Package 'gsl'

Title: Wrapper for the Gnu Scientific Library
Description: An R wrapper for some of the functionality of the Gnu Scientific Library.
Authors: Robin K. S. Hankin [aut, cre] , Andrew Clausen [ctb] (multimin functionality), Duncan Murdoch [ctb] (qrng functions)
Maintainer: Robin K. S. Hankin <[email protected]>
License: GPL-3
Version: 2.1-8
Built: 2024-08-27 09:15:54 UTC
Source: https://github.com/robinhankin/gsl

Help Index

Wrappers for the Gnu Scientific Library

Description

An R wrapper for some of the functionality of the Gnu Scientific Library.

Details

The DESCRIPTION file:

Package: gsl
Version: 2.1-8
Depends: R (>= 4.0.0)
Title: Wrapper for the Gnu Scientific Library
Authors@R: c(person(given=c("Robin", "K. S."), family="Hankin", role = c("aut","cre"), email="[email protected]", comment = c(ORCID = "0000-0001-5982-0415")), person(given="Andrew",family="Clausen",role="ctb",comment="multimin functionality"), person(given="Duncan",family="Murdoch",role="ctb",comment="qrng functions"))
SystemRequirements: Gnu Scientific Library version >= 2.5
Description: An R wrapper for some of the functionality of the Gnu Scientific Library.
Maintainer: Robin K. S. Hankin <[email protected]>
License: GPL-3
URL: https://github.com/RobinHankin/gsl
BugReports: https://github.com/RobinHankin/gsl/issues
NeedsCompilation: yes
Author: Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>), Andrew Clausen [ctb] (multimin functionality), Duncan Murdoch [ctb] (qrng functions)
Date/Publication: 2019-03-25 09:10:03 UTC
Repository: https://robinhankin.r-universe.dev
RemoteUrl: https://github.com/robinhankin/gsl
RemoteRef: HEAD
RemoteSha: ec2ca315b49aea08e087da71490b0d9f4c74d934

Index of help topics: 

Airy                    Airy functions
Bessel                  Bessel functions
Clausen                 Clausen functions
Coulomb                 Coulomb functions
Coupling                Coupling functions
Dawson                  Dawson functions
Debye                   Debye functions
Dilog                   Dilog functions
Ellint                  Elliptic functions
Elljac                  Elliptic functions
Error                   Error functions
Expint                  exponential functions
Fermi_Dirac             Fermi-Dirac functions
Gamma                   gamma functions
Gegenbauer              Gegenbauer functions
Hyperg                  Hypergeometric functions
Laguerre                Laguerre functions
Lambert                 Lambert's W function
Legendre                Legendre functions
Log                     Log functions
Misc                    Argument processing and general info
Poly                    Polynomials
Psi                     Psi (digamma) functions
Qrng                    Quasi-random sequences
Rng                     Random numbers generation
Synchrotron             Synchrotron functions
Transport               Transport functions
Trig                    Trig functions
Zeta                    Zeta functions
gsl-deprecated          gsl-deprecated
gsl-package             Wrappers for the Gnu Scientific Library
multimin                Function minimization
pow_int                 Power functions

The function naming scheme directly copies the GSL manual except that leading gsl_sf_ and, if present, the trailing _e is stripped: thus gsl_sf_Airy_Ai_e goes to R function airy_Ai(); however, some functions retain the prefix to avoid conflicts (viz gsl_sf_sin(), gsl_sf_cos(), gsl_sf_gamma(), gsl_sf_ choose(), gsl_sf_beta()).

R function arguments have the same names as in the GSL reference manual, except for the quasirandom functions documented in the Qrng manpage.

The package is organized into units corresponding to GSL header files; the .c, .R, and .Rd filenames match the GSL header filenames, except that the .Rd files are capitalized. Functions appear in all files in the same order as the GSL reference manual, which precludes the use of the tidying method given in section 3.1 of R-exts. Error forms of GSL functions (_e versions) are used if available.

In general, documentation is limited to: (a), a pointer to the GSL reference book, which would in any case dominate any docs here; and (b), re-productions of some tables and figures in Abramowitz and Stegun (June 1964).

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>), Andrew Clausen [ctb] (multimin functionality), Duncan Murdoch [ctb] (qrng functions)

Maintainer: Robin K. S. Hankin <[email protected]>

References

  • M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

  • M. Galassi et al. 2007. GNU Scientific Library. Reference Manual edition 1.10, for GSL version 1.10; 10 September 2007

  • R. K. S. Hankin 2006. Introducing gsl, a wrapper for the Gnu Scientific Library. Rnews 6(4):24-26

Examples

airy_Ai(1:5)

Airy functions

Description

Airy functions as per the Gnu Scientific Library, reference manual section 7.4 and AMS-55, section 10.4. These functions are declared in header file gsl_sf_airy.h

Usage

airy_Ai(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_deriv(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_deriv(x, mode=0, give=FALSE, strict=TRUE)
airy_Ai_deriv_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_Bi_deriv_scaled(x, mode=0, give=FALSE, strict=TRUE)
airy_zero_Ai(n, give=FALSE, strict=TRUE)
airy_zero_Bi(n, give=FALSE, strict=TRUE)
airy_zero_Ai_deriv(n, give=FALSE, strict=TRUE)
airy_zero_Bi_deriv(n, give=FALSE, strict=TRUE)

Arguments

x

input: real values
n

input: integer values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
mode

input: mode. For GSL_PREC_DOUBLE, GSL_PREC_SINGLE, GSL_PREC_APPROX use 0,1,2 respectively
strict

Boolean, with TRUE meaning to return NaN if status is an error

Details

The zero functions return a status of GSL_EDOM and a value of NA for n≤0n\leq 0.

An example is given in the package vignette.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=1,by=0.01)

f <- function(x){
cbind(x=x, Ai= airy_Ai(x), Aidash= airy_Ai_deriv(x),
Bi=airy_Ai(x),Bidash=airy_Bi_deriv(x))
}

f(x)  #table 10.11, p475
f(-x) #table 10.11, p476


x <- 1:10  #table 10.13, p478
cbind(x,
 airy_zero_Ai(x), airy_Ai_deriv(airy_zero_Ai(x)),
 airy_zero_Ai_deriv(x), airy_Ai(airy_zero_Ai_deriv(x)),
 airy_zero_Bi(x), airy_Bi_deriv(airy_zero_Bi(x)),

 airy_zero_Bi_deriv(x), airy_Bi(airy_zero_Bi_deriv(x))
 )


# Verify 10.4.4 and 10.4.5, p446:
3^(-2/3)/gamma(2/3)   - airy_Ai(0)
3^(-1/3) / gamma(1/3) + airy_Ai_deriv(0) 
3^(-1/6) / gamma(2/3) - airy_Bi(0)
3^(1/6) / gamma(1/3)  - airy_Bi_deriv(0)
#  All should be small

Bessel functions

Description

Bessel functions as per the Gnu Scientific Library, reference manual section 7.5 and AMS-55, chapters 9 and 10. These functions are declared in header file gsl_sf_bessel.h

