NAME

lartg - lartg: generate plane rotation, more accurate than BLAS rot

SYNOPSIS

Functions

subroutine clartg (f, g, c, s, r)
CLARTG generates a plane rotation with real cosine and complex sine. subroutine dlartg (f, g, c, s, r)
DLARTG generates a plane rotation with real cosine and real sine. subroutine slartg (f, g, c, s, r)
SLARTG generates a plane rotation with real cosine and real sine. subroutine zlartg (f, g, c, s, r)
ZLARTG generates a plane rotation with real cosine and complex sine.

Detailed Description

Function Documentation

subroutine clartg (complex(wp) f, complex(wp) g, real(wp) c, complex(wp) s, complex(wp) r)

CLARTG generates a plane rotation with real cosine and complex sine.

Purpose:



 CLARTG generates a plane rotation so that


    [  C         S  ] . [ F ]  =  [ R ]


    [ -conjg(S)  C  ]   [ G ]     [ 0 ]


 where C is real and C**2 + |S|**2 = 1.


 The mathematical formulas used for C and S are


    sgn(x) = {  x / |x|,   x != 0


             {  1,         x  = 0


    R = sgn(F) * sqrt(|F|**2 + |G|**2)


    C = |F| / sqrt(|F|**2 + |G|**2)


    S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)


 Special conditions:


    If G=0, then C=1 and S=0.


    If F=0, then C=0 and S is chosen so that R is real.


 When F and G are real, the formulas simplify to C = F/R and


 S = G/R, and the returned values of C, S, and R should be


 identical to those returned by SLARTG.


 The algorithm used to compute these quantities incorporates scaling


 to avoid overflow or underflow in computing the square root of the


 sum of squares.


 This is the same routine CROTG fom BLAS1, except that


 F and G are unchanged on return.


 Below, wp=>sp stands for single precision from LA_CONSTANTS module.

Parameters



          F is COMPLEX(wp)


          The first component of vector to be rotated.



          G is COMPLEX(wp)


          The second component of vector to be rotated.



          C is REAL(wp)


          The cosine of the rotation.



          S is COMPLEX(wp)


          The sine of the rotation.



          R is COMPLEX(wp)


          The nonzero component of the rotated vector.

Author

Weslley Pereira, University of Colorado Denver, USA

Date

December 2021

Further Details:



 Based on the algorithm from


  Anderson E. (2017)


  Algorithm 978: Safe Scaling in the Level 1 BLAS


  ACM Trans Math Softw 44:1--28


  https://doi.org/10.1145/3061665

Definition at line 115 of file clartg.f90.

subroutine dlartg (real(wp) f, real(wp) g, real(wp) c, real(wp) s, real(wp) r)

DLARTG generates a plane rotation with real cosine and real sine.

Purpose:



 DLARTG generates a plane rotation so that


    [  C  S  ]  .  [ F ]  =  [ R ]


    [ -S  C  ]     [ G ]     [ 0 ]


 where C**2 + S**2 = 1.


 The mathematical formulas used for C and S are


    R = sign(F) * sqrt(F**2 + G**2)


    C = F / R


    S = G / R


 Hence C >= 0. The algorithm used to compute these quantities


 incorporates scaling to avoid overflow or underflow in computing the


 square root of the sum of squares.


 This version is discontinuous in R at F = 0 but it returns the same


 C and S as ZLARTG for complex inputs (F,0) and (G,0).


 This is a more accurate version of the BLAS1 routine DROTG,


 with the following other differences:


    F and G are unchanged on return.


    If G=0, then C=1 and S=0.


    If F=0 and (G .ne. 0), then C=0 and S=sign(1,G) without doing any


       floating point operations (saves work in DBDSQR when


       there are zeros on the diagonal).


 Below, wp=>dp stands for double precision from LA_CONSTANTS module.

Parameters



          F is REAL(wp)


          The first component of vector to be rotated.



          G is REAL(wp)


          The second component of vector to be rotated.



          C is REAL(wp)


          The cosine of the rotation.



          S is REAL(wp)


          The sine of the rotation.



          R is REAL(wp)


          The nonzero component of the rotated vector.

