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Updated: 12 December 1998

OpenVMS VAX RTL Mathematics (MTH$) Manual

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MTH$xATAND

Given the tangent of an angle, the Arc Tangent in Degrees routine returns that angle (in degrees).

Format

MTH$ATAND tangent

MTH$DATAND tangent

MTH$GATAND tangent

Each of the above formats accepts one of the floating-point types as input.

Corresponding JSB Entry Points

MTH$ATAND_R4

MTH$DATAND_R7

MTH$GATAND_R7

Each of the above JSB entry points accepts one of the floating-point types as input.

RETURNS


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: write only
mechanism: by value

Angle in degrees. The angle returned will have a value in the range: 


        -90 <= angle <= 90

MTH$ATAND returns an F-floating number. MTH$DATAND returns a D-floating number. MTH$GATAND returns a G-floating number.

Argument

tangent


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

The tangent of the angle whose value (in degrees) is to be returned. The tangent argument is the address of a floating-point number that is this tangent. For MTH$ATAND, tangent specifies an F-floating number. For MTH$DATAND, tangent specifies a D-floating number. For MTH$GATAND, tangent specifies a G-floating number.

Description

The computation of the arc tangent function is based on the following identities:
  arctan(X) = (180/Pi sign)* (X - X 3/3 + X 5/5 - X 7/7 + ...)
  arctan(X) = 64*X + X*Q(X 2),
where Q(Y) = 180/Pi*[(1- 64*Pi sign/180)] - Y/3 + Y 2/5 - Y 3/7 + Y 4/9
  arctan(X) = X*P(X 2),
where P(Y) = 180/Pi sign*[1 - Y/3 + Y 2/5 - Y 3/7 + Y 4/9 ...]
  arctan(X) = 90 - arctan(1/X)
  arctan(X) = arctan(A) + arctan((X - A)/(1 + A*X))

The angle in degrees whose tangent is X is computed as:
Tangent Angle Returned
X <= 3/32 64*X + X*Q(X 2)
3/32 < X <= 11 ATAND(A) + V*P(V 2) , where A and ATAND(A) are chosen by table lookup and V =(X - A)/(1 + A*X)
11 < X 90 - W * (P(W 2)), where W = 1/X
X < 0 -zATAND(|X|)

See MTH$HATAND for the description of the H-floating point version of this routine.

Condition Value Signaled

SS$_ROPRAND Reserved operand. The MTH$xATAND routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.

MTH$xATAN2

Given sine and cosine, the Arc Tangent in Radians with Two Arguments routine returns the angle (in radians) whose tangent is given by the quotient of sine and cosine (sine/cosine).

Format

MTH$ATAN2 sine ,cosine

MTH$DATAN2 sine ,cosine

MTH$GATAN2 sine ,cosine

Each of the above formats accepts one of the floating-point types as input.

RETURNS


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: write only
mechanism: by value

Angle in radians. MTH$ATAN2 returns an F-floating number. MTH$DATAN2 returns a D-floating number. MTH$GATAN2 returns a G-floating number.

Arguments

sine


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

Dividend. The sine argument is the address of a floating-point number that is this dividend. For MTH$ATAN2, sine specifies an F-floating number. For MTH$DATAN2, sine specifies a D-floating number. For MTH$GATAN2, sine specifies a G-floating number.

cosine


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

Divisor. The cosine argument is the address of a floating-point number that is this divisor. For MTH$ATAN2, cosine specifies an F-floating number. For MTH$DATAN2, cosine specifies a D-floating number. For MTH$GATAN2, cosine specifies a G-floating number.

Description

The angle in radians whose tangent is Y/X is computed as follows, where f is defined in the description of MTH$zCOSH.
Value of Input Arguments Angle Returned
X = 0 or Y/X > 2 (f+1) Pi sign/2* (sign Y)
X > 0 and Y/X <= 2 (f+1) zATAN(Y/X)
X < 0 and Y/X <= 2 (f+1) Pi sign* (sign Y) + zATAN(Y/X)

See MTH$HATAN2 for the description of the H-floating point version of this routine.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$xATAN2 routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.
MTH$_INVARGMAT Invalid argument. Both cosine and sine are zero. LIB$SIGNAL copies the floating-point reserved operand to the mechanism argument vector CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand unless you have written a condition handler to change CHF$L_MCH_SAVR0/R1.

MTH$xATAND2

Given sine and cosine, the Arc Tangent in Degrees with Two Arguments routine returns the angle (in degrees) whose tangent is given by the quotient of sine and cosine (sine/cosine).

Format

MTH$ATAND2 sine ,cosine

MTH$DATAND2 sine ,cosine

MTH$GATAND2 sine ,cosine

Each of the above formats accepts one of the floating-point types as input.

RETURNS


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: write only
mechanism: by value

Angle in degrees. MTH$ATAND2 returns an F-floating number. MTH$DATAND2 returns a D-floating number. MTH$GATAND2 returns a G-floating number.

