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Diffstat (limited to 'libm/ldouble/gammal.c')
-rw-r--r-- | libm/ldouble/gammal.c | 764 |
1 files changed, 0 insertions, 764 deletions
diff --git a/libm/ldouble/gammal.c b/libm/ldouble/gammal.c deleted file mode 100644 index de7ed89a2..000000000 --- a/libm/ldouble/gammal.c +++ /dev/null @@ -1,764 +0,0 @@ -/* gammal.c - * - * Gamma function - * - * - * - * SYNOPSIS: - * - * long double x, y, gammal(); - * extern int sgngam; - * - * y = gammal( x ); - * - * - * - * DESCRIPTION: - * - * Returns gamma function of the argument. The result is - * correctly signed, and the sign (+1 or -1) is also - * returned in a global (extern) variable named sgngam. - * This variable is also filled in by the logarithmic gamma - * function lgam(). - * - * Arguments |x| <= 13 are reduced by recurrence and the function - * approximated by a rational function of degree 7/8 in the - * interval (2,3). Large arguments are handled by Stirling's - * formula. Large negative arguments are made positive using - * a reflection formula. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -40,+40 10000 3.6e-19 7.9e-20 - * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 - * - * Accuracy for large arguments is dominated by error in powl(). - * - */ -/* lgaml() - * - * Natural logarithm of gamma function - * - * - * - * SYNOPSIS: - * - * long double x, y, lgaml(); - * extern int sgngam; - * - * y = lgaml( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of the absolute - * value of the gamma function of the argument. - * The sign (+1 or -1) of the gamma function is returned in a - * global (extern) variable named sgngam. - * - * For arguments greater than 33, the logarithm of the gamma - * function is approximated by the logarithmic version of - * Stirling's formula using a polynomial approximation of - * degree 4. Arguments between -33 and +33 are reduced by - * recurrence to the interval [2,3] of a rational approximation. - * The cosecant reflection formula is employed for arguments - * less than -33. - * - * Arguments greater than MAXLGML (10^4928) return MAXNUML. - * - * - * - * ACCURACY: - * - * - * arithmetic domain # trials peak rms - * IEEE -40, 40 100000 2.2e-19 4.6e-20 - * IEEE 10^-2000,10^+2000 20000 1.6e-19 3.3e-20 - * The error criterion was relative when the function magnitude - * was greater than one but absolute when it was less than one. - * - */ - -/* gamma.c */ -/* gamma function */ - -/* -Copyright 1994 by Stephen L. Moshier -*/ - - -#include <math.h> -/* -gamma(x+2) = gamma(x+2) P(x)/Q(x) -0 <= x <= 1 -Relative error -n=7, d=8 -Peak error = 1.83e-20 -Relative error spread = 8.4e-23 -*/ -#if UNK -static long double P[8] = { - 4.212760487471622013093E-5L, - 4.542931960608009155600E-4L, - 4.092666828394035500949E-3L, - 2.385363243461108252554E-2L, - 1.113062816019361559013E-1L, - 3.629515436640239168939E-1L, - 8.378004301573126728826E-1L, - 