A Set of 74 Test Functions for Nonlinear Equation Solvers

This report gives 74 funct ions appropriate to use to test programs which solve one nonl inear equat ion in ofie real variable . The funct ions are given in a Fortran subrout ine as a two-dimensional array . A SET OF 74 TEST FUNCTIONS FOR WO* 'LINEAR F.OUATION SOLVERS INTRODUCTION AMP SUT1MA n Y: Tliis report «ives 74 functions in FORTRAN code which have been gathered as a set of test funct ions for a nolyalpori thp to solve F(x) = 0 . These functions are arranged in a two-dinensional array whose indices are passed through the bloc): common COMMON /FDATA/ J ,JJ and thus F(x) can be used in its sinnlest form . The ranges of the indices are J = 1 JJ 1 to 14 J = 2 JJ « 1 to 12 J = 3 JJ * 1 to 10 J =» 4 JJ • I to 24 J = 5 JJ 1 to 14 These functions may he classified into overlanpinir T O U O S accord in" to various propert ies the funct ions possess. This classificat ion is given below wi th the notat ion that 3-8 denotes the eiphth elerient of the third group (i .e. , J = 3 , JJ = 8) . -iISIMPLE (25 members) 17 1-11 22 25 26 33 3fi 42 43 4 7 49 4-10 4-11 4-12 4-14 4-15 4-17 4-18 4-19 4-20 4-21 4-22 4-23 4-24 CLUSTERED ROOTS (7 members) 16 1-10 51 52 53 54 55 MULTIPLE ROOTS (14 members) 1 2 1 3 1 4 15 16 19 1-10 4 1 4 4 4 5 46 48 4-16 58 FRACTIONAL ORDER ROOTS (7 members) 13 1-10 28 34 37 5-11 5-12 DISCONTINUITIES (4 members) 1 5 2 9 2-10 32 ASYMPTOTES TO ZERO (6 members) 1-12 56 5 7 58 59 5-10 ROUND OFF SENSITIVE (5 members) 18 1 9 2 4 27 35 NON-FUNCTIONS (3 members) 3 8 3 9 3-10 PATHOLOGICAL (11 members) 1 3 1 5 2 1 24 29 31 32 34 4-13 5-11 5-12 BADLY SCALED (6 members) 11 1-11 21 23 5-13 5-14 FROM ' REAL ' PROBLEMS (2 members) 1-13 27 (A number of other test functions from ' real ' problems they require too much Fortran code for inclusion here) . are avai lable , F U N C T I O N F(X] r0" :j r> ' !/ F n T / J , JJ r^TA P] / co TO 11 < ' »?<", 3C . %400 , ?00) , JJ C T*"ST FI'NCIIC-NS IN US C SUMMFft 19*7 i on rô TiHi.jF PO TC(1»2 ,3 ,4 ,5 ,6 ,7 .0 ,9 ,1C ,11 ,12>13) , J 1 F = ( X . 1 2 F + I F ( A! : 'S (XJo l T . X j E 0 } X = J.rfl F = A ^ S I X Q . L W U 1 r>FTUP> 1 3 F = A p S( J ^ ^ o T ^ ^ ^ S t X + l S ? ? ) * * ] „5*( (X-„ II-O5) / AHSIX, , 1F-10) ) PS i XALO ' ; = 0,. T P t X . N P . P . ) X*L0"= X*AL0r { A^S(X J ) F = ( X l l o ? ^ 7 ! ( . # X 0 1*X A L0G**2 7 . ) PETU ,^; l , * rQMTiMU^ I r I V»S(X). ' : c 0 r >. ,9> TO SOI I F ( X»L c v i ' t ) c = SINtA .^X) I F ( X . f t ) F = .7 I F l Y / ^ ^ . r . a ) r _ ( x + , O 0 0 C 3 ) * ( l „ + C O 5 ( ^ C C a * y . ) / 2 . ) i F I X . ^ c ] ) c = F = A ^ o f X . O * * ? IF( At»S(X) F = A p S ( X ) I F ( 9 A . M D e x B L T „ 4 0 3 o ) F = < X 3 0 0 „ ) s { X ~ 3 1 4 „ ) * ( X ~ 3 9 9 « ) * * 2 RETURN F = APSl X-17 0 )»*?V ° S(X-17 .1 >*»1 u «*(X-?0 . ) 0 F J • ;p N 7 F = I X l J » ( v ' > 1 ) # ( X ' > < } * I X i s ) * I X M * ( X A „ ) DFTijo\i P F = M ( ( (X-? l „ )*X+17* . , ) * X 7 3 5 0 )*X+U,24„ * X + 7 2 0 „ 9 Y = X + 1 , ] ] U 1 C 7 . Y 9 ( ° . Y » ( , Y ) ) ) "FTll^M 7 • v = X 7 * l ? . ! F( ^ S ( Y ) ^ T „ , 1. !T+7 i Y = S I ̂ N ( « 1F+7 1Y 1 ~ = A L O M C ( 1 a+Y*-*? )*( Y + 1 6 . )*SJRT< />PS{ Y-Ro ) ) PT ' JP ' V 1] Y = SO?T t apS(X-*0~ 1 o ) ) IF(A ̂ S ( X)fL r o^0C ( ; ro TO 1110 = = ALOr-10 |17o ) * r O M ] % 0 ) + ( X-400 o ) *( „+3 .*< x-^on . ) ) RETURN 1110 r = ALO^ l Cf l „ + Y * « D * r O S { X / A O . ) PFTIPN 1 ? c = 7 a / < A n s ( X * X * X ) + l . ) Pi iTURN C FORESTRY iNVfSTMFNT RETURN FUNCTION 13 Y = 1 . + X 3 i F = 20 o / Y**!5 •K ,0 C /Y**33 +475 . /Y**^0 "1 . 1 ?* ( Y**40~ l „ ) PFTURfi C T^ST FUNCTIONS OP S^PT . 1967