Usage

bessel_J0(x, give=FALSE, strict=TRUE)
bessel_J1(x, give=FALSE, strict=TRUE)
bessel_Jn(n,x, give=FALSE, strict=TRUE)
bessel_Jn_array(nmin,nmax,x, give=FALSE, strict=TRUE)
bessel_Y0(x, give=FALSE, strict=TRUE)
bessel_Y1(x, give=FALSE, strict=TRUE)
bessel_Yn(n,x, give=FALSE, strict=TRUE)
bessel_Yn_array(nmin, nmax, x, give=FALSE, strict=TRUE)
bessel_I0(x, give=FALSE, strict=TRUE)
bessel_I1(x, give=FALSE, strict=TRUE)
bessel_In(n, x, give=FALSE, strict=TRUE)
bessel_In_array(nmin, nmax, x, give=FALSE, strict=TRUE)
bessel_I0_scaled(x, give=FALSE, strict=TRUE)
bessel_I1_scaled(x, give=FALSE, strict=TRUE)
bessel_In_scaled(n, x, give=FALSE, strict=TRUE)
bessel_In_scaled_array(nmin, nmax, x, give=FALSE, strict=TRUE)
bessel_K0(x, give=FALSE, strict=TRUE)
bessel_K1(x, give=FALSE, strict=TRUE)
bessel_Kn(n, x, give=FALSE, strict=TRUE)
bessel_Kn_array(nmin, nmax, x, give=FALSE, strict=TRUE)
bessel_K0_scaled(x, give=FALSE, strict=TRUE)
bessel_K1_scaled(x, give=FALSE, strict=TRUE)
bessel_Kn_scaled(n, x, give=FALSE, strict=TRUE)
bessel_Kn_scaled_array(nmin, nmax, x, give=FALSE, strict=TRUE)
bessel_j0(x, give=FALSE, strict=TRUE)
bessel_j1(x, give=FALSE, strict=TRUE)
bessel_j2(x, give=FALSE, strict=TRUE)
bessel_jl(l,x, give=FALSE, strict=TRUE)
bessel_jl_array(lmax,x, give=FALSE, strict=TRUE)
bessel_jl_steed_array(lmax, x, give=FALSE, strict=TRUE)
bessel_y0(x, give=FALSE, strict=TRUE)
bessel_y1(x, give=FALSE, strict=TRUE)
bessel_y2(x, give=FALSE, strict=TRUE)
bessel_yl(l, x, give=FALSE, strict=TRUE)
bessel_yl_array(lmax, x, give=FALSE, strict=TRUE)
bessel_i0_scaled(x, give=FALSE, strict=TRUE)
bessel_i1_scaled(x, give=FALSE, strict=TRUE)
bessel_i2_scaled(x, give=FALSE, strict=TRUE)
bessel_il_scaled(l, x, give=FALSE, strict=TRUE)
bessel_il_scaled_array(lmax, x, give=FALSE, strict=TRUE)
bessel_k0_scaled(x, give=FALSE, strict=TRUE)
bessel_k1_scaled(x, give=FALSE, strict=TRUE)
bessel_k2_scaled(x, give=FALSE, strict=TRUE)
bessel_kl_scaled(l,x, give=FALSE, strict=TRUE)
bessel_kl_scaled_array(lmax,x, give=FALSE, strict=TRUE)
bessel_Jnu(nu, x, give=FALSE, strict=TRUE)
bessel_sequence_Jnu(nu, v, mode=0, give=FALSE, strict=TRUE)
bessel_Ynu(nu, x, give=FALSE, strict=TRUE)
bessel_Inu(nu, x, give=FALSE, strict=TRUE)
bessel_Inu_scaled(nu, x, give=FALSE, strict=TRUE)
bessel_Knu(nu, x, give=FALSE, strict=TRUE)
bessel_lnKnu(nu, x, give=FALSE, strict=TRUE)
bessel_Knu_scaled(nu, x, give=FALSE, strict=TRUE)
bessel_zero_J0(s, give=FALSE, strict=TRUE)
bessel_zero_J1(s, give=FALSE, strict=TRUE)
bessel_zero_Jnu(nu, s, give=FALSE, strict=TRUE)

Arguments

x, v, nu

input: real valued
n, nmin, nmax, lmax

input: integer valued
l, s

input: integer valued
mode

Integer, calc mode
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

strict or not

Details

All as for the GSL reference manual section 7.5

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

# Compare native R routine with GSL:
besselK(0.55,4) - bessel_Knu(4,0.55)  # should be small


x <- seq(from=0,to=15,len=1000)
plot(x,bessel_J0(x),xlim=c(0,16),ylim=c(-0.8,1.1),type="l",
           xaxt="n",yaxt="n",bty="n",xlab="",ylab="",
           main="Figure 9.1, p359")
jj.Y0 <- bessel_Y0(x)
jj.Y0[jj.Y0< -0.8] <- NA
lines(x,jj.Y0)
lines(x,bessel_J1(x),lty=2)
jj.Y1 <- bessel_Y1(x)
jj.Y1[jj.Y1< -0.8] <- NA
lines(x,jj.Y1,lty=2)
axis(1,pos=0,at=1:15,
     labels=c("","2","","4","","6","","8","","10","","12","","14","") )
axis(2,pos=0,at=seq(from=-8,to=10,by=2)/10,
labels=c("-.8","-.6","-.4","-.2","0",".2",".4",".6",".8","1.0"))
arrows(0,0,16,0,length=0.1,angle=10)
arrows(0,0,0,1.1,length=0.1,angle=10)
text(1.1, 0.83, expression(J[0]))
text(0.37, 0.3, expression(J[1]))
text(0.34,-0.3, expression(Y[0]))
text(1.7,-0.5, expression(Y[1]))
text(4.2, 0.43, expression(Y[1]))
text(7.2, 0.33, expression(J[0]))
text(8.6, 0.3, expression(J[0],paste("    ,")))
text(9.1, 0.3, expression(Y[0]))

x <- seq(from=0,to=13,len=100)
y <- t(bessel_jl_array(3,x))
y[y>0.6] <- NA
 matplot(x,y,col="black",type="l",xaxt="n",yaxt="n",bty="n",
         xlab="",ylab="",xlim=c(0,16),ylim=c(-0.3,0.75),
         main="Figure 10.1, p438")
axis(1,pos=0,at=2*(1:7))
arrows(0,0,15,0,length=0.1,angle=10)
arrows(0,0,0,0.65,length=0.1,angle=10)
axis(2,pos=0,las=1,at=seq(from=-3,to=6)/10,
         labels=c("-.3","-.2","-.1","0",".1",".2",".3",".4",".5",".6"))
text(0, 0.7, expression(J[n](x)))
text(15.5, 0, expression(x))
text(2.2,0.58,expression(n==0))
text(3.2,0.4,expression(n==1))
text(4.3,0.3,expression(n==2))
text(6.0,0.22,expression(n==3))



x <- seq(from=0 ,to=5,by=0.1)
cbind(x, bessel_J0(x),bessel_J1(x),bessel_Jn(2,x))      #table 9.1, p390
cbind(x, bessel_Y0(x),bessel_Y1(x),bessel_Yn(2,x))      #table 9.2, p391
t(bessel_Jn_array(3,9,x*2))                             #table 9.2, p398



 x <- seq(from=8,to=10,by=0.2)
 jj <- t(bessel_Jn(n=3:9,x=t(matrix(x,11,7))))
colnames(jj) <- paste("J",3:9,"(x)",sep="")
cbind(x,jj)                 #another part of table 9.2, p398


 x <- seq(from=8,to=10,by=0.2)
 jj <- t(bessel_Yn(n=3:9,x=t(matrix(x,11,7))))
colnames(jj) <- paste("J",3:9,"(x)",sep="")
cbind(x,jj)                 #part of table 9.2, p399

cbind(                       x,                         #table 9.8, p416
        exp(-x)*bessel_I0  (x),
        exp(-x)*bessel_I1  (x),
         x^(-2)*bessel_In(2,x)
)

cbind(                      x,                          #table 9.8, p417
        exp(x)*bessel_K0  (x),
        exp(x)*bessel_K1  (x),
         x^(2)*bessel_Kn(2,x)
)

cbind(x,                                                #table 10.1 , p457
    bessel_j0(x),
    bessel_j1(x),
    bessel_j2(x),
    bessel_y0(x),
    bessel_y1(x),
    bessel_y2(x)
)

 cbind(0:9,"x=1"=bessel_yl(l=0:9,x=1), "x=2"=bessel_yl(l=0:9,x=2), "x=5"=bessel_yl(l=0:9,x=5)) 
                                                        #table 10.5, p466, top