1.000000000000000000009E0L, -}; -static long double Q[9] = { --1.397148517476170440917E-5L, - 2.346584059160635244282E-4L, --1.237799246653152231188E-3L, --7.955933682494738320586E-4L, - 2.773706565840072979165E-2L, --4.633887671244534213831E-2L, --2.243510905670329164562E-1L, - 4.150160950588455434583E-1L, - 9.999999999999999999908E-1L, -}; -#endif -#if IBMPC -static short P[] = { -0x434a,0x3f22,0x2bda,0xb0b2,0x3ff0, XPD -0xf5aa,0xe82f,0x335b,0xee2e,0x3ff3, XPD -0xbe6c,0x3757,0xc717,0x861b,0x3ff7, XPD -0x7f43,0x5196,0xb166,0xc368,0x3ff9, XPD -0x9549,0x8eb5,0x8c3a,0xe3f4,0x3ffb, XPD -0x8d75,0x23af,0xc8e4,0xb9d4,0x3ffd, XPD -0x29cf,0x19b3,0x16c8,0xd67a,0x3ffe, XPD -0x0000,0x0000,0x0000,0x8000,0x3fff, XPD -}; -static short Q[] = { -0x5473,0x2de8,0x1268,0xea67,0xbfee, XPD -0x334b,0xc2f0,0xa2dd,0xf60e,0x3ff2, XPD -0xbeed,0x1853,0xa691,0xa23d,0xbff5, XPD -0x296e,0x7cb1,0x5dfd,0xd08f,0xbff4, XPD -0x0417,0x7989,0xd7bc,0xe338,0x3ff9, XPD -0x3295,0x3698,0xd580,0xbdcd,0xbffa, XPD -0x75ef,0x3ab7,0x4ad3,0xe5bc,0xbffc, XPD -0xe458,0x2ec7,0xfd57,0xd47c,0x3ffd, XPD -0x0000,0x0000,0x0000,0x8000,0x3fff, XPD -}; -#endif -#if MIEEE -static long P[24] = { -0x3ff00000,0xb0b22bda,0x3f22434a, -0x3ff30000,0xee2e335b,0xe82ff5aa, -0x3ff70000,0x861bc717,0x3757be6c, -0x3ff90000,0xc368b166,0x51967f43, -0x3ffb0000,0xe3f48c3a,0x8eb59549, -0x3ffd0000,0xb9d4c8e4,0x23af8d75, -0x3ffe0000,0xd67a16c8,0x19b329cf, -0x3fff0000,0x80000000,0x00000000, -}; -static long Q[27] = { -0xbfee0000,0xea671268,0x2de85473, -0x3ff20000,0xf60ea2dd,0xc2f0334b, -0xbff50000,0xa23da691,0x1853beed, -0xbff40000,0xd08f5dfd,0x7cb1296e, -0x3ff90000,0xe338d7bc,0x79890417, -0xbffa0000,0xbdcdd580,0x36983295, -0xbffc0000,0xe5bc4ad3,0x3ab775ef, -0x3ffd0000,0xd47cfd57,0x2ec7e458, -0x3fff0000,0x80000000,0x00000000, -}; -#endif -/* -static long double P[] = { --3.01525602666895735709e0L, --3.25157411956062339893e1L, --2.92929976820724030353e2L, --1.70730828800510297666e3L, --7.96667499622741999770e3L, --2.59780216007146401957e4L, --5.99650230220855581642e4L, --7.15743521530849602425e4L -}; -static long double Q[] = { - 1.00000000000000000000e0L, --1.67955233807178858919e1L, - 8.85946791747759881659e1L, - 5.69440799097468430177e1L, --1.98526250512761318471e3L, - 3.31667508019495079814e3L, - 1.60577839621734713377e4L, --2.97045081369399940529e4L, --7.15743521530849602412e4L -}; -*/ -#define MAXGAML 1755.455L -/*static long double LOGPI = 1.14472988584940017414L;*/ - -/* Stirling's formula for the gamma function -gamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) -z(x) = x -13 <= x <= 1024 -Relative error -n=8, d=0 -Peak error = 9.44e-21 -Relative error spread = 8.8e-4 -*/ -#if UNK -static long double STIR[9] = { - 7.147391378143610789273E-4L, --2.363848809501759061727E-5L, --5.950237554056330156018E-4L, - 6.989332260623193171870E-5L, - 7.840334842744753003862E-4L, --2.294719747873185405699E-4L, --2.681327161876304418288E-3L, - 