Class: NumRu::GPhys

Inherits:
Object
  • Object
show all
Defined in:
lib/numru/ganalysis/fitting.rb,
lib/numru/gphys/grib.rb,
lib/numru/ganalysis/eof.rb,
lib/numru/ganalysis/met.rb,
lib/numru/gphys/ep_flux.rb,
lib/numru/gphys/version.rb,
lib/numru/gphys/gphys_io.rb,
lib/numru/ganalysis/log_p.rb,
lib/numru/gphys/gphys_fft.rb,
lib/numru/gphys/derivative.rb,
lib/numru/gphys/interpolate.rb,
lib/numru/gphys/gphys_dim_op.rb,
lib/numru/ganalysis/histogram.rb,
lib/numru/gphys/gphys_grib_io.rb,
lib/numru/ganalysis/covariance.rb,
lib/numru/gphys/coordtransform.rb,
lib/numru/gphys/gphys_grads_io.rb,
lib/numru/gphys/gphys_gtool3_io.rb,
lib/numru/gphys/gphys_io_common.rb,
lib/numru/gphys/gphys_netcdf_io.rb,
lib/numru/gphys/gphys_nusdas_io.rb,
lib/numru/gphys/gphys_hdfeos5_io.rb,
ext/numru/gphys/dim_op.c,
ext/numru/gphys/interpo.c

Overview

GPhys extension with GAnalysis::Fitting

Defined Under Namespace

Modules: Derivative, EP_Flux, GrADS_IO, GribUtils, Grib_IO, Gtool3_IO, HE5_IO, IO, IO_Common, NetCDF_IO, NuSDaS_IO Classes: Grib, GribDim, GribVar

Constant Summary collapse

VERSION =

Add alpha while under development; remove it to release

"1.5.0"
COS_TAPER_SP_FACTOR = 
1.0 / 0.875
BC_SIMPLE =

enum in convol_filter.c

10
BC_CYCLIC =

enum in convol_filter.c

11
BC_TRIM =

enum in convol_filter.c

12
@@fft_forward = 
-1
@@fft_backward = 
1
@@fft_ignore_missing = 
false
@@fft_missing_replace_val = 
nil
@@interpo_previous_cutter = 
nil
@@interpo_previous_modifier = 
nil
@@interpo_missval =

NC_FILL_DOUBLE/FLOAT ~15*2^119

9.9692099683868690e+36
@@interpo_extrapolation = 
false
@@default_missval =

NC_FILL_DOUBLE/FLOAT ~15*2^119

9.9692099683868690e+36

Class Method Summary collapse

Instance Method Summary collapse

Class Method Details

.c_cap_by_boundary ⇒ Object

cap_by_boundary : Cap (insert) a NArray with boundary values

Restriction; data alignment is restricted so that the beginning of the out data is always valid (within the domain). To ensure it, it should be either zcrd is increasing and upper==true or zcrd is decreasing and upper==false.

RETURN VALUES

fe: f capped by the boundary values. The dimension zdim is

extended by 1; i.e., f[:,nz,:] --> fe[:,nz+1,:], where ":" respresent
arbitrary number of dimensions. The elements of fe are equal to
those of f where inside the domain (simple copies), and they are equal
to the elements of fs at the bondary (simple copies if fs is given;
if not, guessed by interpolation or naive extension).

ze: grid points of fe along zdim. It is a mixture of zcrd and zs;

it is zcrd inside the domain (where f is copied), and it is zs
at the boundary (where fs is copied).
Same shape as fe.

nze: The number of valid data along zdim of fe. Shaped as ze,

according to the notation above. For example, when fe is 4D and
zdim==2, fe[i,j,k,l] is valid for k = 0,1,...,nze[i,j,l]-1,
where the boundary is at nze[i,j,l]-1. Thus, nze is always
smaller than or equal to the length of zdim of fe (which is nz+1)



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# File 'ext/numru/gphys/dim_op.c', line 1023

static VALUE
cap_by_boundary(obj, f, zdim, zcrd, upper, zb, fb)
VALUE obj;

.c_cell_integ_irreg ⇒ Object

cell_integ_irreg: trapezoidal numerical integration over coordinate cells, supporting irregular grid

Description

Suppose a multi-dimensional NArray f, where colon represents any number of dimensions, and k is the “z” dimension along which integration is made. We write its real space representation as f(z; x), where x symbolically represents all of the independent variables other than z, and for simplicity, we further write it as f(z).

z is sampled at z_k, k=0,1,…,nzbound-1. This method allows z_k to be defined for each z column, so it requires a multi-D NArray argument z (having the same shape as f). Optionally, nzbound can also vary as nzbound. If, instead, nil is given to nzbound, the entire z grid is used; nzbound is set to f.shape(zdim).

