent(in) and optional, and is equal to stdlib_meshgrid_xy by default.
program example_meshgrid
use stdlib_math, only: meshgrid, linspace, stdlib_meshgrid_ij
use stdlib_kinds, only: sp
implicit none
integer, parameter :: nx = 3, ny = 2
real(sp) :: x(nx), y(ny), &
xm_cart(ny, nx), ym_cart(ny, nx), &
xm_mat(nx, ny), ym_mat(nx, ny)
x = linspace(0_sp, 1_sp, nx)
y = linspace(0_sp, 1_sp, ny)
call meshgrid(x, y, xm_cart, ym_cart)
print *, "xm_cart = "
call print_2d_array(xm_cart)
print *, "ym_cart = "
call print_2d_array(ym_cart)
call meshgrid(x, y, xm_mat, ym_mat, indexing=stdlib_meshgrid_ij)
print *, "xm_mat = "
call print_2d_array(xm_mat)
print *, "ym_mat = "
call print_2d_array(ym_mat)
contains
subroutine print_2d_array(array)
real(sp), intent(in) :: array(:, :)
integer :: i
do i = 1, size(array, dim=1)
print *, array(i, :)
end do
end subroutine
end program example_meshgrid