3.472222222230075327854E-3L, - 8.333333333333331800504E-2L, -}; -#endif -#if IBMPC -static short STIR[] = { -0x6ede,0x69f7,0x54e3,0xbb5d,0x3ff4, XPD -0xc395,0x0295,0x4443,0xc64b,0xbfef, XPD -0xba6f,0x7c59,0x5e47,0x9bfb,0xbff4, XPD -0x5704,0x1a39,0xb11d,0x9293,0x3ff1, XPD -0x30b7,0x1a21,0x98b2,0xcd87,0x3ff4, XPD -0xbef3,0x7023,0x6a08,0xf09e,0xbff2, XPD -0x3a1c,0x5ac8,0x3478,0xafb9,0xbff6, XPD -0xc3c9,0x906e,0x38e3,0xe38e,0x3ff6, XPD -0xa1d5,0xaaaa,0xaaaa,0xaaaa,0x3ffb, XPD -}; -#endif -#if MIEEE -static long STIR[27] = { -0x3ff40000,0xbb5d54e3,0x69f76ede, -0xbfef0000,0xc64b4443,0x0295c395, -0xbff40000,0x9bfb5e47,0x7c59ba6f, -0x3ff10000,0x9293b11d,0x1a395704, -0x3ff40000,0xcd8798b2,0x1a2130b7, -0xbff20000,0xf09e6a08,0x7023bef3, -0xbff60000,0xafb93478,0x5ac83a1c, -0x3ff60000,0xe38e38e3,0x906ec3c9, -0x3ffb0000,0xaaaaaaaa,0xaaaaa1d5, -}; -#endif -#define MAXSTIR 1024.0L -static long double SQTPI = 2.50662827463100050242E0L; - -/* 1/gamma(x) = z P(z) - * z(x) = 1/x - * 0 < x < 0.03125 - * Peak relative error 4.2e-23 - */ -#if UNK -static long double S[9] = { --1.193945051381510095614E-3L, - 7.220599478036909672331E-3L, --9.622023360406271645744E-3L, --4.219773360705915470089E-2L, - 1.665386113720805206758E-1L, --4.200263503403344054473E-2L, --6.558780715202540684668E-1L, - 5.772156649015328608253E-1L, - 1.000000000000000000000E0L, -}; -#endif -#if IBMPC -static short S[] = { -0xbaeb,0xd6d3,0x25e5,0x9c7e,0xbff5, XPD -0xfe9a,0xceb4,0xc74e,0xec9a,0x3ff7, XPD -0x9225,0xdfef,0xb0e9,0x9da5,0xbff8, XPD -0x10b0,0xec17,0x87dc,0xacd7,0xbffa, XPD -0x6b8d,0x7515,0x1905,0xaa89,0x3ffc, XPD -0xf183,0x126b,0xf47d,0xac0a,0xbffa, XPD -0x7bf6,0x57d1,0xa013,0xa7e7,0xbffe, XPD -0xc7a9,0x7db0,0x67e3,0x93c4,0x3ffe, XPD -0x0000,0x0000,0x0000,0x8000,0x3fff, XPD -}; -#endif -#if MIEEE -static long S[27] = { -0xbff50000,0x9c7e25e5,0xd6d3baeb, -0x3ff70000,0xec9ac74e,0xceb4fe9a, -0xbff80000,0x9da5b0e9,0xdfef9225, -0xbffa0000,0xacd787dc,0xec1710b0, -0x3ffc0000,0xaa891905,0x75156b8d, -0xbffa0000,0xac0af47d,0x126bf183, -0xbffe0000,0xa7e7a013,0x57d17bf6, -0x3ffe0000,0x93c467e3,0x7db0c7a9, -0x3fff0000,0x80000000,0x00000000, -}; -#endif -/* 1/gamma(-x) = z P(z) - * z(x) = 1/x - * 0 < x < 0.03125 - * Peak relative error 5.16e-23 - * Relative error spread = 2.5e-24 - */ -#if UNK -static long double SN[9] = { - 1.133374167243894382010E-3L, - 7.220837261893170325704E-3L, - 9.621911155035976733706E-3L, --4.219773343731191721664E-2L, --1.665386113944413519335E-1L, --4.200263503402112910504E-2L, - 6.558780715202536547116E-1L, - 5.772156649015328608727E-1L, --1.000000000000000000000E0L, -}; -#endif -#if IBMPC -static short SN[] = { -0x5dd1,0x02de,0xb9f7,0x948d,0x3ff5, XPD -0x989b,0xdd68,0xc5f1,0xec9c,0x3ff7, XPD -0x2ca1,0x18f0,0x386f,0x9da5,0x3ff8, XPD -0x783f,0x41dd,0x87d1,0xacd7,0xbffa, XPD -0x7a5b,0xd76d,0x1905,0xaa89,0xbffc, XPD -0x7f64,0x1234,0xf47d,0xac0a,0xbffa, XPD -0x5e26,0x57d1,0xa013,0xa7e7,0x3ffe, XPD -0xc7aa,0x7db0,0x67e3,0x93c4,0x3ffe, XPD -0x0000,0x0000,0x0000,0x8000,0xbfff, XPD -}; -#endif -#if MIEEE -static long SN[27] = { -0x3ff50000,0x948db9f7,0x02de5dd1, -0x3ff70000,0xec9cc5f1,0xdd68989b, -0x3ff80000,0x9da5386f,0x18f02ca1, -0xbffa0000,0xacd787d1,0x41dd783f, -0xbffc0000,0xaa891905,0xd76d7a5b, -0xbffa0000,0xac0af47d,0x12347f64, -0x3ffe0000,0xa7e7a013,0x57d15e26, -0x3ffe0000,0x93c467e3,0x7db0c7aa, -0xbfff0000,0x80000000,0x00000000, -}; -#endif - -int sgngaml = 0; -extern int sgngaml; -extern long double MAXLOGL, MAXNUML, PIL; -/* #define PIL 3.14159265358979323846L */ -/* #define MAXNUML 1.189731495357231765021263853E4932L */ - -#ifdef ANSIPROT -extern long double fabsl ( long double ); -extern long double lgaml ( long double ); -extern long double logl ( long double ); -extern long double expl ( long double ); -extern long double gammal ( long double ); -extern long double sinl ( long double ); -extern long double floorl ( long double ); -extern long double powl ( long double, long double ); -extern long double polevll ( long double, void *, int ); -extern long double p1evll ( long double, void *, int ); -extern int isnanl ( long double ); -extern int isfinitel ( long double ); -static long double stirf ( long double ); -#else -long double fabsl(), lgaml(), logl(), expl(), gammal(), sinl(); -long double floorl(), powl(), polevll(), p1evll(), isnanl(), isfinitel(); -static long double stirf(); -#endif -#ifdef INFINITIES -extern long double INFINITYL; -#endif -#ifdef NANS -extern long double NANL; -#endif - -/* Gamma function computed by Stirling's formula. - */ -static long double stirf(x) -long double x; -{ -long double y, w, v; - -w = 1.0L/x; -/* For large x, use rational coefficients from the analytical expansion. */ -if( x > 1024.0L ) - w = (((((6.97281375836585777429E-5L * w - + 7.84039221720066627474E-4L) * w - - 2.29472093621399176955E-4L) * w - - 2.68132716049382716049E-3L) * w - + 3.47222222222222222222E-3L) * w - + 8.33333333333333333333E-2L) * w - + 1.0L; -else - w = 1.0L + w * polevll( w, STIR, 8 ); -y = expl(x); -if( x > MAXSTIR ) - { /* Avoid overflow in pow() */ - v = powl( x, 0.5L * x - 0.25L ); - y = v * (v / y); - } -else - { - y = powl( x, x - 0.5L ) / y; - } -y = SQTPI * y * w; -return( y ); -} - - - -long double gammal(x) -long double x; -{ -long double p, q, z; -int i; - -sgngaml = 1; -#ifdef NANS -if( isnanl(x) ) - return(NANL); -#endif -#ifdef INFINITIES -if(x == INFINITYL) - return(INFINITYL); -#ifdef NANS -if(x == -INFINITYL) - goto gamnan; -#endif -#endif -q = fabsl(x); - -if( q > 13.0L ) - { - if( q > MAXGAML ) - goto goverf; - if( x < 0.0L ) - { - p = floorl(q); - if( p == q ) - { -gamnan: -#ifdef NANS - mtherr( "gammal", DOMAIN ); - return (NANL); -#else - goto goverf; -#endif - } - i = p; - if( (i & 1) == 0 ) - sgngaml = -1; - z = q - p; - if( z > 0.5L ) - { - p += 1.0L; - z = q - p; - } - z = q * sinl( PIL * z ); - z = fabsl(z) * stirf(q); - if( z <= PIL/MAXNUML ) - { -goverf: -#ifdef INFINITIES - return( sgngaml * INFINITYL); -#else - mtherr( "gammal", OVERFLOW ); - return( sgngaml * MAXNUML); -#endif - } - z = PIL/z; - } - else - { - z = stirf(x); - } - return( sgngaml * z ); - } - -z = 1.0L; -while( x >= 3.0L ) - { - x -= 1.0L; - z *= x; - } - -while( x < -0.03125L ) - { - z /= x; - x += 1.0L; - } - -if( x <= 0.03125L ) - goto small; - -while( x < 2.0L ) - { - z /= x; - x += 1.0L; - } - -if( x == 2.0L ) - return(z); - -x -= 2.0L; -p = polevll( x, P, 7 ); -q = polevll( x, Q, 8 ); -return( z * p / q ); - -small: -if( x == 0.0L ) - { - goto gamnan; - } -else - { - if( x < 0.0L ) - { - x = -x; - q = z / (x * polevll( x, SN, 8 )); - } - else - q = z / (x * polevll( x, S, 8 )); - } -return q; -} - - - -/* A[]: Stirling's formula expansion of log gamma - * B[], C[]: log gamma function between 2 and 3 - */ - - -/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x A(1/x^2) - * x >= 8 - * Peak relative error 1.51e-21 - * Relative spread of error peaks 5.67e-21 - */ -#if UNK -static long double A[7] = { - 4.885026142432270781165E-3L, --1.880801938119376907179E-3L, - 8.412723297322498080632E-4L, --5.952345851765688514613E-4L, - 7.936507795855070755671E-4L, --2.777777777750349603440E-3L, - 8.333333333333331447505E-2L, -}; -#endif -#if IBMPC -static short A[] = { -0xd984,0xcc08,0x91c2,0xa012,0x3ff7, XPD -0x3d91,0x0304,0x3da1,0xf685,0xbff5, XPD -0x3bdc,0xaad1,0xd492,0xdc88,0x3ff4, XPD -0x8b20,0x9fce,0x844e,0x9c09,0xbff4, XPD -0xf8f2,0x30e5,0x0092,0xd00d,0x3ff4, XPD -0x4d88,0x03a8,0x60b6,0xb60b,0xbff6, XPD -0x9fcc,0xaaaa,0xaaaa,0xaaaa,0x3ffb, XPD -}; -#endif -#if MIEEE -static long A[21] = { -0x3ff70000,0xa01291c2,0xcc08d984, -0xbff50000,0xf6853da1,0x03043d91, -0x3ff40000,0xdc88d492,0xaad13bdc, -0xbff40000,0x9c09844e,0x9fce8b20, -0x3ff40000,0xd00d0092,0x30e5f8f2, -0xbff60000,0xb60b60b6,0x03a84d88, -0x3ffb0000,0xaaaaaaaa,0xaaaa9fcc, -}; -#endif - -/* log gamma(x+2) = x B(x)/C(x) - * 0 <= x <= 1 - * Peak relative error 7.16e-22 - * Relative spread of error peaks 4.78e-20 - */ -#if UNK -static long double B[7] = { --2.163690827643812857640E3L, --8.723871522843511459790E4L, --1.104326814691464261197E6L, --6.111225012005214299996E6L, --1.625568062543700591014E7L, --2.003937418103815175475E7L, --8.875666783650703802159E6L, -}; -static long double C[7] = { -/* 1.000000000000000000000E0L,*/ --5.139481484435370143617E2L, --3.403570840534304670537E4L, --6.227441164066219501697E5L, --4.814940379411882186630E6L, --1.785433287045078156959E7L, --3.138646407656182662088E7L, --2.099336717757895876142E7L, -}; -#endif -#if IBMPC -static short B[] = { -0x9557,0x4995,0x0da1,0x873b,0xc00a, XPD -0xfe44,0x9af8,0x5b8c,0xaa63,0xc00f, XPD -0x5aa8,0x7cf5,0x3684,0x86ce,0xc013, XPD -0x259a,0x258c,0xf206,0xba7f,0xc015, XPD -0xbe18,0x1ca3,0xc0a0,0xf80a,0xc016, XPD -0x168f,0x2c42,0x6717,0x98e3,0xc017, XPD -0x2051,0x9d55,0x92c8,0x876e,0xc016, XPD -}; -static short C[] = { -/*0x0000,0x0000,0x0000,0x8000,0x3fff, XPD*/ -0xaa77,0xcf2f,0xae76,0x807c,0xc008, XPD -0xb280,0x0d74,0xb55a,0x84f3,0xc00e, XPD -0xa505,0xcd30,0x81dc,0x9809,0xc012, XPD -0x3369,0x4246,0xb8c2,0x92f0,0xc015, XPD -0x63cf,0x6aee,0xbe6f,0x8837,0xc017, XPD -0x26bb,0xccc7,0xb009,0xef75,0xc017, XPD -0x462b,0xbae8,0xab96,0xa02a,0xc017, XPD -}; -#endif -#if MIEEE -static long B[21] = { -0xc00a0000,0x873b0da1,0x49959557, -0xc00f0000,0xaa635b8c,0x9af8fe44, -0xc0130000,0x86ce3684,0x7cf55aa8, -0xc0150000,0xba7ff206,0x258c259a, -0xc0160000,0xf80ac0a0,0x1ca3be18, -0xc0170000,0x98e36717,0x2c42168f, -0xc0160000,0x876e92c8,0x9d552051, -}; -static long C[21] = { -/*0x3fff0000,0x80000000,0x00000000,*/ -0xc0080000,0x807cae76,0xcf2faa77, -0xc00e0000,0x84f3b55a,0x0d74b280, -0xc0120000,0x980981dc,0xcd30a505, -0xc0150000,0x92f0b8c2,0x42463369, -0xc0170000,0x8837be6f,0x6aee63cf, -0xc0170000,0xef75b009,0xccc726bb, -0xc0170000,0xa02aab96,0xbae8462b, -}; -#endif - -/* log( sqrt( 2*pi ) ) */ -static long double LS2PI = 0.91893853320467274178L; -#define MAXLGM 1.04848146839019521116e+4928L - - -/* Logarithm of gamma function */ - - -long double lgaml(x) -long double x; -{ -long double p, q, w, z, f, nx; -int i; - -sgngaml = 1; -#ifdef NANS -if( isnanl(x) ) - return(NANL); -#endif -#ifdef INFINITIES -if( !isfinitel(x) ) - return(INFINITYL); -#endif -if( x < -34.0L ) - { - q = -x; - w = lgaml(q); /* note this modifies sgngam! */ - p = floorl(q); - if( p == q ) - { -#ifdef INFINITIES - mtherr( "lgaml", SING ); - return (INFINITYL); -#else - goto loverf; -#endif - } - i = p; - if( (i & 1) == 0 ) - sgngaml = -1; - else - sgngaml = 1; - z = q - p; - if( z > 0.5L ) - { - p += 1.0L; - z = p - q; - } - z = q * sinl( PIL * z ); - if( z == 0.0L ) - goto loverf; -/* z = LOGPI - logl( z ) - w; */ - z = logl( PIL/z ) - w; - return( z ); - } - -if( x < 13.0L ) - { - z = 1.0L; - nx = floorl( x + 0.5L ); - f = x - nx; - while( x >= 3.0L ) - { - nx -= 1.0L; - x = nx + f; - z *= x; - } - while( x < 2.0L ) - { - if( fabsl(x) <= 0.03125 ) - goto lsmall; - z /= nx + f; - nx += 1.0L; - x = nx + f; - } - if( z < 0.0L ) - { - sgngaml = -1; - z = -z; - } - else - sgngaml = 1; - if( x == 2.0L ) - return( logl(z) ); - x = (nx - 2.0L) + f; - p = x * polevll( x, B, 6 ) / p1evll( x, C, 7); - return( logl(z) + p ); - } - -if( x > MAXLGM ) - { -loverf: -#ifdef INFINITIES - return( sgngaml * INFINITYL ); -#else - mtherr( "lgaml", OVERFLOW ); - return( sgngaml * MAXNUML ); -#endif - } - -q = ( x - 0.5L ) * logl(x) - x + LS2PI; -if( x > 1.0e10L ) - return(q); -p = 1.0L/(x*x); -q += polevll( p, A, 6 ) / x; -return( q ); - - -lsmall: -if( x == 0.0L ) - goto loverf; -if( x < 0.0L ) - { - x = -x; - q = z / (x * polevll( x, SN, 8 )); - } -else - q = z / (x * polevll( x, S, 8 )); -if( q < 0.0L ) - { - sgngaml = -1; - q = -q; - } -else - sgngaml = 1; -q = logl( q ); -return(q); -} |