We define the integration of f as

           {    \int_za^zb f(z) dz,  when za<=zb,
I(za,zb) = {
           {   -\int_za^zb f(z) dz,  otherwise.

In other words, our integration is always made from the smaller end to the greater end.

In the normal use case (when w is given nil), we define the cell integration as,

I(-\infty, zc_0), I(zc_0, zc_1), I(zc_1, zc_2),...,

The cell boundaries zc_m (m=0,1,..) are specified by the 1D NArray argument “ccell”; ccell must be aligned in the increasing order.

This method allows coordinate transformation by specifying another coordinate variable w (having the same shape as f). In this case, the ccell argument specifies a coordinate with respect to w: wc_m (m=0,1,…; wc_m must be in the increasing order). The integration is still taken with respect to z, so the cell integration is expressed as

I(-\infty, z(wc_0)), I(z(wc_0), z(wc_1)), I(z(wc_1), z(wc_2)),...,

The grid values z and w do not have to be monotonic; the numerical integration properly treats the contribution from multiple ranges along k. Mathematically, the coordinate-transferred integration over the w bin (-infty, wc] is expressed as

\int_-\infty^+\infty H(wc-w(z)) f(z) dz,

where H is the Heaviside function. The normal use case (without w) is simply when w is z itself, which is exploited in implementation.



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# File 'ext/numru/gphys/dim_op.c', line 790

static VALUE
cell_integ_irreg(obj, f, z, zdim, nzbound, ccell, w)
VALUE obj;

.c_cum_integ_irreg ⇒ Object

cum_integ_irreg : similar to cell_integ_irreg but it sums up along the axis. – This method acutually uses cell_integ_irreg and make sumation.



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# File 'ext/numru/gphys/dim_op.c', line 979

static VALUE
cum_integ_irreg(obj, f, z, zdim, nzbound, ccell, w)
VALUE obj;

.extrapolation=(extrapo) ⇒ Object

Change the behavior of the interpolation methods to extrapolate outside the grid coverage.

ARGUMENTS

  • extrapo : true or false — the default behaviour is false (not to extrapolate), so use this method if you want to set it to true.



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# File 'lib/numru/gphys/interpolate.rb', line 28

def self.extrapolation=(extrapo)
  @@interpo_extrapolation = extrapo
end

.fft_ignore_missing(ignore = true, replace_val = nil) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 331

def self.fft_ignore_missing( ignore=true, replace_val=nil )
  @@fft_ignore_missing = ignore 
  @@fft_missing_replace_val = replace_val
end

.interpo_find_loc_1D ⇒ Object

to make “find loc” methods available outside GPhys as class methods



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# File 'ext/numru/gphys/interpo.c', line 311

static VALUE
interpo_find_loc_1D(obj, X, x, missval, extrapo)
VALUE obj;

.interpo_find_loc_1D_MD ⇒ Object

To apply interpo_find_loc_1D multi-dimensionally



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# File 'ext/numru/gphys/interpo.c', line 372

static VALUE
interpo_find_loc_1D_MD(obj, X, x, dimc, missval, extrapo)
VALUE obj;

Instance Method Details

#bin_mean(dim, len, nminvalid = 1) ⇒ Object

Binning along a dimension (mean)

The values are averaged every “len” grids; unlike running_mean the number of grids is reduced to 1/len. Currently, the only available boundary condition is BC_TRIM.

ARGUMENTS

  • dim (Integer or String) : the dimension

  • len (Integer): length of the bin

  • nminvalid (Integer; optional; defualt=1): Effective only for data with missing. Minimum number of grid points needed for averaging (1 to len).

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/gphys_dim_op.rb', line 113

def bin_mean(dim, len, nminvalid=1)
  dim = dim_index(dim)  # to handle String or negative specification
  GPhys.new( grid.binning(dim, len), data.bin_mean(dim, len, nminvalid) )
end

#bin_sum(dim, len, nminvalid = 1) ⇒ Object

Binning along a dimension (summation)

Similar to bin_mean, but the values are simply summed without averaging

ARGUMENTS

  • dim (Integer or String) : the dimension

  • len (Integer): length of the bin

  • nminvalid (Integer; optional; defualt=1): Effective only for data with missing. Minimum number of grid points needed for averaging (1 to len).

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/gphys_dim_op.rb', line 131

def bin_sum(dim, len, nminvalid=1)
  dim = dim_index(dim)  # to handle String or negative specification
  GPhys.new( grid.binning(dim, len), data.bin_sum(dim, len, nminvalid) ) 
end

#cderiv(*args) ⇒ Object 



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# File 'lib/numru/gphys/derivative.rb', line 91

def cderiv(*args)
  Derivative::cderiv(self,*args)
end

#coord_data_reverse(axname, pos) ⇒ Object

Reverse the main data (i.e., the dependent variable) and one of the coordinates (an independent variable) through interpolation.

