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

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subroutineclartg(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. ParametersF 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 FurtherDetails: 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 subroutinedlartg(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. ParametersF 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 FurtherDetails: 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 subroutineslartg(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. ParametersF 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 FurtherDetails: 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 subroutinezlartg(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. ParametersF 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 FurtherDetails: 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

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

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.