hadamard(2)
LIBRARY(PRACMA)
library(pracma)
hadamard(2)
hadamard(4)
H12 <- hadamard(12)
t(H12) %*% H12
diag(20, 20, 20)
base::diag(20, 20, 20)
H12 <- hadamard(12)#
all.equal(t(H12) %*% H12,#
          diag(12, 12, 12))#
H20 <- hadamard(20)#
all.equal(t(H20) %*% H20,#
          diag(20, 20, 20))
toeplitz(c(1, 2, 3, 4, 5))#
toeplitz(c(1, 2, 3, 4, 5), c(1.5, 2.5, 3.5, 4.5, 5.5))
toeplitz(c(1, 2, 4, 6, 8), c(1, 3, 5, 7, 9))
setwd('/Users/HwB/Work/myRforge/optimist/pkg/pracma/tests')
all.equal(hadamard(2),#
          matrix(c(1, 1, 1, -1), 2, 2))#
all.equal(hadamard(4),#
          matrix(c(1, 1, 1, -1,#
                   1, -1, 1, -1,#
                   1, 1, -1, -1,#
                   1, -1, -1, 1), 2, 2))#
H12 <- hadamard(12)#
all.equal(t(H12) %*% H12,#
          diag(12, 12, 12))#
H20 <- hadamard(20)#
all.equal(t(H20) %*% H20,#
          diag(20, 20, 20))#
#
all.equal(toeplitz(c(1, 2, 4, 6, 8), c(1, 3, 5, 7, 9)),#
          matrix(c(1, 3, 5, 7, 9,#
                   2, 1, 3, 5, 7,#
                   4, 2, 1, 3, 5,#
                   6, 4, 2, 1, 3,#
                   8, 6, 4, 2, 1), 5, 5, byrow = TRUE))
setwd('/Users/HwB/Work/myRforge/optimist/pkg/pracma/tests')
all.equal(hadamard(2),#
          matrix(c(1, 1, 1, -1), 2, 2))#
all.equal(hadamard(4),#
          matrix(c(1, 1, 1, -1,#
                   1, -1, 1, -1,#
                   1, 1, -1, -1,#
                   1, -1, -1, 1), 4, 4))#
H12 <- hadamard(12)#
all.equal(t(H12) %*% H12,#
          diag(12, 12, 12))#
H20 <- hadamard(20)#
all.equal(t(H20) %*% H20,#
          diag(20, 20, 20))#
#
all.equal(toeplitz(c(1, 2, 4, 6, 8), c(1, 3, 5, 7, 9)),#
          matrix(c(1, 3, 5, 7, 9,#
                   2, 1, 3, 5, 7,#
                   4, 2, 1, 3, 5,#
                   6, 4, 2, 1, 3,#
                   8, 6, 4, 2, 1), 5, 5, byrow = TRUE))
hadamard(4)
setwd('/Users/HwB/Work/myRforge/optimist/pkg/pracma/tests')
all.equal(hadamard(2),#
          matrix(c(1, 1, 1, -1), 2, 2))#
all.equal(hadamard(4),#
          matrix(c(1,  1,  1,  1,#
                   1, -1,  1, -1,#
                   1,  1, -1, -1,#
                   1, -1, -1,  1), 4, 4))#
H12 <- hadamard(12)#
all.equal(t(H12) %*% H12,#
          diag(12, 12, 12))#
H20 <- hadamard(20)#
all.equal(t(H20) %*% H20,#
          diag(20, 20, 20))#
#
all.equal(toeplitz(c(1, 2, 4, 6, 8), c(1, 3, 5, 7, 9)),#
          matrix(c(1, 3, 5, 7, 9,#
                   2, 1, 3, 5, 7,#
                   4, 2, 1, 3, 5,#
                   6, 4, 2, 1, 3,#
                   8, 6, 4, 2, 1), 5, 5, byrow = TRUE))
library(gsl)
?fact
?lngamma_complex
lngamma_complex(1+seq(from=0,to=5,by=0.1)*1i)
lngamma_complex(1+seq(from=0,to=5,by=0.5)*1i)
y <- seq(from=0,to=5,by=0.1)#
exp(lngamma_complex(1+y*1i))#
#
gammaz(1+y*1i)
y <- seq(from=0,to=5,by=0.5)#
# z0 <- exp(lngamma_complex(1+y*1i))#
z0 <- c(0.0000000+0.0000000i, -0.1909455-0.2440583i, -0.6509232-0.3016403i,#
       -1.2344831-0.1629398i, -1.8760788+0.1296463i, -2.5499068+0.5426044i,#