Clausen functions

Description

Clausen functions as per the Gnu Scientific Library section 7.6. These functions are declared in header file gsl_sf_clausen.h

Usage

clausen(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- (0:30)*pi/180
clausen(x)          #table 27.8, p1006

Coulomb functions

Description

Coulomb functions as per the Gnu Scientific Library, reference manual section 7.7 and AMS-55, chapter 14. These functions are declared in header file gsl_sf_coulomb.h

Usage

hydrogenicR_1(Z, r, give=FALSE, strict=TRUE)
hydrogenicR(n, l, Z, r, give=FALSE, strict=TRUE)
coulomb_wave_FG(eta, x, L_F, k, give=FALSE, strict=TRUE)
coulomb_wave_F_array(L_min, kmax, eta, x, give=FALSE,strict=TRUE)
coulomb_wave_FG_array(L_min, kmax, eta, x, give=FALSE,strict=TRUE)
coulomb_wave_FGp_array(L_min, kmax, eta, x, give=FALSE,strict=TRUE)
coulomb_wave_sphF_array(L_min, kmax, eta, x, give=FALSE,strict=TRUE)
coulomb_CL(L,eta, give=FALSE,strict=TRUE)
coulomb_CL_array(L_min, kmax, eta, give=FALSE,strict=TRUE)

Arguments

n, l, kmax

input: integers
Z, r, eta, x, L_F, L_min, k, L

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=14,len=300)
jj <- coulomb_wave_FG(1,10,x,0)
plot(x,jj$val_F,type="l",xaxt="n",yaxt="n",bty="n",xlab="",ylab="",
       main="Figure 14.1, p539")
lines(x,jj$val_G,type="l",lty=2)
axis(1,pos=0,at=1:14,
       labels=c("","2","","4","","6","","8","","10","","12","","14"))
lines(c(0,1),c(0,0))
axis(2,pos=0)
text(9.5, 0.63, expression(F[L]))
text(8.5, 1.21, expression(G[L]))





x <- seq(from=0,to=24,len=400)
plot(x,coulomb_wave_FG(eta=1,x,L_F=0,k=0)$val_F,type="l",
     ylim=c(-1.3,1.7), xlim=c(0,26),
     xaxt="n",yaxt="n",bty="n",xlab="",ylab="",main="Figure 14.3, p541",lty=3)
lines(x,coulomb_wave_FG(eta= 0,x,L_F=0,k=0)$val_F,type="l",lty=1)
lines(x,coulomb_wave_FG(eta= 5,x,L_F=0,k=0)$val_F,type="l",lty=6)
lines(x,coulomb_wave_FG(eta=10,x,L_F=0,k=0)$val_F,type="l",lty=6)
lines(x,coulomb_wave_FG(eta=x/2,x,L_F=0,k=0)$val_F,type="l",lty="F3")
axis(1,pos=0,at=1:24,
       labels=c("","2","","4","","","","8","","10","","12",
                "","14","","","","18","","","","22","","24"))
lines(c(0,26),c(0,0))
axis(2,pos=0,at=0.2*(-6:9),
       labels=c("","-1.2","","-.8","","-.4","","0","",".4",
                "",".8","","1.2","","1.6"))
text(2.5, -0.8, expression(eta == 0))
text(4.5,1.1,adj=0, expression(eta == 1))
text(14,1.4,adj=0, expression(eta == 5))
text(22,1.4,adj=0, expression(eta == 10))






x <- seq(from=0.5,to=10,by=0.5)
jj <- coulomb_wave_FG(eta=t(matrix(x,20,5)), x=1:5,0,0)
jj.F <- t(jj$val_F)
jj.G <- t(jj$val_G)
colnames(jj.F) <- 1:5
colnames(jj.G) <- 1:5
cbind(x,jj.F)        #table 14.1, p 546, top bit.
cbind(x,jj.G)        #table 14.1, p 547, top bit.

Coupling functions

Description

Coupling functions as per the Gnu Scientific Library, reference manual section 7.8. These functions are declared in header file gsl_sf_coupling.h

Usage

coupling_3j(two_ja, two_jb, two_jc, two_ma, two_mb, two_mc, give=FALSE, strict=TRUE)
coupling_6j(two_ja, two_jb, two_jc, two_jd, two_je, two_jf, give=FALSE, strict=TRUE)
coupling_9j(two_ja, two_jb, two_jc, two_jd, two_je, two_jf,
      two_jg, two_jh, two_ji, give=FALSE, strict=TRUE)

Arguments

two_ja, two_jb, two_jc, two_jd, two_je, two_jf, two_jg, two_jh, two_ji, two_ma, two_mb, two_mc

Arguments as per the GSL manual
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

coupling_3j(1,2,3,4,5,6)
coupling_6j(1,2,3,4,5,6)
coupling_9j(1,2,3,4,5,6,7,8,9)

Dawson functions

Description

Dawson functions as per the Gnu Scientific Library, reference manual section 7.9. These functions are declared in header file gsl_sf_dawson.h

Usage

dawson(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=2,by=0.01)
dawson(x)   #table 7.5 of Ab and St

Debye functions

Description

Debye functions as per the Gnu Scientific Library, section 7.10 of the reference manual. These functions are declared in header file gsl_sf_debye.h

Usage

debye_1(x, give=FALSE, strict=TRUE)
debye_2(x, give=FALSE, strict=TRUE)
debye_3(x, give=FALSE, strict=TRUE)
debye_4(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=10,by=0.1)
cbind(x,debye_1(x),debye_2(x),debye_3(x),debye_4(x))  #table 27.1

Dilog functions

Description

Dilog functions as per the Gnu Scientific Library reference manual section 7.11. These functions are declared in header file gsl_sf_dilog.h

Usage

dilog(x, give=FALSE, strict=TRUE)
complex_dilog(r, theta, give=FALSE, strict=TRUE)

Arguments

x

input: real values
r, theta

In complex_dilog(), input values. If theta takes its default value of NULL, interpret r as a complex-valued object. If theta is non-null, interpret r as the Modulus, and theta as the argument, of the complex object passed to gsl_sf_complex_dilog_e()
give

Boolean, with default FALSE meaning to return just the answers, and TRUE meaning to return a status vector as well
strict

Boolean, with TRUE meaning to return NaN if nonzero status is returned by the GSL function (FALSE means to return the value: use with caution)

Details

All functions as documented in the GSL reference manual section 7.11.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0, to=0.1,by=0.01)
cbind(x,"f(x)"=dilog(1-x))   #table 27.7, p1005

Elliptic functions

Description

Elliptic functions as per the Gnu Scientific Library, reference manual section 7.13 and AMS-55, chapter 17. These functions are declared in header file gsl_sf_ellint.h

Usage

ellint_Kcomp(k, mode=0, give=FALSE,strict=TRUE)
ellint_Ecomp(k, mode=0, give=FALSE,strict=TRUE)
ellint_F(phi,k, mode=0, give=FALSE,strict=TRUE)
ellint_E(phi,k, mode=0, give=FALSE,strict=TRUE)
ellint_P(phi,k,n, mode=0, give=FALSE,strict=TRUE)
ellint_D(phi,k, mode=0, give=FALSE,strict=TRUE)
ellint_RC(x, y, mode=0, give=FALSE,strict=TRUE)
ellint_RD(x, y, z, mode=0, give=FALSE,strict=TRUE)
ellint_RF(x, y, z, mode=0, give=FALSE,strict=TRUE)
ellint_RJ(x, y, z, p, mode=0, give=FALSE,strict=TRUE)