Arguments

sine


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

Dividend. The sine argument is the address of a floating-point number that is this dividend. For MTH$ATAND2, sine specifies an F-floating number. For MTH$DATAND2, sine specifies a D-floating number. For MTH$GATAND2, sine specifies a G-floating number.

cosine


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

Divisor. The cosine argument is the address of a floating-point number that is this divisor. For MTH$ATAND2, cosine specifies an F-floating number. For MTH$DATAND2, cosine specifies a D-floating number. For MTH$GATAND2, cosine specifies a G-floating number.

Description

The angle in degrees whose tangent is Y/X is computed below and where f is defined in the description of MTH$zCOSH.
Value of Input Arguments Angle Returned
X = 0 or Y/X > 2 (f+1) 90* (sign Y)
X > 0 and Y/X <= 2 (f+1) zATAND(Y/X)
X < 0 and Y/X <= 2 (f+1) 180 * (sign Y) + zATAND(Y/X)

See MTH$HATAND2 for the description of the H-floating point version of this routine.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$xATAND2 routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.
MTH$_INVARGMAT Invalid argument. Both cosine and sine are zero. LIB$SIGNAL copies the floating-point reserved operand to the mechanism argument vector CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand unless you have written a condition handler to change CHF$L_MCH_SAVR0/R1.

MTH$xATANH

Given the hyperbolic tangent of an angle, the Hyperbolic Arc Tangent routine returns the hyperbolic arc tangent of that angle.

Format

MTH$ATANH hyperbolic-tangent

MTH$DATANH hyperbolic-tangent

MTH$GATANH hyperbolic-tangent

Each of the above formats accepts one of the floating-point types as input.

RETURNS


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: write only
mechanism: by value

The hyperbolic arc tangent of hyperbolic-tangent. MTH$ATANH returns an F-floating number. MTH$DATANH returns a D-floating number. MTH$GATANH returns a G-floating number.

Argument

hyperbolic-tangent


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: read only
mechanism: by reference

Hyperbolic tangent of an angle. The hyperbolic-tangent argument is the address of a floating-point number that is this hyperbolic tangent. For MTH$ATANH, hyperbolic-tangent specifies an F-floating number. For MTH$DATANH, hyperbolic-tangent specifies a D-floating number. For MTH$GATANH, hyperbolic-tangent specifies a G-floating number.

Description

The hyperbolic arc tangent function is computed as follows:
Value of x Value Returned
|X| < 1 zATANH(X) = zLOG((1+X)/(1-X))/2
|X| => 1 An invalid argument is signaled

See MTH$HATANH for the description of the H-floating point version of this routine.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$xATANH routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.
MTH$_INVARGMAT Invalid argument: |X| => 1 . LIB$SIGNAL copies the floating-point reserved operand to the mechanism argument vector CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand unless you have written a condition handler to change CHF$L_MCH_SAVR0/R1.

MTH$CxABS

The Complex Absolute Value routine returns the absolute value of a complex number (r,i).

Format

MTH$CABS complex-number

MTH$CDABS complex-number

MTH$CGABS complex-number

Each of the above formats accepts one of the floating-point complex types as input.

RETURNS


OpenVMS usage: floating_point
type: F_floating, D_floating, G_floating
access: write only
mechanism: by value

The absolute value of a complex number. MTH$CABS returns an F-floating number. MTH$CDABS returns a D-floating number. MTH$CGABS returns a G-floating number.

Argument

complex-number


OpenVMS usage: complex_number
type: F_floating complex, D_floating complex, G_floating complex
access: read only
mechanism: by reference

A complex number (r,i), where r and i are both floating-point complex values. The complex-number argument is the address of this complex number. For MTH$CABS, complex-number specifies an F-floating complex number. For MTH$CDABS, complex-number specifies a D-floating complex number. For MTH$CGABS, complex-number specifies a G-floating complex number.

Description

The complex absolute value is computed as follows, where MAX is the larger of |r| and |i|, and MIN is the smaller of |r| and |i|:
result = MAX * SQRT((MIN/MAX)2 + 1)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$CxABS routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.
MTH$_FLOOVEMAT Floating-point overflow in Math Library when both r and i are large.