       -3.2441443+1.0533508i, -3.9524671+1.6461926i, -4.6710996+2.3096981i,#
       -5.3976062+3.0351970i, -6.1303241-2.4672867i)#
#
all.equal(gammaz(1+y*1i), z0)
abs(gammaz(1+y*1i) - z0)
gammaz(1+y*1i)
y <- seq(from=0,to=5,by=0.1)
z1 <- gammaz(1+y*1i)
y <- seq(from=0,to=5,by=0.5)#
# z0 <- exp(lngamma_complex(1+y*1i))#
z0 <- c(0.0000000+0.0000000i, -0.1909455-0.2440583i, -0.6509232-0.3016403i,#
       -1.2344831-0.1629398i, -1.8760788+0.1296463i, -2.5499068+0.5426044i,#
       -3.2441443+1.0533508i, -3.9524671+1.6461926i, -4.6710996+2.3096981i,#
       -5.3976062+3.0351970i, -6.1303241-2.4672867i)#
#
all.equal(gammaz(1+y*1i), exp(z0))
y <- seq(from=0,to=5,by=0.5)#
# z0 <- exp(lngamma_complex(1+y*1i))#
z0 <- c(0.0000000+0.0000000i, -0.1909455-0.2440583i, -0.6509232-0.3016403i,#
       -1.2344831-0.1629398i, -1.8760788+0.1296463i, -2.5499068+0.5426044i,#
       -3.2441443+1.0533508i, -3.9524671+1.6461926i, -4.6710996+2.3096981i,#
       -5.3976062+3.0351970i, -6.1303241-2.4672867i)#
#
all.equal(gammaz(1+y*1i), exp(z0), tolerance = 1e-7)
?lambertWp
?lambertW_p
??lambertW
x <- seq(0, 10, len=21)
x
lambert_W0(x)
lambertW(x)
lambertWp(x)
x <- seq(0, 10, len=21)#
# y <- lambert_W0(x)#
y0 <- c(0.0000000, 0.3517337, 0.5671433, 0.7258614, 0.8526055, 0.9585864,#
        1.0499089, 1.1302893, 1.2021679, 1.2672378, 1.3267247, 1.3815454,#
        1.4324048, 1.4798568, 1.5243452, 1.5662310, 1.6058120, 1.6433371,#
        1.6790164, 1.7130288, 1.7455280)#
#
all.equal(lambertWp(x), y0)
x <- seq(0, 10, len=21)#
# y <- lambert_W0(x)#
y0 <- c(0.0000000, 0.3517337, 0.5671433, 0.7258614, 0.8526055, 0.9585864,#
        1.0499089, 1.1302893, 1.2021679, 1.2672378, 1.3267247, 1.3815454,#
        1.4324048, 1.4798568, 1.5243452, 1.5662310, 1.6058120, 1.6433371,#
        1.6790164, 1.7130288, 1.7455280)#
#
all.equal(lambertWp(x), y0, tolerance = 1e-7)
x <- c(1, 2, 3, 4, 5, 6)#
y <- c(16, 18, 21, 17, 15, 12)#
plot(x, y, col="red", xlim=c(0,7), ylim=c(10,22),#
     main = "Spline and 'pchip' Interpolation Polynomials")#
grid()
xs <- seq(1, 6, len=51)#
ys <- interp1(x, y, xs, "spline")#
lines(xs, ys, col="cyan")#
yp <- pchip(x, y, xs)#
lines(xs, yp, col = "magenta")}
xs <- seq(1, 6, len=51)#
ys <- interp1(x, y, xs, "spline")#
lines(xs, ys, col="cyan")#
yp <- pchip(x, y, xs)#
lines(xs, yp, col = "magenta")}
xs <- seq(1, 6, len=51)#
ys <- interp1(x, y, xs, "spline")#
lines(xs, ys, col="cyan")#
yp <- pchip(x, y, xs)#
lines(xs, yp, col = "magenta")
pchip(x, y, 1.15)
xs <- c(1.5, 2.5, 3.5, 4.5, 5.5)#
ys <- pchip(x, y, xs)
ys
?interp1
y2 <- interp1(x, y, xs, method="cubic")
y2
x <- c(1, 2, 3, 4, 5, 6)#
y <- c(16, 18, 21, 17, 15, 12)#
#
xs <- c(1.5, 2.5, 3.5, 4.5, 5.5)#
ys <- pchip(x, y, xs)#
# ys <- interp1(x, y, xs, method="cubic")       # the same#
# 16.88750 19.80000 19.33333 15.96667 13.63750#
#
yml <- c(16.887499999999999, 19.800000000000001, 19.333333333333332,#
         15.966666666666667, 13.637499999999999)#
#
all.equal(ys, yml, tolerance = 1e-7)