Arguments

phi, k, n, p, x, y, z

input: real values
give

Boolean, with default FALSE meaning to return just the answers, and TRUE meaning to return a status vector as well
strict

Boolean
mode

input: mode. For GSL_PREC_DOUBLE, GSL_PREC_SINGLE, GSL_PREC_APPROX use 0,1,2 respectively.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

ellint_Kcomp(0.3)
ellint_Ecomp(0.3)
ellint_F(0.4,0.7)
ellint_E(0.4,0.7)
ellint_P(0.4,0.7,0.3)
ellint_D(0.4,0.3)
ellint_RC(0.5,0.6)
ellint_RD(0.5,0.6,0.7)
ellint_RF(0.5,0.6,0.7)
ellint_RJ(0.5,0.6,0.7,0.1)


x <- seq(from=0,to=0.5,by=0.01)
col1 <- ellint_Kcomp(sqrt(x))
col2 <- ellint_Kcomp(sqrt(1-x))
col3 <- exp(-pi*col2/col1)
cbind(x,col1,col2,col3)         #table 17.1, p608

x <- 0:45
col1 <- ellint_Kcomp(sin(pi/180*x))
col2 <- ellint_Kcomp(sin(pi/2-pi/180*x))
col3 <- exp(-pi*col2/col1)
cbind(x,col1,col2,col3)       #table 17.2, p610

x <- seq(from=0,to=90,by=2)
f <- function(a){ellint_F(phi=a*pi/180,sin(x*pi/180))}
g <- function(a){ellint_E(phi=a*pi/180,sin(x*pi/180))}
h <- function(a,n){ellint_P(phi=a*pi/180,sin( a*15*pi/180),n)}
i <- function(x){ellint_P(phi=x*pi/180, k=sin((0:6)*15*pi/180),  n= -0.6)}


cbind(x,f(5),f(10),f(15),f(20),f(25),f(30))          #table 17.5, p613
cbind(x,g(5),g(10),g(15),g(20),g(25),g(30))          #table 17.6, p616



cbind(i(15),i(30),i(45),i(60),i(75),i(90))           #table 17.9,
                                                     #(BOTTOM OF p625)

Elliptic functions

Description

Elljac functions as per the Gnu Scientific Library, reference manual section 7.14 and AMS-55, chapter 16. These functions are declared in header file gsl_sf_elljac.h

Usage

elljac(u, m, give=FALSE, strict=TRUE)
gsl_sn(z,m)
gsl_cn(z,m)
gsl_dn(z,m)
gsl_ns(z,m)
gsl_nc(z,m)
gsl_nd(z,m)
gsl_sc(z,m)
gsl_sd(z,m)
gsl_cs(z,m)
gsl_cd(z,m)
gsl_ds(z,m)
gsl_dc(z,m)

Arguments

u, m

input: real values
z

input: complex values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Details

A straightforward wrapper for the gsl_sf_elljac_e function of the GSL library, except for gsl_sn(), gsl_cn(), and gsl_dn(), which implement 16.21.1 to 16.21.4 (thus taking complex arguments); and gsl_ns() et seq which are the minor elliptic functions.

Function sn_cn_dn() is not really intended for the end-user.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

K <- ellint_F(phi=pi/2,k=sqrt(1/2))  #note the sqrt: m=k^2
u <- seq(from=0,to=4*K,by=K/24)
jj <- elljac(u,1/2)
plot(u,jj$sn,type="l",xaxt="n",yaxt="n",bty="n",ylab="",xlab="",main="Fig 16.1, p570")
lines(u,jj$cn,lty=2)
lines(u,jj$dn,lty=3)
axis(1,pos=0,at=c(K,2*K,3*K,4*K),labels=c("K","2K","3K","4K"))
abline(0,0)
axis(2,pos=0,at=c(-1,1))
text(1.8*K,0.6,"sn u")
text(1.6*K,-0.5,"cn u")
text(2.6*K,0.9,"dn u")



 a <- seq(from=-5,to=5,len=100)
jj <- outer(a,a,function(a,b){a})
z <- jj+1i*t(jj)
e <- Re(gsl_cd(z,m=0.2))
e[abs(e)>10] <- NA
contour(a,a,e,nlev=55)

Error functions

Description

Error functions as per the Gnu Scientific Library, reference manual section 7.15 and AMS-55, chapter 7. Thes functions are declared in header file gsl_sf_error.h

Usage

erf(x, mode=0, give=FALSE, strict=TRUE)
erfc(x, mode=0, give=FALSE, strict=TRUE)
log_erfc(x, mode=0, give=FALSE, strict=TRUE)
erf_Q(x, mode=0, give=FALSE, strict=TRUE)
hazard(x, mode=0, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
mode

input: mode. For GSL_PREC_DOUBLE , GSL_PREC_SINGLE, GSL_PREC_APPROX use 0,1,2 respectively
strict

Boolean, with TRUE meaning to return NaN if status is an error

Details

The zero functions return a status of GSL_EDOM and a value of NA for n≤0n\leq 0

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

erf(0.745) # Example 1, page 304

exponential functions

Description

Expint functions as per the Gnu Scientific Library, reference manual section 7.17 and AMS-55, chapter 5. These functions are declared in header file gsl_sf_expint.h.

Usage

expint_E1(x, give=FALSE, strict=TRUE)
expint_E2(x, give=FALSE, strict=TRUE)
expint_En(n, x, give=FALSE, strict=TRUE)
expint_Ei(x, give=FALSE, strict=TRUE)
Shi(x, give=FALSE, strict=TRUE)
Chi(x, give=FALSE, strict=TRUE)
expint_3(x, give=FALSE, strict=TRUE)
Si(x, give=FALSE, strict=TRUE)
Ci(x, give=FALSE, strict=TRUE)
atanint(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
n

input: integer values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Note

Function expint_En() requires GSL version 1.8 or later.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0.5, to=1, by=0.01)
cbind(x,Si(x),Ci(x),expint_Ei(x),expint_E1(x))  #table 5.1 of AS, p239

x <- seq(from=0, to=12, len=100)
plot(x,Ci(x),col="black",type="l",xaxt="n",yaxt="n",bty="n",
         xlab="",ylab="",main="Figure 5.6, p232",
         xlim=c(0,12),ylim=c(-1,2.0))
lines(x,Si(x))
axis(1,pos=0)
axis(2,pos=0)
 abline(h=pi/2,lty=2)


# Table 5.4, page 245:
 xvec <- seq(from=0,by=0.01,len=20)
 nvec <- c(2,3,4,10,20)
 x <- kronecker(xvec,t(rep(1,5)))
 n <- kronecker(t(nvec),rep(1,20))
 ans <- cbind(x=xvec,expint_En(n,x))
 rownames(ans) <- rep(" ",length(xvec))
 colnames(ans) <- c("x",paste("n=",nvec,sep=""))
 class(ans) <- "I do not understand the first column"

 ans

Fermi-Dirac functions

Description

Fermi-Dirac functions as per the Gnu Scientific Library, reference manual section 7.18. These functions are declared in header file gsl_sf_fermi_dirac.h