Examples

#1 

C+ 
C    This Fortran example forms the absolute value of an 
C    F-floating complex number using MTH$CABS and the 
C    Fortran random number generator RAN. 
C 
C    Declare Z as a complex value and MTH$CABS as a REAL*4 value. 
C    MTH$CABS will return the absolute value of Z:   Z_NEW = MTH$CABS(Z). 
C- 
 
        COMPLEX Z 
        COMPLEX CMPLX 
        REAL*4 Z_NEW,MTH$CABS 
        INTEGER M 
        M = 1234567 
 
C+ 
C    Generate a random complex number with the Fortran generic CMPLX. 
C- 
 
        Z = CMPLX(RAN(M),RAN(M)) 
 
C+ 
C    Z is a complex number (r,i) with real part "r" and 
C    imaginary part "i". 
C- 
 
       TYPE *, ' The complex number z is',z 
       TYPE *, ' It has real part',REAL(Z),'and imaginary part',AIMAG(Z) 
       TYPE *, ' ' 
 
C+ 
C    Compute the complex absolute value of Z. 
C- 
 
        Z_NEW = MTH$CABS(Z) 
        TYPE *, ' The complex absolute value of',z,' is',Z_NEW 
        END

This example uses an F-floating complex number for complex-number. The output of this Fortran example is as follows: 


The complex number z is (0.8535407,0.2043402) 
It has real part  0.8535407    and imaginary part  0.2043402 
 
The complex absolute value of (0.8535407,0.2043402) is  0.8776597

#2 

C+ 
C    This Fortran example forms the absolute 
C    value of a G-floating complex number using 
C    MTH$CGABS and the Fortran random number 
C    generator RAN. 
C 
C    Declare Z as a complex value and MTH$CGABS as a 
C    REAL*8 value. MTH$CGABS will return the absolute 
C    value of Z: Z_NEW = MTH$CGABS(Z). 
C- 
 
 
        COMPLEX*16 Z 
        REAL*8 Z_NEW,MTH$CGABS 
 
C+ 
C    Generate a random complex number with the Fortran 
C    generic CMPLX. 
C- 
 
        Z = (12.34567890123,45.536376385345) 
        TYPE *, ' The complex number z is',z 
        TYPE *, ' ' 
 
C+ 
C    Compute the complex absolute value of Z. 
C- 
 
        Z_NEW = MTH$CGABS(Z) 
        TYPE *, ' The complex absolute value of',z,' is',Z_NEW 
        END

This Fortran example uses a G-floating complex number for complex-number. Because this example uses a G-floating number, it must be compiled as follows: 


$ Fortran/G MTHEX.FOR

Notice the difference in the precision of the output generated: 


The complex number z is (12.3456789012300,45.5363763853450) 
The complex absolute value of (12.3456789012300,45.5363763853450) is 
 47.1802645376230

MTH$CCOS

The Cosine of a Complex Number (F-Floating Value) routine returns the cosine of a complex number as an F-floating value.

Format

MTH$CCOS complex-number

RETURNS


OpenVMS usage: complex_number
type: F_floating complex
access: write only
mechanism: by value

The complex cosine of the complex input number. MTH$CCOS returns an F-floating complex number.

Argument

complex-number


OpenVMS usage: complex_number
type: F_floating complex
access: read only
mechanism: by reference

A complex number (r,i) where r and i are floating-point numbers. The complex-number argument is the address of this complex number. For MTH$CCOS, complex-number specifies an F-floating complex number.

Description

The complex cosine is calculated as follows:
result = (COS(r) * COSH(i), -SIN(r) * SINH(i))

See MTH$CxCOS for the descriptions of the D- and G-floating point versions of this routine.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$CCOS routine encountered a floating-point reserved operand due to incorrect user input. A floating-point reserved operand is a floating-point datum with a sign bit of 1 and a biased exponent of 0. Floating-point reserved operands are reserved for future use by Digital.
MTH$_FLOOVEMAT Floating-point overflow in Math Library: the absolute value of i is greater than about 88.029 for F-floating values.

Example


C+ 
C    This Fortran example forms the complex 
C    cosine of an F-floating complex number using 
C    MTH$CCOS and the Fortran random number 
C    generator RAN. 
C 
C    Declare Z and MTH$CCOS as complex values. 
C    MTH$CCOS will return the cosine value of 
C    Z:         Z_NEW = MTH$CCOS(Z) 
C- 
 
 
        COMPLEX Z,Z_NEW,MTH$CCOS 
        COMPLEX CMPLX 
        INTEGER M 
        M = 1234567 
 
C+ 
C    Generate a random complex number with the 
C    Fortran generic CMPLX. 
C- 
 
        Z = CMPLX(RAN(M),RAN(M)) 
 
C+ 
C    Z is a complex number (r,i) with real part "r" and 
C    imaginary part "i". 
C- 
 
        TYPE *, ' The complex number z is',z 
        TYPE *, ' It has real part',REAL(Z),'and imaginary part',AIMAG(Z) 
        TYPE *, ' ' 
 
C+ 
C    Compute the complex cosine value of Z. 
C- 
 
        Z_NEW = MTH$CCOS(Z) 
        TYPE *, ' The complex cosine value of',z,' is',Z_NEW 
        END

This Fortran example demonstrates the use of MTH$CCOS, using the MTH$CCOS entry point. The output of this program is as follows: 


The complex number z is (0.8535407,0.2043402) 
It has real part  0.8535407    and imaginary part  0.2043402 
The complex cosine value of (0.8535407,0.2043402) is (0.6710899,-0.1550672)

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