lartg(3)

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

    !>          F is COMPLEX(wp)
    !>          The first component of vector to be rotated.
    !>

    G

    !>          G is COMPLEX(wp)
    !>          The second component of vector to be rotated.
    !>

    C

    !>          C is REAL(wp)
    !>          The cosine of the rotation.
    !>

    S

    !>          S is COMPLEX(wp)
    !>          The sine of the rotation.
    !>

    R

    !>          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
    !>
    !>

  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

    !>          F is REAL(wp)
    !>          The first component of vector to be rotated.
    !>

    G

    !>          G is REAL(wp)
    !>          The second component of vector to be rotated.
    !>

    C

    !>          C is REAL(wp)
    !>          The cosine of the rotation.
    !>

    S

    !>          S is REAL(wp)
    !>          The sine of the rotation.
    !>

    R

    !>          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
    !>
    !>

  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

    !>          F is REAL(wp)
    !>          The first component of vector to be rotated.
    !>

    G

    !>          G is REAL(wp)
    !>          The second component of vector to be rotated.
    !>

    C

    !>          C is REAL(wp)
    !>          The cosine of the rotation.
    !>

    S

    !>          S is REAL(wp)
    !>          The sine of the rotation.
    !>

    R

    !>          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
    !>
    !>

  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

    !>          F is COMPLEX(wp)
    !>          The first component of vector to be rotated.
    !>

    G

    !>          G is COMPLEX(wp)
    !>          The second component of vector to be rotated.
    !>

    C

    !>          C is REAL(wp)
    !>          The cosine of the rotation.
    !>

    S

    !>          S is COMPLEX(wp)
    !>          The sine of the rotation.
    !>

    R

    !>          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
    !>
    !>

Author

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