Returns a GPhys in which the main data is the specfied coordinate (argument: axname) sampled at specified locations (argument: pos) in terms of the main data of self. The main data of self is expected to be quai-monotonic with respect to the specfied coordinate.

ARGUMENTS

  • axname [String] : one of the names of the axes (i.e. main coordinates. Auxiliary coordinates are not supported as the target.)

  • pos [NArray] : grid locations. For example, if the current data is potential temperature theta, pos consists of the theta levels to make sampling.

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/interpolate.rb', line 231

def coord_data_reverse(axname,pos)
  gp = self.axis(axname).to_gphys
  gp = self.shape_coerce_full(gp)[0]   # conform the shape to that of self
  gp = GPhys.new( gp.grid.copy, gp.data )  # copy grid to avoid side effect
                                           # on the grid of self
  gp.set_assoc_coords([self])
  pos = NArray[*pos].to_type(NArray::FLOAT) if pos.is_a?(Array)
  newcrd = VArray.new(pos,self.data,self.name)  # succeeds the attributes
  gp.interpolate(axname=>newcrd)
end

#coordtransform(coordmapping, axes_to, *dims) ⇒ Object 



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# File 'lib/numru/gphys/coordtransform.rb', line 41

def coordtransform( coordmapping, axes_to, *dims )

  rankmp = coordmapping.rank

  #< check arguments >
  if axes_to.length != rankmp
    raise ArgumentError,
      "length of axes_to must be equal to the rank of coordmapping"
  end
  if self.rank == rankmp
    dims = (0...rankmp).collect{|i| i}
  elsif self.rank < rankmp
    raise ArgumentError,"rank of coordmapping is greater than self.rank"
  elsif dims.length != rankmp
    raise ArguemntError,
      "# of dimensions speficied is not equal to the rank of coordmapping"
  elsif dims != dims.sort
    raise ArguementErroor,"dims must be in the increasing order"
  end

  #< get grid points >
  vt = coordmapping.map_grid( *dims.collect{|d| axes_to[d].pos.val} )
  x = dims.collect{|d| self.grid.axis(d).pos.val}
  #< prepare the output object >
  axes = (0...self.rank).collect{|i| grid.axis(i)}
  dims.each_with_index{|d,j| axes[d]=axes_to[j]}
  grid_to = Grid.new( *axes )
  vnew = VArray.new( NArray.new( self.data.ntype, *grid_to.shape ),
                    self.data, self.name )

  #< do interpolation (so far only 2D is supported) >
  case dims.length
  when 2
    if !HAVE_NUMRU_SSL2

      p "interpolation without SSL2"
#         raise "Sorry, so far I need SSL2 (ruby-ssl2)"
      self.each_subary_at_dims_with_index( *dims ){ |fxy,idx|

        wgts = Array.new
        idxs = Array.new

        for d in 0..dims.length-1
          wgt = vt[d].dup.fill!(-1.0)
          idx0 = vt[d].dup.to_i.fill!(-1)
          idx1 = idx0.dup.fill!(x[d].length)

          xsort = x[d].sort
          xsortindex = x[d].sort_index
          for i in 0..x[d].length-1
            idx0[ xsort[i] <= vt[d] ] = xsortindex[i]
            idx1[ xsort[-1-i] >= vt[d] ] = xsortindex[-1-i]
          end

          # where idx0=idx1
          wgt[ idx0.eq(idx1) ] = 1.0

          # where vt[d] < x[d].min
          wgt[ idx0 <= -1 ] = 1.0
          idx0[ idx0 <= -1 ] = 0

          # where vt[d] > x[d].max
          wgt[ idx1 >= x[d].length ] = 0.0
          idx1[ idx1 >= x[d].length ] = x[d].length-1

          # normal points
          mask = wgt.eq(-1.0)
          wgt[mask] = (vt[d][mask]-x[d][idx0[mask]])/(x[d][idx1[mask]]-x[d][idx0[mask]])

          wgts.push(wgt)
          idxs[d*2] = idx0
          idxs[d*2+1] = idx1

        end

        case dims.length
#            when 1
#              f =   fxy.data.val[idxs[0]]*(1-wgts[0]) + 
#                    fxy.data.val[idxs[1]]*wgts[0]
#              f = f.to_na if( f.class.to_s == "NArrayMiss" )
        when 2
          lx = fxy.shape[0]
          f =   ( fxy.data.val[idxs[0]+idxs[2]*lx]*(1-wgts[0]) + 
                  fxy.data.val[idxs[1]+idxs[2]*lx]*wgts[0]
                ) * (1-wgts[1]) + 
                ( fxy.data.val[idxs[0]+idxs[3]*lx]*(1-wgts[0]) + 
                  fxy.data.val[idxs[1]+idxs[3]*lx]*wgts[0] 
                ) * wgts[1]
          f = f.to_na if( f.class.to_s == "NArrayMiss" )
        else
          raise "Sorry, #{v.length}D interpolation is yet to be supported"
        end