Usage

fermi_dirac_m1(x, give=FALSE, strict=TRUE)
fermi_dirac_0(x, give=FALSE, strict=TRUE)
fermi_dirac_1(x, give=FALSE, strict=TRUE)
fermi_dirac_2(x, give=FALSE, strict=TRUE)
fermi_dirac_int(j, x, give=FALSE, strict=TRUE)
fermi_dirac_mhalf(x, give=FALSE, strict=TRUE)
fermi_dirac_half(x, give=FALSE, strict=TRUE)
fermi_dirac_3half(x, give=FALSE, strict=TRUE)
fermi_dirac_inc_0(x, b, give=FALSE, strict=TRUE)

Arguments

x, j, b

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=2,by=0.01)
fermi_dirac_m1(x)   #table 7.5 of Ab and St

gamma functions

Description

Gamma functions as per the Gnu Scientific Library reference manual section 7.19. These functions are declared in header file gsl_sf_gamma.h

Usage

gsl_sf_gamma(x,give=FALSE,strict=TRUE)
lngamma(x,give=FALSE,strict=TRUE)
lngamma_sgn(x,give=FALSE,strict=TRUE)
gammastar(x,give=FALSE,strict=TRUE)
gammainv(x,give=FALSE,strict=TRUE)
lngamma_complex(zr, zi=NULL, r.and.i=TRUE, give=FALSE, strict=TRUE)
taylorcoeff(n,x,give=FALSE,strict=TRUE)
fact(n,give=FALSE,strict=TRUE)
doublefact(n,give=FALSE,strict=TRUE)
lnfact(n,give=FALSE,strict=TRUE)
lndoublefact(n,give=FALSE,strict=TRUE)
gsl_sf_choose(n,m,give=FALSE,strict=TRUE)
lnchoose(n,m,give=FALSE,strict=TRUE)
poch(a,x,give=FALSE,strict=TRUE)
lnpoch(a,x,give=FALSE,strict=TRUE)
lnpoch_sgn(a,x,give=FALSE,strict=TRUE)
pochrel(a,x,give=FALSE,strict=TRUE)
gamma_inc_Q(a,x,give=FALSE,strict=TRUE)
gamma_inc_P(a,x,give=FALSE,strict=TRUE)
gamma_inc(a,x,give=FALSE,strict=TRUE)
gsl_sf_beta(a,b,give=FALSE,strict=TRUE)
lnbeta(a,b,give=FALSE,strict=TRUE)
beta_inc(a,b,x,give=FALSE,strict=TRUE)

Arguments

x, a, b

input: real values
m, n

input: integer value
zr

In gamma_complex(), the real part of the argument
zi

In gamma_complex(), the imaginary part of the argument. If missing (ie takes the default value of NULL), interpret zr as complex, even if real
r.and.i

In gamma_complex(), Boolean variable with default value of TRUE meaning to return a complex variable as per the details section below; and FALSE meaning to return the values as advertised in the GSL manual
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Details

All functions as documented in the GSL reference manual section 7.19.

Note that gamma_inc_P() gives the area of the left tail of the gamma distribution so, for example, gamma_inc_P(1.8, 5) = pgamma(5, 1.8) to numerical accuracy.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

gsl_sf_gamma(3)

lngamma_complex(1+seq(from=0,to=5,by=0.1)*1i)  #table 6.7, p 277 (LH col)
                                               #note 2pi phase diff


jj <- expand.grid(1:10,2:5)
x <- taylorcoeff(jj$Var1,jj$Var2)
dim(x) <- c(10,4)
x    #table 23.5, p818


jj <- expand.grid(36:50,9:13)
x <- gsl_sf_choose(jj$Var1,jj$Var2)
dim(x) <- c(15,5)
x     #table 24.1, p829  (bottom bit)

gamma_inc(1.2,1.3)
beta(1.2, 1.3)
lnbeta(1.2,1.55)
beta_inc(1.2,1.4,1.6)

gamma_inc_P(1.8, 5) - pgamma(5, 1.8)  # should be small

Gegenbauer functions

Description

Gegenbauer functions as per the Gnu Scientific Library reference manual section 7.20, and AMS-55, chapter 22. These functions are declared in header file gsl_sf_gegenbauer.h

Usage

gegenpoly_1(lambda, x, give=FALSE,strict=TRUE)
gegenpoly_2(lambda, x, give=FALSE,strict=TRUE)
gegenpoly_3(lambda, x, give=FALSE,strict=TRUE)
gegenpoly_n(n,lambda, x, give=FALSE,strict=TRUE)
gegenpoly_array(nmax,lambda, x, give=FALSE,strict=TRUE)

Arguments

lambda, x

input: real values
n, nmax

input: integer value
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=-1 ,to=1,len=300)
y <- gegenpoly_array(6,0.5,x)
matplot(x,t(y[-(1:2),]), xlim=c(-1,1.2),ylim=c(-0.5,1.5),
       type="l",xaxt="n",yaxt="n",bty="n",xlab="",ylab="",
       main="Figure 22.5, p777",col="black")
axis(1,pos=0)
axis(2,pos=0)


plot(x, gegenpoly_n(5,lambda=0.2, x,give=FALSE,strict=TRUE),
xlim=c(-1,1),ylim=c(-1.5,1.5),main="Figure 22.5, p777",
type="n",xaxt="n",yaxt="n",bty="n",xlab="",ylab="")
lines(x, gegenpoly_n(5,lambda=0.2, x,give=FALSE,strict=TRUE))
lines(x, gegenpoly_n(5,lambda=0.4, x,give=FALSE,strict=TRUE))
lines(x, gegenpoly_n(5,lambda=0.6, x,give=FALSE,strict=TRUE))
lines(x, gegenpoly_n(5,lambda=0.8, x,give=FALSE,strict=TRUE))
lines(x, gegenpoly_n(5,lambda=1.0, x,give=FALSE,strict=TRUE))
axis(1,pos=0)
axis(2,pos=0,las=1)

gsl-deprecated

Description

Deprecated Legendre functions as per the Gnu Scientific Library reference manual section 7.24.

Usage

legendre_Plm_array(...)
legendre_Plm_deriv_array(...)
legendre_sphPlm_array(...)
legendre_sphPlm_deriv_array(...)
legendre_array_size(...)
deprecated_legendre(...)

Arguments

...

(ignored)

Note

As of GSL-2.1, functions

  • gsl_sf_legendre_Plm_array

  • gsl_sf_legendre_Plm_deriv_array

  • gsl_sf_legendre_sphPlm_array

  • gsl_sf_legendre_sphPlm_deriv_array

  • gsl_sf_legendre_array_size

are deprecated. This functionality is now provided in GSL by the gsl_sf_legendre_array suite of functions; in R, use one of:

  • legendre_array()

  • legendre_deriv_array()

  • legendre_deriv_alt_array()

  • legendre_deriv2_array()

  • legendre_deriv2_alt_array().

These are documented under ?Legendre.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

See Also

Legendre

Hypergeometric functions

Description

Hypergeometric functions as per the Gnu Scientific Library reference manual section 7.21 and AMS-55, chapters 13 and 15. These functions are declared in header file gsl_sf_hyperg.h

Usage

hyperg_0F1(c, x, give=FALSE, strict=TRUE)
hyperg_1F1_int(m, n, x, give=FALSE, strict=TRUE)
hyperg_1F1(a, b, x, give=FALSE, strict=TRUE)
hyperg_U_int(m, n, x, give=FALSE, strict=TRUE)
hyperg_U(a, b, x, give=FALSE, strict=TRUE)
hyperg_2F1(a, b, c, x, give=FALSE, strict=TRUE)
hyperg_2F1_conj(aR, aI, c, x, give=FALSE, strict=TRUE)
hyperg_2F1_renorm(a, b, c, x, give=FALSE, strict=TRUE)
hyperg_2F1_conj_renorm(aR, aI, c, x, give=FALSE, strict=TRUE)
hyperg_2F0(a, b, x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
a, b, c

input: real values
m, n

input: integer values
aR, aI

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number.
strict

Boolean, with TRUE meaning to return NaN if status is an error

Note

“The circle of convergence of the Gauss hypergeometric series is the unit circle ∣z∣=1|z|=1” (AMS, page 556).