Author

Edward Anderson, Lockheed Martin

Date

July 2016

Contributors:

Weslley Pereira, University of Colorado Denver, USA

Further Details:



  Anderson E. (2017)


  Algorithm 978: Safe Scaling in the Level 1 BLAS


  ACM Trans Math Softw 44:1--28


  https://doi.org/10.1145/3061665

Definition at line 110 of file dlartg.f90.

subroutine slartg (real(wp) f, real(wp) g, real(wp) c, real(wp) s, real(wp) r)

SLARTG generates a plane rotation with real cosine and real sine.

Purpose:



 SLARTG generates a plane rotation so that


    [  C  S  ]  .  [ F ]  =  [ R ]


    [ -S  C  ]     [ G ]     [ 0 ]


 where C**2 + S**2 = 1.


 The mathematical formulas used for C and S are


    R = sign(F) * sqrt(F**2 + G**2)


    C = F / R


    S = G / R


 Hence C >= 0. The algorithm used to compute these quantities


 incorporates scaling to avoid overflow or underflow in computing the


 square root of the sum of squares.


 This version is discontinuous in R at F = 0 but it returns the same


 C and S as CLARTG for complex inputs (F,0) and (G,0).


 This is a more accurate version of the BLAS1 routine SROTG,


 with the following other differences:


    F and G are unchanged on return.


    If G=0, then C=1 and S=0.


    If F=0 and (G .ne. 0), then C=0 and S=sign(1,G) without doing any


       floating point operations (saves work in SBDSQR when


       there are zeros on the diagonal).


 Below, wp=>sp stands for single precision from LA_CONSTANTS module.

Parameters



          F is REAL(wp)


          The first component of vector to be rotated.



          G is REAL(wp)


          The second component of vector to be rotated.



          C is REAL(wp)


          The cosine of the rotation.



          S is REAL(wp)


          The sine of the rotation.



          R is REAL(wp)


          The nonzero component of the rotated vector.

Author

Edward Anderson, Lockheed Martin

Date

July 2016

Contributors:

Weslley Pereira, University of Colorado Denver, USA

Further Details:



  Anderson E. (2017)


  Algorithm 978: Safe Scaling in the Level 1 BLAS


  ACM Trans Math Softw 44:1--28


  https://doi.org/10.1145/3061665

Definition at line 110 of file slartg.f90.

subroutine zlartg (complex(wp) f, complex(wp) g, real(wp) c, complex(wp) s, complex(wp) r)

ZLARTG generates a plane rotation with real cosine and complex sine.

Purpose:



 ZLARTG generates a plane rotation so that


    [  C         S  ] . [ F ]  =  [ R ]


    [ -conjg(S)  C  ]   [ G ]     [ 0 ]


 where C is real and C**2 + |S|**2 = 1.


 The mathematical formulas used for C and S are


    sgn(x) = {  x / |x|,   x != 0


             {  1,         x  = 0


    R = sgn(F) * sqrt(|F|**2 + |G|**2)


    C = |F| / sqrt(|F|**2 + |G|**2)


    S = sgn(F) * conjg(G) / sqrt(|F|**2 + |G|**2)


 Special conditions:


    If G=0, then C=1 and S=0.


    If F=0, then C=0 and S is chosen so that R is real.


 When F and G are real, the formulas simplify to C = F/R and


 S = G/R, and the returned values of C, S, and R should be


 identical to those returned by DLARTG.


 The algorithm used to compute these quantities incorporates scaling


 to avoid overflow or underflow in computing the square root of the


 sum of squares.


 This is the same routine ZROTG fom BLAS1, except that


 F and G are unchanged on return.


 Below, wp=>dp stands for double precision from LA_CONSTANTS module.

Parameters



          F is COMPLEX(wp)


          The first component of vector to be rotated.



          G is COMPLEX(wp)


          The second component of vector to be rotated.



          C is REAL(wp)


          The cosine of the rotation.



          S is COMPLEX(wp)


          The sine of the rotation.



          R is COMPLEX(wp)


          The nonzero component of the rotated vector.

Author

Weslley Pereira, University of Colorado Denver, USA

Date

December 2021

Further Details:



 Based on the algorithm from


  Anderson E. (2017)


  Algorithm 978: Safe Scaling in the Level 1 BLAS


  ACM Trans Math Softw 44:1--28


  https://doi.org/10.1145/3061665

Definition at line 115 of file zlartg.f90.

Author

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