        if(idx==false)
          vnew[] = f
        else
          vnew[*idx] = f
        end
      }

    else
      ix=iy=0
      m=3
      self.each_subary_at_dims_with_index( *dims ){ |fxy,idx|
        c,xt = SSL2.bicd3(x[0],x[1],fxy.val,m)
        begin
          ix,iy,f = SSL2.bifd3(x[0],x[1],m,c,xt,0,vt[0],ix,0,vt[1],iy)
        rescue
          $stderr.print "Interpolation into", vt[0].inspect, vt[1].inspect
          raise $!
        end
        vnew[*idx] = f
      }
    end
  else
    raise "Sorry, #{v.length}D interpolation is yet to be supported"
  end

  #< finish >
  GPhys.new( grid_to, vnew )
end

#corelation(other, *dims) ⇒ Object Also known as: correlation 



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# File 'lib/numru/ganalysis/covariance.rb', line 93

def corelation(other, *dims)
  GAnalysis.corelation(self, other, *dims)
end

#cos_taper(*dims) ⇒ Object

Spectral factor for the cosine taper. Specta should be multiplied by this.



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# File 'lib/numru/gphys/gphys_fft.rb', line 339

def cos_taper(*dims)
  if dims.length < 1
  raise ArgumentError,'You have to specify one or more dimensions'
  end
  dims.sort!.uniq!
  val = self.data.val
  dims.each{|dim|
  dim = dim_index(dim) if dim.is_a?(String)
  dim += rank if dim < 0
  raise ArgumentError,"dim #{dim} does not exist" if dim<0 || dim>rank
    nx = shape[dim]
  wgt = NArray.float(nx).fill!(1)
    x = 10.0 / nx * (NArray.float(nx).indgen!+0.5) 
  wskl = x.lt(1).where
  wskr = x.gt(9).where
  wgt[wskl] = 0.5*( 1.0 - NMath::cos(Math::PI*x[wskl]) )
  wgt[wskr] = 0.5*( 1.0 - NMath::cos(Math::PI*x[wskr]) )
  wgt.reshape!( *([1]*dim + [nx] + [1]*(rank-dim-1)) )
  val = val*wgt
  }
  to_ret = self.copy
  to_ret.data.val = val
  to_ret
end

#covariance(other, *dims) ⇒ Object 



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# File 'lib/numru/ganalysis/covariance.rb', line 89

def covariance(other, *dims)
  GAnalysis.covariance(self, other, *dims)
end

#dcl_fig_cut(dimx, dimy, ux, uy) ⇒ Object

Interpolation on the DCL window (automatic iso-interval interpolation along a poly line that can be drawn in the current viewport of the DCL window). Used in mouse_cut.

ARGUMENTS

  • dimx [Integer or String] : specifies the dimension corresponding to the UX coordinate. (Here, the UX coordinate is the X coordinate of the DCL’s USER coordinate. For exapmle, longitude if map projection.)

  • dimy [Integer or String] : specifies the dimension corresponding to the UY coordinate. (Here, the UY coordinate is the Y coordinate of the DCL’s USER coordinate. For exapmle, latitude if map projection.)

  • ux [Array] : x values in terms of the UX coordinate

  • uy [Array] : y values in terms of the UY coordinate Lengths of ux and uy must be the same and greter or equal to 2.



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# File 'lib/numru/gphys/interpolate.rb', line 89