There is a known issue in hyperg_2F1() in GSL-2.6, https://savannah.gnu.org/bugs/?54998 and the package returns the erroneous value given by GSL.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

hyperg_0F1(0.1,0.55)

hyperg_1F1_int(2,3,0.555)
hyperg_1F1(2.12312,3.12313,0.555)
hyperg_U_int(2, 3, 0.555)
hyperg_U(2.234, 3.234, 0.555)

Laguerre functions

Description

Laguerre functions as per the Gnu Scientific Library reference manual section 7.22. These functions are declared in header file gsl_sf_laguerre.h

Usage

laguerre_1(a, x, give=FALSE, strict=TRUE)
laguerre_2(a, x, give=FALSE, strict=TRUE)
laguerre_3(a, x, give=FALSE, strict=TRUE)
laguerre_n(n, a, x, give=FALSE, strict=TRUE)

Arguments

a, x

input: real values
n

input: integer values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=6,len=100)
plot(x,laguerre_n(2,0,x),xlim=c(0,6),ylim=c(-2,3),
          type="l",xaxt="n",yaxt="n",bty="n",xlab="",ylab="",
          main="Figure 22.9, p780")

lines(x,laguerre_n(3,0,x))
lines(x,laguerre_n(4,0,x))
lines(x,laguerre_n(5,0,x))
axis(1,pos=0)
axis(2,pos=0)

Lambert's W function

Description

Lambert's W function as per the Gnu Scientific Library reference manual section 7.23. These functions are declared in header file gsl_sf_lambert.h

Usage

lambert_W0(x, give=FALSE, strict=TRUE)
lambert_Wm1(x, give=FALSE,strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

a <- runif(6)
L <- lambert_W0(a)
print(L*exp(L) - a)

Legendre functions

Description

Legendre functions as per the Gnu Scientific Library reference manual section 7.24, and AMS-55, chapter 8. These functions are declared in header file gsl_sf_legendre.h

Usage

legendre_P1(x, give=FALSE, strict=TRUE)
legendre_P2(x, give=FALSE, strict=TRUE)
legendre_P3(x, give=FALSE, strict=TRUE)
legendre_Pl(l, x, give=FALSE, strict=TRUE)
legendre_Pl_array(lmax, x, give=FALSE, strict=TRUE)
legendre_Q0(x, give=FALSE, strict=TRUE)
legendre_Q1(x, give=FALSE, strict=TRUE)
legendre_Ql(l, x, give=FALSE, strict=TRUE)
legendre_array_n(lmax)
legendre_array_index(l,m)
legendre_check_args(x,lmax,norm,csphase)
legendre_array(x, lmax, norm=1, csphase= -1)
legendre_deriv_array(x, lmax, norm=1, csphase= -1)
legendre_deriv_alt_array(x, lmax, norm=1, csphase= -1)
legendre_deriv2_array(x, lmax, norm=1, csphase= -1)
legendre_deriv2_alt_array(x, lmax, norm=1, csphase= -1)
legendre_Plm(l, m, x, give=FALSE, strict=TRUE)
legendre_sphPlm(l, m, x, give=FALSE, strict=TRUE)
conicalP_half(lambda, x, give=FALSE, strict=TRUE)  
conicalP_mhalf(lambda, x, give=FALSE, strict=TRUE)  
conicalP_0(lambda, x, give=FALSE, strict=TRUE)  
conicalP_1(lambda, x, give=FALSE, strict=TRUE)  
conicalP_sph_reg(l, lambda, x, give=FALSE, strict=TRUE)  
conicalP_cyl_reg(m, lambda, x, give=FALSE, strict=TRUE)
legendre_H3d_0(lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d_1(lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d(l, lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d_array(lmax, lambda, eta, give=FALSE, strict=TRUE)

Arguments

eta, lambda, x

input: real values
l, m, lmax

input: integer values
csphase, norm

Options for use with legendre_array()
give

Boolean, with default FALSE meaning to return just the answers, and TRUE meaning to return a status vector as well
strict

Boolean, with TRUE meaning to return NaN if nonzero status is returned by the GSL function (FALSE means to return the value: use with caution)

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

theta <- seq(from=0,to=pi/2,len=100)
 plot(theta,legendre_P1(cos(theta)),type="l",ylim=c(-0.5,1), main="Figure 8.1, p338")
 abline(1,0)
 lines(theta,legendre_P2(cos(theta)),type="l")
 lines(theta,legendre_P3(cos(theta)),type="l")

x <- seq(from=0,to=1,len=600)
plot(x, legendre_Plm(3,1,x), type="l",lty=3,main="Figure 8.2, p338: note sign error")
lines(x,legendre_Plm(2,1,x), type="l",lty=2)
lines(x,legendre_Plm(1,1,x), type="l",lty=1)
abline(0,0)


plot(x,legendre_Ql(0,x),xlim=c(0,1), ylim=c(-1,1.5), type="l",lty=1,
main="Figure 8.4, p339")
lines(x,legendre_Ql(1,x),lty=2)
lines(x,legendre_Ql(2,x),lty=3)
lines(x,legendre_Ql(3,x),lty=4)
abline(0,0)

#table 8.1 of A&S:
t(legendre_Pl_array(10, seq(from=0,to=1,by=0.01))[1+c(2,3,9,10),])

#table 8.3:
f <- function(n){legendre_Ql(n, seq(from=0,to=1,by=0.01))}
sapply(c(0,1,2,3,9,10),f)


# Some checks for the legendre_array() series:

# P_6^1(0.3):
legendre_array(0.3,7)[7,2]         # MMA:  LegendreP[6,1,0.3]; note off-by-one issue

# d/dx  P_8^5(x) @ x=0.2:
legendre_deriv_array(0.2,8)[9,6]   # MMA: D[LegendreP[8,5,x],x] /. {x -> 0.2}


# alternative derivatives:
 legendre_deriv_alt_array(0.4,8)[9,6]  # D[LegendreP[8,5,Cos[x]],x] /. x -> ArcCos[0.4]

Log functions

Description

Log functions as per the Gnu Scientific Library, reference manual section 7.25 and AMS-55, chapter 4. These functions are declared in header file gsl_sf_log.h

Usage

gsl_sf_log(x, give=FALSE, strict=TRUE)
log_abs(x, give=FALSE, strict=TRUE)
complex_log(zr, zi=NULL, r.and.i=TRUE, give=FALSE, strict=TRUE)
log_1plusx(x, give=FALSE, strict=TRUE)
log_1plusx_mx(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
zr

In complex_log(), the real part of the argument
zi

In complex_log(), the imaginary part of the argument. If missing (ie takes the default value of NULL), interpret zr as complex, even if real
r.and.i

In complex_log(), Boolean variable with default value of TRUE meaning to return a complex variable as per the details section below; and FALSE meaning to return the values as advertised in the GSL manual
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0.1,to=2,by=0.01)
log(x)   #table 7.5 of Ab and St

Argument processing and general info

Description

Various widely used functions in the package

Usage

process.args(...)
strictify(val,status)

Arguments

...

Argument list to be coerced to the same length
val

Value component of &result
status

status integer

Details

Function process.args() is an internal function used to massage the arguments into a form suitable for passing to .C(). For example, in function hyperg_0F1(c,x), one wants each of hyperg_0F1(0.1, c(0.3,0.4)) and hyperg_0F1(c(0.1,0.2), 0.3) and hyperg_0F1(c(0.1,0.2),c(0.3,0.4)) to behave sensibly.

