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Precision Control and GRACE

Precision Control and GRACE. J. Fujimoto(KEK), N. Hamaguchi(Hitachi Co. Ltd.),T. Ishikawa(KEK), T. Kaneko(KEK), H. Morita(Hitachi Co. Ltd.), D. Perret-Gallix(LAPP), A. Tokura(Hitachi Co. Ltd.) and Y. Shimizu(The Graduate University for Advanced Studies) ACAT05

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Precision Control and GRACE

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  1. Precision Control and GRACE J. Fujimoto(KEK), N. Hamaguchi(Hitachi Co. Ltd.),T. Ishikawa(KEK), T. Kaneko(KEK), H. Morita(Hitachi Co. Ltd.), D. Perret-Gallix(LAPP), A. Tokura(Hitachi Co. Ltd.) and Y. Shimizu(The Graduate University for Advanced Studies) ACAT05 2005.５.２５, Zeuthenpresented byJ. Fujmoto(KEK)

  2. Introduction High-precision experiments like LC Requires high prevision theoretical prediction => higher order calculation 　 Automatic System like GRACE isneededto perform1-loop correctionto e+e- 3, 4-bodies Great progress since Sept. 2002

  3. Full 1-loop calculations available GRACE, PLB559(2003)252Denner et al., NPB660(2003)289 GRACE, PLB571(2003)163You et al., PLB571(2003)85Denner et al., PLB575(2003)290 GRACE, PLB576(2003)152Zhang et al., PLB578(2004)349 GRACE, PLB600(2004)65 GRACE, NIM A534(2004)334 GRACE, Talk by K. Kato at LCWS05(Mar.2005) Denner etal., hep-ph/0502063

  4. How to check the results? One solution is check gauge invariance Independence on gauge parameters Non-linear gauge fixing terms in GRACE

  5. Linear Gauge vs Non-Linear Gauge check of NLG Cuv=0 Cuv=100 LG Cuv=0

  6. Alternative solution • Use higher computation precision. • Is Quadruple precision good enough?  see ‘Simple Example’ • Hitachi has developed a new FORTRAN library of High-speed Multiprecision operations in collaboration with the GRACE group. • This library provides information on ‘lost-bits’ during the calculation.

  7. Simple Example • f = 333.75*(b**6) + (a**2)*{11*(a**2)*(b**2) - (b**6)- 121*(b**4) - 2} + 5.5*(b**8) + a/(2*b), where a=77617.0、b=33096.0. by C. Hu, S. Xu and X. Yang • f = 1.1726039400531786318588349045201801 w/ Quadruple precision • Analytical result = - 54767/66192 = -0.82739605994682136814116509547981370 • Using the new Octuple precision library in HMLib: f = -0.827396059946821368141165095479816 with lost bits = 121

  8. HMLIb: New FORTRAN Library for High-speed Multiprecision operations • Library in FORTRAN  available for any architectures • Based on Integer operations  fast & “lost bits” information for subtraction. • For the octuple floating point operations: based on IEEE754, 1 bit for sign,15bits for exponent, 240 bits for mantissa. • For example call Q4ADDSUB(A,B,C,I,IBIT) : for add/sub in Quad. call Q8ADDSUB(A,B,C,I.IBIT) : for add/sub in Octuple • MULT/DIV,SQRT,LOG,ATAN2 … are also available • Please contact to Mr. Hashimoto from Hitachi; ko-hashimoto@itg.hitachi.co.jp lost bits information

  9. Number of “lost-bits” Suppose A=2**T1*(1.F), B=2**T2*(1.G), where A≧B>0. Construct two positive integers, IA=1.F*2**242､ IB=1.G*2**(242 – (T1 – T2)). In the case of subtraction, we can get IA – IB=2**T3+N (0≦N≦2**T3 – 1). Then, the number of lost bits is given by 242 – T3.

  10. Performance Comparison Ratio of the execution time (Double precision:1) Pen4 Intel Fortran V8.1 -O0 only 4 times slower FPU ALU HMLIb w/ lost bits HMLib w/ lost bits

  11. Actual application QED corrections to t Quadruple precision is required in some phase space points due to the Gram determinant mass of happens in the reduction algorithm. mass of photon

  12. Loop integrals for box diagrams where and Using

  13. LHS is given by 2-dim integrals since RHS is presented by are given by and 2-dim integrals. So Similarly, are given by , 2-dim/1-dim integrals. Reduction Algorithm Determinant to solve the system is nothing but the ‘Gram Determinant’

  14. in Double precision ReJ[1]= -1.49368718239238 ReJ[x] = -6.86111482424926E-0002 ReJ[y] = -6.86785270067264E-0002 ReJ[w] = -1.39799775179174 ReJ[w**2] = - 1.36472026946296 ReJ[w*x] = - 2.708863236843683E-0002 ReJ[x*y] = - 3.048903558925384E-0002 … ReJ[w**3] = 93763.26727997246 … Blow up !! in Quadruple precision ReJ[1]=-1.49368718238777512062307539882045 ReJ[x] =-6.861114708877389206553392789958382E-0002 ReJ[y] =-6.867852585600575199171661642779842E-0002 ReJ[w] =-1.39799775496536042464289674154150 ReJ[w**2] = - 1.34746346742190735627641191119128 ReJ[w*x] =-3.334744118868393382280835719751654E-0002 ReJ[x*y] =-2.822377826411337874789947823777159E-0002 … ReJ[w**3] =-1.60389378482142986480454883491878 …

  15. How does HMLib work? • Call HMLib in the codes. • At the point of J[w**3], HMLib reported 70-bit lost in total, which means 12 decimal digits of the result is guaranteed. • More precisely, Gram determ. for J[w**3] is 5.038066E-5. Before J[W**3], already 43bits were lost,

  16. Summary • Precision control is mandatory for large scale calculations. • GRACE relies on gauge independent checks for 1-loop calculations. • High precision computation provides an alternative approach. • HMLib(Hitachi in collaboration with GRACEgroup) is a FORTRAN library for Multiprecision operations. • HMLib is fast due to the integer operations and gives the number of “lost-bits” in the computations. • HMLib has been applied to1-loop corrections; • We have shown that higher precision computationsandHMLib guarantees the precision of the results.

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