def dcl_fig_cut(dimx,dimy,ux,uy)
  len = ux.length
  raise("ux and uy must be arrays with the (same) length >= 2") if len<=1
  raise("ux's len (#{len}) != uy's len (#{uy.length})") if uy.length != len
  vx=Array.new; vy=Array.new
  for i in 0...len
    vx[i],vy[i] = NumRu::DCL.stftrf(ux[i],uy[i]) 
  end
  kx = Array.new
  ky = Array.new
  cut = [true]*rank
  for i in 0...len
    cut[dimx] = ux[i]
    cut[dimy] = uy[i]
    dummy, sl = grid.cut(*cut)
    kx[i] = sl[dimx]
    ky[i] = sl[dimy]
  end
  ndiv = Array.new
  ndsum = [0]
  for i in 0...len-1
    ndiv[i] = Math.sqrt( (kx[i+1]-kx[i])**2 + (ky[i+1]-ky[i])**2).to_i
    ndiv[i] += 1 if i==len-2
    ndsum.push ndsum[-1] + ndiv[i]   # 0, ndiv[0], ndiv[0]+ndiv[1], ...
  end
  ndtot = ndsum[-1]
  vxdiv = NArray.float(ndtot)
  vydiv = NArray.float(ndtot)
  for i in 0...len-1
    if i!=len-2
      a = NArray.float(ndiv[i]).indgen / ndiv[i]
    else
      a = NArray.float(ndiv[i]).indgen / (ndiv[i]-1)
    end
    vxdiv[ndsum[i]...ndsum[i+1]] = (1.0-a)*vx[i] + a*vx[i+1]
    vydiv[ndsum[i]...ndsum[i+1]] = (1.0-a)*vy[i] + a*vy[i+1]
  end
  uxdiv = NArray.float(ndtot)
  uydiv = NArray.float(ndtot)
  for i in 0...ndtot
    uxdiv[i], uydiv[i] = DCL.stitrf(vxdiv[i], vydiv[i])
  end
  cx = coord(dimx)
  xcrd = VArray.new(uxdiv, cx, cx.name)
  cy = coord(dimy)
  ycrd = VArray.new(uydiv, cy, cy.name)
  if (vxdiv[-1]-vxdiv[0]).abs > (vydiv[-1]-vydiv[0]).abs
    cutter = [xcrd,ycrd]  # x will be the main coord var if not map proj
    crd = xcrd
  else
    cutter = [ycrd,xcrd]  # x will be the main coord var if not map proj
    crd = ycrd
  end
  axnm = crd.name
  itr = DCL.sgqtrn
  if itr>=10 and itr<=40
    newcrd = __sp_dist(xcrd,ycrd)
    modifier = Proc.new{|gp|
      newax = Axis.new.set_pos(newcrd)
      gp.grid.set_axis(axnm,newax)
      g = Grid.new( newax )
      gxcrd = GPhys.new(g,xcrd)
      gycrd = GPhys.new(g,ycrd)
      gp.set_assoc_coords([gxcrd, gycrd])
      gp
    }
  else
    modifier = nil
  end
  @@interpo_previous_cutter = cutter
  @@interpo_previous_modifier = modifier

  # < do the job >

  gpnew = interpolate(cutter)
  gpnew = modifier[gpnew] if modifier
  gpnew
end

#deriv2nd(*args) ⇒ Object 



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# File 'lib/numru/gphys/derivative.rb', line 97

def deriv2nd(*args)
  Derivative::deriv2nd(self,*args)
end

#detrend(*dims) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 364

def detrend(*dims)
  if dims.length < 1
  raise ArgumentError,'You have to specify one or more dimensions'
  end
  dims.sort!.uniq!
  val = self.data.val
  dims.each{|dim|
  dim = dim_index(dim) if dim.is_a?(String)
  dim += rank if dim < 0
  raise ArgumentError,"dim #{dim} does not exist" if dim<0 || dim>rank
  if val.is_a?(NArray)
    x = self.coord(dim).val
    x.reshape!( *([1]*dim + [x.length] + [1]*(rank-dim-1)) )
    vmean = val.mean(dim)
    vxmean = (val*x).mean(dim)
    xmean = x.mean(dim)
    x2mean = (x*x).mean(dim)
    denom = x2mean-xmean**2
    if denom != 0
 a = (vxmean - vmean*xmean)/denom
 b = (vmean*x2mean - vxmean*xmean)/denom
    else
 a = 0
 b = vmean
    end
  elsif val.is_a?(NArrayMiss)
    x = self.coord(dim).val
    x.reshape!( *([1]*dim + [x.length] + [1]*(rank-dim-1)) )
    x = NArrayMiss.to_nam( NArray.new(x.typecode, *val.shape) + x,
 val.get_mask ) 
    vmean = val.mean(dim)
    vxmean = (val*x).mean(dim)
    xmean = x.mean(dim)
    x2mean = (x*x).mean(dim)
    denom = x2mean-xmean**2
    meq0 = denom.eq(0).to_na(0)    # ==0 and not masked
    mne0 = denom.ne(0).to_na(0)    # !=0 and not masked
      denom.set_mask(mne0)    # only nonzero part will be used to divide:
    a = (vxmean - vmean*xmean)/denom
    b = (vmean*x2mean - vxmean*xmean)/denom
    a[meq0] = 0
    b[meq0] = vmean[meq0]
  end
  a.newdim!(dim) if !a.is_a?(Numeric)
  b.newdim!(dim) if !b.is_a?(Numeric)
  val = val - a*x-b
  }
  to_ret = self.copy
  to_ret.data.val = val
  to_ret
end

#eof(*args) ⇒ Object 



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# File 'lib/numru/ganalysis/eof.rb', line 229

def eof(*args)
  GAnalysis.eof(self, *args)
end

#fft(backward = false, *dims) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 416

def fft(backward=false, *dims)
  fftw3 = false
  if defined?(FFTW3)
  fftw3 = true
  elsif !defined?(FFTW)
  raise "Both FFTW3 and FFTW are not installed."
  end
  if backward==true
  dir = @@fft_backward
  elsif !backward
  dir = @@fft_forward
  else
  raise ArgumentError,"1st arg must be true or false (or, equivalenty, nil)"
  end

