functions: Functions for testing and other
In cwhmisc: Miscellaneous Functions for Math, Plotting, Printing, Statistics, Strings, and Tools

Description Usage Arguments Details Value Note Author(s) References Examples

Functions for testing on equality, exactly or with a tolerance, functions usable as parameters in other functions, pythagorean sums, etc.

  chsvd( s )
  chsvd( s )
  divmod( i, n )
  divmodL( i, n )
  dsm( x, w )
  equal( x, y )
  equalFuzzy( x, y, prec=8*.Machine$double.eps, rel=TRUE )
  exch( x, L, R )
  frac(x,d)
  int( x )
  inrange( x, y )
  Ko(z)
  Km(z)
  last( x )
  LE ( x )
  loop.vp( ve, overlap=1 )
  LS ( )
  lV ( x )
  mod( x, y )
  modR( x, y )
  modS( x, y )
  norm2( x )
  one ( x )
  onebyx( x )
  powr( a, x )
  pythag( a, b )
  quotmean( x, y )
  safeDiv( num, den )
  signp( x )
  solveQeq( a, b, c )
  sqr( x )
  sqrtH( x )
  submod( x, v )
  zero ( x )

`prec,L,R`	Real
`a,b,c,z`	Complex
`i`	Integer vector
`d`	If not missing, 'frac' shows 'd' decimals after "." as integer
`n,num,den`	Integer
`rel`	Boolean
`s`	square matrix, result of `svd`
`v`	real vector > 0, preferably cumsum of some other positive vector
`ve`	real any vector or matrix
`overlap`	integer vector, giving element indices/column numbers to be appended at the end, see examples
`x,y`	Real vector
`w`	Real vector > 0

BEWARE of NAs !!
chsvd Check for svd to reproduce matrix.
divmod rbind(div, mod) for ease of use.
divmodL list(d = div, m = mod)
dsm combination of divmod and submod, used in Jul2Dat
equalFuzzy One can choose between relative and absolute precision
equal x == y, of same length.
inrange Check if 'x' (scalar) is in the range (min(y),max(y)).
int returns 'x' as integer in fix format
last return the last element of a vector.
LE short for 'length(x)'.
LS short for '.Last.value'.
loop.vr: loop around vector with overlap.
LS short for '.Last.value'.
modS: same as 'mod', symmetric to 0.
mod = x %% y, x and y with same number of elements.
onebyx = 1.0/x
one returns 1.0, same length as 'x'
powr = x^y, with 0^0 := 1, 0^y := 0, any y
quotmean Compute quotient of means of non-NA elements of x by y
safeDiv Compute quotient, set 0/0 -> 1, and safeguard r/0 <- c3Q otherwise
signp(0) -> 1, signp(complex) -> NA !
solveQeq Solve the quadratic equation a*z^2+ b*z+c given
the coefficients a,b,c OR c(a,b,c), returns always *two*
solutions; if a = b = 0 returns c(Inf, Inf)
sqr = x^2
submod analog to divmod for unequally spaced data, c(greatest index gi of v s.t. v < x, x - v['gi']
zero returns 0.0, same length as 'x'

exch: Exchanges elements 'L' and 'R': x[which( x == L )] <- R; x[which( x == R )] <- L
K: Cayley transform (z - i)/(z + i)
Km: (1 + z)i/(1 - z), inverse transformation of K norm2: 2-norm.
last: last element of vector LE: = length of vector pythag: c(A,B,C), A=final a' = sqrt( a^2 + b^2 ) without squaring and
taking the square root, avoiding overflow and underflow, B=final b',
C=residual = final (b'/a')^2, see note.
signp: ifelse( is.na(x) | (!is.finite(x) | x>=0),1,-1 ), avoiding
NA, NaN and 0 in the result.
sqrtH: Square root with Halley's hyperbolical method.

see also examples of date
Note that 1 results with signp( 0 ) ;
It is not possible to discriminate between Inf and -Inf, by definition in R,
but: as.character(-Inf) = "-Inf". pythag: The invariant of the iteration is sqrt(a^2 + b^2), iterating a':=max(a,b) and reducing b':=min(a,b).

Christian W. Hoffmann <christian@echoffmann.ch>

Moler, C. and Morrison. D, 1983 Replacing Square Roots by Pythagorean Sums, IBM J.Res.Devel., 27, 6, 577–589.
jjam(Prime numbers); Mathpath(Formula for Halley); Wikipedia(Derivation of Halley)

int (c(0,pi,2*pi,30*pi))    # 0  3  6 94
frac(c(0,pi,2*pi,30*pi))    # 0.000000 0.141593 0.283185 0.247780
frac(c(0,pi,2*pi,30*pi), 3) # 0 142 283 248
y <- c( Inf, -Inf,NA,  NaN, -NaN,-1, 0, 1 )
signp(c(-1:1,NA,NaN,Inf, -Inf)) # -1   1   1  1  1     1  1
#  instead of sign() =            -1   0   1  NA NaN   1  -1
mod((-3:5),4 ) # 1 2 3 0 1 2 3 0 1
modS((-3:5),4) # -3 -2 -1  0  1  2  3  0  1
x <- 200; y <- x + 0.1
equalFuzzy(x,y,0.1*c(10^(-3:0))) # FALSE  TRUE  TRUE  TRUE
equalFuzzy(x,y,0.1*c(10^(-3:0)),FALSE) # FALSE FALSE FALSE  TRUE
loop.vp(1:4) #  1 2 3 4 1
loop.vp(matrix(1:12,nrow=3),c(2,4))
#     [,1] [,2] [,3] [,4] [,5] [,6]
#[1,]    1    4    7   10    4   10
#[2,]    2    5    8   11    5   11
#[3,]    3    6    9   12    6   12
safeDiv(0:3,c(0,0:2)) # 1.552518e+231
signp(c(-1:1,NA,NaN,Inf, -Inf)) # -1   1   1  1  1     1  1
#  instead of sign() =            -1   0   1  NA NaN   1  -1
solveQeq(0,0,1) # NA NA
solveQeq(0,1,0) # 0
solveQeq(0,1,1) # -1
solveQeq(1,0,0) #  0 0
solveQeq(1,0,1) # 0-1i 0+1i
solveQeq(1,1,0) #  -1  0
solveQeq(1,1,1) #  -0.5-0.866025i  -0.5+0.866025i
solveQeq(sample(1:4,1),sample(1:4,1),sample(1:4,1))
x <- matrix(rnorm(9),3,3)
s <- svd(x)
lV(s$d)
norm(chsvd(s) - x) # 9.4368957e-16
submod(8.1,c(10.3, 31) )  # 0.0 8.1
submod(18.1,c(10.3, 31) )  # 1.0 7.8
exch(LETTERS, "A", "Y") # "Y" "B" ... "W" "X" "A" "Z"
exch(1:5, "2", "Y") # "1" "Y" "3" "4" "5"
pythag(19,180) # 1.8100000e+02 3.8414499e-23