Function process.args() is used widely in the package, taking an arbitrary number of arguments and returning a list whose elements are vectors of the same length. Most of the special functions use process.args() to ensure that the returned value takes the attributes of the input argument with most elements where possible.

Function strictify() uses the status value returned by the “error” form of the GSL special functions to make values returned with a nonzero error a NaN. In most of the special functions, strictify() is called if argument strict takes its default value of TRUE. Setting it to FALSE sometimes returns a numerical value as per the GSL reference manual.

In most of the special functions, if argument give takes its default value of FALSE, only a numerical value is returned. If TRUE, error information and the status (see preceding paragraph) is also returned.

Following tips found on R-devel:

  1. Download and extract source code of R-package gsl

  2. Use gsl-config --libs to get the path to GSL's lib directory (-L<path-to-lib>), use gsl-config --cflags to get the path to GSL's include directory (-I<path-to-include>)

  3. Change Makevars in gsl/src:

    • Add -L<path-to-lib> to PKG_LIBS

    • Add (new) line: PKG_CPPFLAGS=-I<path-to-include>

  4. Install gsl via

    LDFLAGS=-L<path-to-lib>; export LDFLAGS

    CPPFLAGS=-I<path-to-include>;export CPPFLAGS

    R CMD INSTALL gsl

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Function minimization

Description

These functions have been removed from the package temporarily, pending a permanent fix.

Function minimization using the Gnu Scientific Library, reference manual section 35. These functions are declared in header file gsl_multimin.h

Several algorithms for finding (local) minima of functions in one or more variables are provided. All of the algorithms operate locally, in the sense that they maintain a best guess and require the function to be continuous. Apart from the Nelder-Mead algorithm, these algorithms also use a derivative.

Usage

multimin(..., prec=0.0001)
multimin.init(x, f, df=NA, fdf=NA, method=NA, step.size=NA, tol=NA)
multimin.iterate(state)
multimin.restart(state)
multimin.fminimizer.size(state)

Arguments

...

In function multimin(), the argument list passed to multimin.init()
x

A starting point. These algorithms are faster with better initial guesses
f

The function to minimize. This function must take a single numeric vector as input, and output a numeric scalar
df

The derivative of f. This is required for all algorithms except Nelder-Mead
fdf

A function that evaluates f and df simultaneously. This is optional, and is only useful if simultaneous evaluation is faster
method

The algorithm to use, which is one of “conjugate-fr”, “conjugate-pr”, “bfgs”, “steepest-descent” and “nm”
step.size

This step size guides the algorithm to pick a good distance between points in its search
tol

This parameter is relevant for gradient-based methods. It controls how much the gradient should flatten out in each line search. More specifically, let u(t)=f(x+st)u(t) = f(x + st) be the function restricted to the search ray. Then a point tt is tolerable if u′(t)<tolu′(0)u'(t) < tol u'(0). Higher values give more lax linesearches. This parameter trades-off searching intensively in the outer loop (finding search directions) versus the inner loop (finding a good point in a particular direction)
prec

The stopping-rule precision parameter. For the derivative-based methods, a solution is good enough if the norm of the gradient is smaller than prec. For the non-derivative-based methods, a solution is good enough if the norm of successive solutions is smaller than prec
state

This stores all information relating to the progress of the optimization problem

Details

There are two ways to call multimin. The simple way is to merely call multimin directly. A more complicated way is to call multimin.init first, and then repeatedly call multimin.iterate until the guess gets good enough. In addition, multimin.restart can be used with the second approach to discard accumulated information (such as curvature information) if that information turns out to be unhelpful. This is roughly equivalent to calling multimin.init by setting the starting point to be the current best guess.

All of the derivative-based methods consist of iterations that pick a descent direction, and conduct a line search for a better point along the ray in that direction from the current point. The Fletcher-Reeves and Polak-Ribiere conjugate gradient algorithms maintain a a vector that summarizes the curvature at that point. These are useful for high-dimensional problems (eg: more than 100 dimensions) because they don't use matrices which become expensive to keep track of. The Broyden-Fletcher-Goldfarb-Shanno is better for low-dimensional problems, since it maintains an approximation of the Hessian of the function as well, which gives better curvature information. The steepest-descent algorithm is a naive algorithm that does not use any curvature information. The Nelder-Mead algorithm which does not use derivatives.

Value

All of these functions return a state variable, which consists of the following items:

internal.state

Bureaucratic stuff for communicating with GSL
x

The current best guess of the optimal solution
f

The value of the function at the best guess
df

The derivative of the function at the best guess
is.fdf

TRUE if the algorithm is using a derivative
code

The GSL return code from the last iteration

Note

The source code for the functions documented here conditionalizes on WIN32; under windows there is a slight memory leak.

Author(s)

Andrew Clausen [email protected]

References

https://www.gnu.org/software/gsl/

See Also

optim and nlm are the standard optimization functions in R.

deriv and D are the standard symbolic differentation functions in R. Ryacas provides more extensive differentiation support using Yet Another Computer Algebra System.

numericDeriv is the standard numerical differentation function in R. GSL can also do numerical differentiation, but no-one has written an R interface yet.

multimin requires the objective function to have a single (vector) argument. unlist and relist are useful for converting between more convenient forms.

Examples

# COMMENTED OUT PENDING PERMANENT FIX
# The Rosenbrock function:

# x0 <- c(-1.2, 1)
# f <- function(x) (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
# df <- function(x) c(-2*(1 - x[1]) + 100 * 2 * (x[2] - x[1]^2) * (-2*x[1]),
#                     100 * 2 * (x[2] - x[1]^2))
# 
# # The simple way to call multimin.
# state <- multimin(x0, f, df)
# print(state$x)
# 
# # The fine-control way to call multimin.
# state <- multimin.init(x0, f, df, method="conjugate-fr")
# for (i in 1:200)
# 	state <- multimin.iterate(state)
# print(state$x)

Polynomials

Description

Polynomial functions as per the Gnu Scientific Library, reference manual section 6.1. These functions are defined in header file gsl_poly.h

Usage

gsl_poly(c_gsl,x)

Arguments

c_gsl

Coefficients of the poynomial (c in the function definition and the GSL ref manual) starting at the constant term and ending in the highest power; see details section. This argument is called “c_gsl” (and not “c”) to avoid confusion with R function c()
x

input: real values

Details

One must be careful to avoid off-by-one errors. In C idiom, the function evaluates the polynomial

c[0]+c[1]x+c[2]x2+…+c[len−1]xlen−1c[0]+c[1]x+c[2]x^2+\ldots+c[\mathrm{len}-1]x^{\mathrm{len}-1}

where len is the second argument of GSL function gsl_poly_eval().

The R idiom would be

c[1]+c[2]x+c[3]x2+…+c[len]xlen−1.c[1]+c[2]x+c[3]x^2+\ldots+c[\mathrm{len}]x^{\mathrm{len}-1}.