  # <FFT>

  gfc = self.copy  # make a deep clone
  if fftw3
  val = gfc.data.val
  if @@fft_ignore_missing and val.is_a?(NArrayMiss)
    if @@fft_missing_replace_val
 val = val.to_na(@@fft_missing_replace_val)
    else
 val = val.to_na 
    end
    elsif val.is_a?(NArrayMiss) && val.count_invalid == 0
      val = val.to_na 
  end
  fcoef = FFTW3.fft( val, dir, *dims )
  else
  # --> always FFT for all dimensions
  if dims.length == 0
    raise ArgumentError,
 "dimension specification is available only if FFTW3 is installed"
  end
  val = gfc.data.val
  if @@fft_ignore_missing and val.is_a?(NArrayMiss)
    if @@fft_missing_replace_val
 val = val.to_na(@@fft_missing_replace_val)
    else
 val = val.to_na 
    end
    elsif val.is_a?(NArrayMiss) && val.count_invalid == 0
      val = val.to_na 
  end
  fcoef = FFTW.fftw( val, dir )
  end
  if dir == @@fft_forward
  if dims.length == 0
    fcoef = fcoef / fcoef.length.to_f   # normalized if forward FT
  else
    sh = fcoef.shape
    len = 1
    dims.each{|d|
 raise ArgumentError, "dimension out of range" if sh[d] == nil
 len *= sh[d]
      }
    fcoef = fcoef / len
    end
  end
  gfc.data.replace_val( fcoef )

  # <coordinate variables>
  for i in 0...gfc.rank
  if dims.length == 0 || dims.include?(i) || dims.include?(i+rank)
    __predefined_coord_units_conversion(gfc.coord(i))
    cv = gfc.coord(i).val
    n = cv.length
    clen = (cv.max - cv.min) * n / (n-1)
    wn = (2*Math::PI/clen) * NArray.new(cv.typecode,cv.length).indgen!.to_f
    if (!backward)
 gfc.coord(i).set_att('origin_in_real_space',cv[0..0])
    else 
 if ( org = gfc.coord(i).get_att('origin_in_real_space') )
   wn += org[0]
   ###gfc.coord(i).del_att('origin_in_real_space')
 end
    end
    gfc.coord(i).replace_val(wn)
    gfc.coord(i).units = gfc.coord(i).units**(-1)
    __coord_name_conversion(gfc.coord(i), backward)
  end
  end

  # <fini>
  gfc
end

#fft_deriv(dim) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 505

def fft_deriv(dim)
  tp = self.data.typecode
  fc = self.fft(false,dim)
  wn = fc.coord(dim)
  k = wn.val.to_type(NArray::Complex)
  n = k.length
  n2a = (n-1)/2
  n2b = [n/2 + 1, n-1].min  # min to avoid error if n=2 (though meaningless)
  kmx = k[-1]+k[1]
  ik = NArray.complex(n)
  ik[0..n2a] = k[0..n2a]*Complex::I
  ik[n2b..-1] = (k[n2b..-1]-kmx) * Complex::I
  dim.times{ik.newdim!(0)}
  (self.rank-dim-1).times{ik.newdim!(-1)}
  fc.replace_val(fc.val*ik)
  deriv = fc.fft(true,dim)
  deriv.units = deriv.units * wn.units
  if tp >= NArray::SCOMPLEX
    deriv
  else
    deriv.real
  end
end

#histogram(opts = Hash.new) ⇒ Object Also known as: histogram1D 



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# File 'lib/numru/ganalysis/histogram.rb', line 155

def histogram(opts=Hash.new)
  GAnalysis.histogram(self, opts)
end

#interpolate(*coords) ⇒ Object

Wide-purpose multi-dimensional linear interpolation

This method supports interpolation regarding combinations of 1D and 2D coordinate variables. For instance, suppose self is 4D with coordinates named [“x”, “y”, “z”, “t”] and associated coordinates “sigma” (“sigma” is 1D and its axis is “z”), “p”, “q” (“p” and “q” are 2D having the coordinates “x” and “y”). You can make interpolation by specifying 1D VArrays whose names are among “x”, “y”, “z”, “t”, “sigma”, “p”, “q”. You can also use a Hash like => 1.0 to specify a single point along the “x” coordinate.

If the units of the target coordinate and the current coordinate are different, a converstion was made so that slicing is made correctly, as long as the two units are comvertible; if the units are not convertible, it is just warned.

If you specify only “x”, “y”, and “t” coordinates for interpolation, the remaining coordinates “z” is simply retained. So the result will be 4 dimensional with coordinates named [“x”, “y”, “z”, “t”], but the lengths of “x”, “y”, and “t” dimensions are changed according to the specification. Note that the result could be 3-or-smaller dimensional – see below.

