COT
This function evaluates the cotangent.
Function Value Return
COT — Function value. (Output)
Required Arguments
X — Angle in radians for which the cotangent is desired. (Input)
FORTRAN 90 Interface
Generic: COT (X)
Specific: The specific interface names are COT, DCOT, CCOT, and ZCOT.
FORTRAN 77 Interface
Single: COT (X)
Double: The double precision function name is DCOT.
Complex: The complex name is CCOT.
Double Complex: The double complex name is ZCOT.
Description
For real x, the magnitude of x must not be so large that most of the computer word contains the integer part of x. Likewise, x must not be too near an integer multiple of π, although x close to the origin causes no accuracy loss. Finally, x must not be so close to the origin that COT(X) ≈ 1/x overflows.
For complex arguments, let z = x + iy. If ∣sin z∣2 is very small, that is, if x is very close to a multiple of π and if ∣y∣ is small, then cot z is nearly singular and a fatal error condition is reported. If ∣sin z∣2 is somewhat larger but still small, then the result will be less accurate than half precision. When ∣2x∣ is so large that sin 2x cannot be evaluated accurately to even zero precision, the following situation results. If ∣y∣ < 3/2, then CCOT cannot be evaluated accurately to be better than one significant figure. If 3/2 ≤∣y∣ < ‑1/2 ln ɛ/2, where ɛ = AMACH(4) is the machine precision, then CCOT can be evaluated by ignoring the real part of the argument; however, the answer will be less accurate than half precision. Finally, ∣z∣ must not be so small that cot z ≈ 1/z overflows.
Comments
1. Informational error for Real arguments
Type | Code | Description |
|---|
3 | 2 | Result of COT(X) is accurate to less than one‑half precision because ABS(X) is too large, or X is nearly a multiple of π. |
2. Informational error for Complex arguments
Type | Code | Description |
|---|
3 | 2 | Result of CCOT(Z) is accurate to less than one‑half precision because the real part of Z is too near a multiple of π when the imaginary part of Z is zero, or because the absolute value of the real part is very large and the absolute value of the imaginary part is small. |
3. Referencing COT(X) is not the same as computing 1.0/TAN(X) because the error conditions are quite different. For example, when X is near π/2, TAN(X) cannot be evaluated accurately and an error message must be issued. However, COT(X) can be evaluated accurately in the sense of absolute error.
Examples
Example 1
In this example, cot(0.3) is computed and printed.
USE COT_INT
USE UMACH_INT
IMPLICIT NONE
! Declare variables
INTEGER NOUT
REAL VALUE, X
! Compute
X = 0.3
VALUE = COT(X)
! Print the results
CALL UMACH (2, NOUT)
WRITE (NOUT,99999) X, VALUE
99999 FORMAT (' COT(', F6.3, ') = ', F6.3)
END
Output
COT( 0.300) = 3.233
Example 2
In this example, cot(1 + i) is computed and printed.
USE COT_INT
USE UMACH_INT
IMPLICIT NONE
! Declare variables
INTEGER NOUT
COMPLEX VALUE, Z
! Compute
Z = (1.0, 1.0)
VALUE = COT(Z)
! Print the results
CALL UMACH (2, NOUT)
WRITE (NOUT,99999) Z, VALUE
99999 FORMAT (' COT((', F6.3, ',', F6.3, ')) = (', &
F6.3, ',', F6.3, ')')
END
Output
COT(( 1.000, 1.000)) = ( 0.218,-0.868)
