rotg - rotg: generate plane rotation (cf. lartg)

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subroutinecrotg(complex(wp)a,complex(wp)b,real(wp)c,complex(wp)s)CROTG generates a Givens rotation with real cosine and complex sine. Purpose: CROTG constructs a plane rotation [ c s ] [ a ] = [ r ] [ -conjg(s) c ] [ b ] [ 0 ] where c is real, s is complex, and c**2 + conjg(s)*s = 1. The computation uses the formulas |x| = sqrt( Re(x)**2 + Im(x)**2 ) sgn(x) = x / |x| if x /= 0 = 1 if x = 0 c = |a| / sqrt(|a|**2 + |b|**2) s = sgn(a) * conjg(b) / sqrt(|a|**2 + |b|**2) r = sgn(a)*sqrt(|a|**2 + |b|**2) When a and b are real and r /= 0, the formulas simplify to c = a / r s = b / r the same as in SROTG when |a| > |b|. When |b| >= |a|, the sign of c and s will be different from those computed by SROTG if the signs of a and b are not the same. Seealsolartg:generateplanerotation,moreaccuratethanBLASrot, lartgp:generateplanerotation,moreaccuratethanBLASrotParametersA A is COMPLEX On entry, the scalar a. On exit, the scalar r. B B is COMPLEX The scalar b. C C is REAL The scalar c. S S is COMPLEX The scalar s. 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 subroutinedrotg(real(wp)a,real(wp)b,real(wp)c,real(wp)s)DROTGPurpose: DROTG constructs a plane rotation [ c s ] [ a ] = [ r ] [ -s c ] [ b ] [ 0 ] satisfying c**2 + s**2 = 1. The computation uses the formulas sigma = sgn(a) if |a| > |b| = sgn(b) if |b| >= |a| r = sigma*sqrt( a**2 + b**2 ) c = 1; s = 0 if r = 0 c = a/r; s = b/r if r != 0 The subroutine also computes z = s if |a| > |b|, = 1/c if |b| >= |a| and c != 0 = 1 if c = 0 This allows c and s to be reconstructed from z as follows: If z = 1, set c = 0, s = 1. If |z| < 1, set c = sqrt(1 - z**2) and s = z. If |z| > 1, set c = 1/z and s = sqrt( 1 - c**2). Seealsolartg:generateplanerotation,moreaccuratethanBLASrot, lartgp:generateplanerotation,moreaccuratethanBLASrotParametersA A is DOUBLE PRECISION On entry, the scalar a. On exit, the scalar r. B B is DOUBLE PRECISION On entry, the scalar b. On exit, the scalar z. C C is DOUBLE PRECISION The scalar c. S S is DOUBLE PRECISION The scalar s. Author Edward Anderson, Lockheed Martin 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 subroutinesrotg(real(wp)a,real(wp)b,real(wp)c,real(wp)s)SROTGPurpose: SROTG constructs a plane rotation [ c s ] [ a ] = [ r ] [ -s c ] [ b ] [ 0 ] satisfying c**2 + s**2 = 1. The computation uses the formulas sigma = sgn(a) if |a| > |b| = sgn(b) if |b| >= |a| r = sigma*sqrt( a**2 + b**2 ) c = 1; s = 0 if r = 0 c = a/r; s = b/r if r != 0 The subroutine also computes z = s if |a| > |b|, = 1/c if |b| >= |a| and c != 0 = 1 if c = 0 This allows c and s to be reconstructed from z as follows: If z = 1, set c = 0, s = 1. If |z| < 1, set c = sqrt(1 - z**2) and s = z. If |z| > 1, set c = 1/z and s = sqrt( 1 - c**2). Seealsolartg:generateplanerotation,moreaccuratethanBLASrot, lartgp:generateplanerotation,moreaccuratethanBLASrotParametersA A is REAL On entry, the scalar a. On exit, the scalar r. B B is REAL On entry, the scalar b. On exit, the scalar z. C C is REAL The scalar c. S S is REAL The scalar s. Author Edward Anderson, Lockheed Martin 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 subroutinezrotg(complex(wp)a,complex(wp)b,real(wp)c,complex(wp)s)ZROTG generates a Givens rotation with real cosine and complex sine. Purpose: ZROTG constructs a plane rotation [ c s ] [ a ] = [ r ] [ -conjg(s) c ] [ b ] [ 0 ] where c is real, s is complex, and c**2 + conjg(s)*s = 1. The computation uses the formulas |x| = sqrt( Re(x)**2 + Im(x)**2 ) sgn(x) = x / |x| if x /= 0 = 1 if x = 0 c = |a| / sqrt(|a|**2 + |b|**2) s = sgn(a) * conjg(b) / sqrt(|a|**2 + |b|**2) r = sgn(a)*sqrt(|a|**2 + |b|**2) When a and b are real and r /= 0, the formulas simplify to c = a / r s = b / r the same as in DROTG when |a| > |b|. When |b| >= |a|, the sign of c and s will be different from those computed by DROTG if the signs of a and b are not the same. Seealsolartg:generateplanerotation,moreaccuratethanBLASrot, lartgp:generateplanerotation,moreaccuratethanBLASrotParametersA A is DOUBLE COMPLEX On entry, the scalar a. On exit, the scalar r. B B is DOUBLE COMPLEX The scalar b. C C is DOUBLE PRECISION The scalar c. S S is DOUBLE COMPLEX The scalar s. 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

rotg - rotg: generate plane rotation (cf. lartg)

Functions subroutine crotg (a, b, c, s) CROTG generates a Givens rotation with real cosine and complex sine. subroutine drotg (a, b, c, s) DROTG subroutine srotg (a, b, c, s) SROTG subroutine zrotg (a, b, c, s) ZROTG generates a Givens rotation with real cosine and complex sine.