Suppose you have two 1D VArrays, xnew and ynew, having names “x” and “y”, respectively, and the lengths of xnew and the ynew are the same. Then, you can give an array of the two, [xnew, ynew], for coord0 as

gp_int = gp_org.interpolate( [xnew, ynew] )

(Here, gp_org represents a GPhys object, and the return value pointed by gp_int is also a GPhys.) In this case, the 1st dimension of the result (gp_int) will be sampled at the points [xnew,ynew], [xnew,ynew], [xnew,ynew], …, while the 2nd and the third dimensions are “z” and “t” (no interpolation). This way, the rank of the result will be reduced from that of self.

If you instead give xnew to coord0 and ynew to coord1 as

gp_int = gp_org.interpolate( xnew, ynew )

The result will be 4-dimensional with the first coordinate sampled at xnew, xnew, xnew,… and the second coordinate sampled at ynew, ynew, ynew,…

You can also cut regarding 2D coordinate variable as

gp_int = gp_org.interpolate( pnew, qnew )
gp_int = gp_org.interpolate( xnew, qnew )
gp_int = gp_org.interpolate( [pnew, qnew] )
gp_int = gp_org.interpolate( [xnew, qnew] )

In any case, the desitination VArrays such as xnew ynew pnew qnew must be one-dimensional.

Note that

gp_int = gp_org.interpolate( qnew )

fails (exception raised), since it is ambiguous. If you tempted to do so, perhaps what you want is covered by the following special form:

As a special form, you can specify a particular dimension like this:

gp_int = gp_org.interpolate( "x"=>pnew )

Here, interpolation along “x” is made, while other axes are retained. This is useful if pnew corresponds to a multi-D coordinate variable where there are two or more corresponding axes (otherwise, this special form is not needed.)

See the test part at the end of this file for more examples.

LIMITATION

Currently associated coordinates expressed by 3D or greater dimensional arrays are not supported.

Computational efficiency of pure two-dimensional coordinate support should be improved by letting C extensions cover deeper and improving the search algorithm for grid (which is usually ordered quasi-regularly).

COVERAGE

Extrapolation is covered for 1D coordinates, but only interpolation is covered for 2D coordinates (which is limited by gt2dlib in DCL – exception will be raised if you specify a grid point outside the original 2D grid points.).

MATHEMATICAL SPECIFICATION

The multi-dimensional linear interpolation is done by supposing a (hyper-) “rectangular” grid, where each dimension is independently sampled one-dimensionally. In case of interpolation along two dimensional coordinates such as “p” and “q” in the example above, a mapping from a rectangular grid is assumed, and the corresponding points in the rectangular grid is solved inversely (currently by using gt2dlib in DCL).

For 1D and 2D cases, linear interpolations may be expressed as

1D:  zi = (1-a)*z0 + a*z1
2D:  zi = (1-a)*(1-b)*z00 + a*(1-b)*z10 + (1-a)*b*z01 + a*b*z11

This method is extended to arbitrary number of dimensions. Thus, if the number of dimensions to interpolate is S, then 2**S grid points are used for each interpolation (8 points for 3D, 16 points for 4D,…). Thus, the linearity of this interpolation is only along each dimension, not over the whole dimensionality.

USAGE

interpolate(coord0, coord1, ...)

ARGUMENTS

  • coord0, coord1,… [ 1D VArray, or Array of 1D VArray, or a 1-element Hash as => slice_loc_value(Numeric) ] : locations to which interpolation is made. Names of all the VArray’s in the arguments must exist among the names of the coordinates of self (including associated coordinates), since the dimension finding is made in terms of coordinate names. If an argument is an Array of VArray’s, the first VArray will become the main coordinate variable, and the rest will be associated coordinates.

  • SPECIAL CASE

    You can specfify a one-element Hash as the only argument such as

    gphys.interpolate("x"=>varray)

    where varray is a coordinate onto which interpolation is made. This is espcially useful if varray is multi-D. If varray’s name “p” (name of a 2D coordnate var), for example, you can interpolate only regarding “x” by retaining other axes. If varray is 1-diemnsional, the same thing can be done simply by

    gphys.interpolate(varray)

    since the corresponding 1D coordinate is found aotomatically.