This section is work-in-progress and more will be added when I have the time/need for the other functions here.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

a <- matrix(1:4,2,2)
rownames(a) <- letters[1:2]
(jj <- gsl_poly(1:3,a))

jj-(1 + 2*a + 3*a^2)  #should be small

Power functions

Description

Power functions as per the Gnu Scientific Library reference manual section 7.27. These functions are declared in the header file gsl_sf_pow_int.h

Usage

pow_int(x, n, give=FALSE, strict=TRUE)

Arguments

x

input: real values
n

input: integer values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

pow_int(pi/2,1:10)

Psi (digamma) functions

Description

Psi (digamma) functions as per the Gnu Scientific Library, reference manual section 7.27. These functions are declared in header file gsl_sf_psi.h

Usage

psi_int(n, give=FALSE, strict=TRUE)
psi(x, give=FALSE, strict=TRUE)
psi_1piy(y, give=FALSE, strict=TRUE)
psi_1_int(n, give=FALSE, strict=TRUE)
psi_1(x, give=FALSE, strict=TRUE)
psi_n(m, x, give=FALSE, strict=TRUE)

Arguments

m, n

input: integer values
x, y

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with default TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=1.2,to=1.25,by=0.005)
cbind(x,psi(x),psi_1(x))
#tabe 6.1, p267, bottom bit

psi_int(1:6)
psi(pi+(1:6))
psi_1piy(pi+(1:6))
psi_1_int(1:6)
psi_n(m=5,x=c(1.123,1.6523))

Quasi-random sequences

Description

Quasi-random sequences as per the Gnu Scientific Library, reference manual section 18. These functions are declared in header file gsl_qrng.h

Usage

qrng_alloc(type = c("niederreiter_2", "sobol"), dim)
qrng_clone(q)
qrng_init(q)
qrng_name(q)
qrng_size(q)
qrng_get(q, n = 1)

Arguments

type

Type of sequence
dim

Dimension of sequence
q

Generator from qrng_alloc or qrng_clone
n

How many vectors to generate

Details

These are wrappers for the quasi-random sequence functions from the GSL https://www.gnu.org/software/gsl/ with arguments corresponding to those from the library, with a few exceptions. In particular: I have used dim where the GSL uses just d; I have added the n argument to the qrng_get function, so that a single call can generate n vectors; I have not provided R functions corresponding to qrng_free (because R will automatically free the generator when it is garbage collected) or qrng_state or qrng_memcpy (because these don't make sense within R.)

Value

qrng_alloc, qrng_clone and qrng_init return an external pointer to the C structure representing the generator. The internals of this structure are not accessible from within R.

qrng_name returns a character vector giving the name of the generator.

qrng_size returns an integer value giving the internal memory usage of the generator.

qrng_get returns a matrix with n rows and dim columns. Each row is a vector in the quasi-random sequence.

Author(s)

Duncan Murdoch

References

https://www.gnu.org/software/gsl/

Examples

q <- qrng_alloc(dim = 2)
qrng_name(q)
qrng_get(q, 10)

Random numbers generation

Description

Random number generation with the Gnu Scientific Library, as per the reference manual section 17

Usage

rng_alloc(type)
rng_clone(r)
rng_name(r)
rng_max(r)
rng_min(r)
rng_set(r, seed)
rng_get(r, length)
rng_uniform(r, length)
rng_uniform_int(r, N, length)
rng_uniform_pos(r, length)

Arguments

type

In function rng_alloc(), type of random number generator. This argument is taken to be a character string which is matched to the names of the random number generators given in the GSL manual section 17.9, with the initial “gsl_rng_” removed (for example, to use generator gsl_rng_ranlux, set type to ranlux). Partial matching is used; a null string is interpreted as mt19937.
r

Instance of a random number generator. Generate this using function rng_alloc().
seed

Random number seed
length

Length of vector of random numbers to create
N

In function rng_uniform_int(), upper bound of uniform distribution

Details

These are wrappers for the random number generator functions from the GSL https://www.gnu.org/software/gsl/ with arguments corresponding to those from the library. Calling rng_free is not necessary as R performs garbage collection automatically.

The functions that return random numbers (rng_get, rng_uniform, rng_uniform_int, rng_uniform_pos) take an extra argument that specifies the length of the vector of random numbers to be returned.

Value

Function rng_alloc() returns an external pointer to a GSL random number generator.

Author(s)

Max Bruche

References

https://www.gnu.org/software/gsl/

Examples

r <- rng_alloc("cmrg")
rng_set(r, 100)
rng_uniform(r, 10)

Synchrotron functions

Description

Synchrotron functions as per the Gnu Scientific Library, reference section 7.29. These functions are declared in header file gsl_sf_synchrotron.h

Usage

synchrotron_1(x, give=FALSE, strict=TRUE)
synchrotron_2(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

Examples

x <- seq(from=0,to=2,by=0.01)
synchrotron_1(x)
synchrotron_2(x)

Transport functions

Description

Transport functions as per the Gnu Scientific Library, reference manual section 7.29. These functions are defined in header file gsl_sf_transport.h

Usage

transport_2(x, give=FALSE, strict=TRUE)
transport_3(x, give=FALSE, strict=TRUE)
transport_4(x, give=FALSE, strict=TRUE)
transport_5(x, give=FALSE, strict=TRUE)

Arguments

x

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number.
strict

Boolean, with TRUE meaning to return NaN if status is an error.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=2,by=0.01)
transport_2(x)
transport_3(x)

Trig functions

Description

Trig functions as per the Gnu Scientific Library, reference manual section 7.30. These functions are declared in header file gsl_sf_trig.h

Usage

gsl_sf_sin(x, give=FALSE, strict=TRUE)
gsl_sf_cos(x, give=FALSE, strict=TRUE)
hypot(x, y, give=FALSE, strict=TRUE)
sinc(x, give=FALSE, strict=TRUE)
complex_sin(zr, zi=NULL, r.and.i=TRUE, give=FALSE, strict=TRUE)
complex_cos(zr, zi=NULL, r.and.i=TRUE, give=FALSE, strict=TRUE)
lnsinh(x, give=FALSE, strict=TRUE)
lncosh(x, give=FALSE, strict=TRUE)

Arguments

x, y

input: real values
zr

In gamma_complex(), the real part of the argument
zi

In complex_sin() et seq, the imaginary part of the argument. If missing (ie takes the default value of NULL), interpret zr as complex, even if real
r.and.i

In complex_sin() et seq, Boolean variable with default value of TRUE meaning to return a complex variable as per the details section below; and FALSE meaning to return the values as advertised in the GSL manual
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number
strict

Boolean, with TRUE meaning to return NaN if status is an error

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

x <- seq(from=0,to=2,by=0.01)
gsl_sf_sin(x)   #table xx of Ab and St
gsl_sf_cos(x)   #table xx of Ab and St

f <- function(x){abs(sin(x+1)-sin(x)*cos(1)-cos(x)*sin(1))}
g <-
function(x){abs(gsl_sf_sin(x+1)-gsl_sf_sin(x)*gsl_sf_cos(1)-gsl_sf_cos(x)*gsl_sf_sin(1))}

f(100000:100010)
g(100000:100010)

Zeta functions

Description

Zeta functions as per the Gnu Scientific Library 7.31 and AMS-55, section 23.2. These functions are declared in header file gsl_sf_zeta.h

Usage

zeta_int(n, give=FALSE, strict=TRUE)
zeta(s, give=FALSE, strict=TRUE)
zetam1_int(n, give=FALSE, strict=TRUE)
zetam1(s, give=FALSE, strict=TRUE)
hzeta(s, q, give=FALSE, strict=TRUE)
eta_int(n, give=FALSE, strict=TRUE)
eta(s, give=FALSE, strict=TRUE)

Arguments

n

input: integer values
s, q

input: real values
give

Boolean with TRUE meaning to return a list of three items: the value, an estimate of the error, and a status number.
strict

Boolean, with TRUE meaning to return NaN if status is an error.

Author(s)

Robin K. S. Hankin

References

https://www.gnu.org/software/gsl/

Examples

n <- 1:10
cbind(n,zeta(n),eta(n))   #table 23.3, p 811


zeta_int(1:5)
zeta(c(pi,pi*2))
zetam1_int(1:5)
zetam1(c(pi,pi*2))
hzeta(1.1,1.2)
eta_int(1:5)
eta(c(pi,pi*2))