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/interpolate.rb', line 389

def interpolate(*coords)
  coords, org_coords, org_dims, newgrid = _interpo_match_coords(coords)
  crdmap = _interpo_reorder_2crdmap(coords, org_coords, org_dims)
  idxmap = _interpo_find_position(crdmap)

  z = val
  if z.is_a?(NArrayMiss)
    missval = ( (a=get_att('_FillValue')) ? a[0] : nil ) || 
              ( (a=get_att('missing_value')) ? a[0] : nil ) || 
              @@interpo_missval
    z = z.to_na(missval)
    input_nomiss = false
  else
    input_nomiss = true
    if @@interpo_extrapolation
      missval = nil
    else
      missval = @@interpo_missval
    end
  end

  na = c_interpo_do(newgrid.shape, idxmap, z, missval,
                    @@interpo_extrapolation)   # [C-extension]

  if !input_nomiss || !@@interpo_extrapolation
    mask = na.ne(missval)
    if !input_nomiss || mask.min == 0
      na = NArrayMiss.to_nam_no_dup(na,mask)
    end
  end

  va = VArray.new(na, data, name)

  ret = GPhys.new(newgrid, va)
  ret.grid.set_lost_axes(self.lost_axes)
  ret
end

#least_square_fit(functions, ensemble_dims = nil, indep_dims = nil) ⇒ Object 



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# File 'lib/numru/ganalysis/fitting.rb', line 460

def least_square_fit(functions, ensemble_dims=nil, indep_dims=nil)

  #< preparation >

  no_fitting_dims = Array.new
  if ensemble_dims
    ensemble_dims = ensemble_dims.collect{|d| @grid.dim_index(d)}
    no_fitting_dims += ensemble_dims
  end
  if indep_dims
    indep_dims = indep_dims.collect{|d| @grid.dim_index(d)}
    no_fitting_dims += indep_dims
  end
  fitting_dims = (0...rank).collect{|i| i} - no_fitting_dims
  grid_locs = fitting_dims.collect{|d| coord(d).val}
  data = self.val

  #< fitting >
  c, bf, diff = GAnalysis::Fitting.least_square_fit(data, grid_locs, 
                                      functions, ensemble_dims, indep_dims)

  #< make a GPhys of the best fit >

  if !ensemble_dims
    grid = self.grid
  else
    axes = Array.new
    (0...rank).each{|d| 
      axes.push(self.axis(d)) unless ensemble_dims.include?(d)
    }
    grid = Grid.new(*axes)
    shape = bf.shape
    ensemble_dims.sort.reverse_each{|d| shape.delete_at(d)}
    bf = bf.reshape(*shape)
  end

  va = VArray.new(bf, self.data, self.name)
  bf = GPhys.new(grid, va)

  [c, bf, diff]
end

#logp_coord_p2z(pdim = nil) ⇒ Object

Convert the pressure coordinate in self to log-pressure height (after duplicating self)

Return value: a GPhys



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# File 'lib/numru/ganalysis/log_p.rb', line 84

def logp_coord_p2z(pdim=nil)
  pdim = GAnalysis::Met.find_prs_d(self) if !pdim
  p = self.coord(pdim)
  z = GAnalysis::LogP.p2z(p)
  ax = self.axis(pdim).copy
  ax.set_pos(z)
  ax.name = z.name
  grid = self.grid.copy.set_axis(pdim, ax)
  GPhys.new(grid,self.data)
end

#mouse_cut(dimx, dimy, num = 2, line_type = 1, line_index = 1) ⇒ Object

Makes a subset interactively by specifying a (poly-)line on the DCL viewport

ARGUMENTS

  • dimx {String] : name of number (0,1,..) of the dimension corresponding to the X coordinate in the current window of DCL

  • dimy {String] : name of number (0,1,..) of the dimension corresponding to the Y coordinate in the current window of DCL

  • num {Integer] : the number of points along the (poly-)line (2 or greater – if 2, a single line segment; if 3 or more, a poly-line)

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/interpolate.rb', line 45

def mouse_cut(dimx, dimy, num=2, line_type=1, line_index=1)

  # < preparation >

  dimx = dim_index(dimx)
  dimy = dim_index(dimy)

  rundef = DCL.glpget("rundef")
  line = nil
  while(true)
    puts "\n*** Waiting for mouse click. Click #{num} points in the current viewport."
    line = DCLMouseLine.new(num)
    if line.ux.include?(rundef)
      puts "** The points specified include one(s) outside the U window. Do it again."
    else
      break
    end
  end
  line.draw(line_type, line_index)
  vx = line.vx
  vy = line.vy
  ux = line.ux
  uy = line.uy
  gpnew = dcl_fig_cut(dimx,dimy,ux,uy)
  [gpnew, line]
end

#mouse_cut_repeat ⇒ Object

Interpolation onto grid points specified by the previous call of GPhys#mouse_cut



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# File 'lib/numru/gphys/interpolate.rb', line 169

def mouse_cut_repeat
  if @@interpo_previous_cutter.nil?
    raise("You must first use GPhys#mouse_cut. This method repeats it") 
  end
  gpnew = interpolate(@@interpo_previous_cutter)
  gpnew = @@interpo_previous_modifier[gpnew] if @@interpo_previous_modifier
  gpnew
end

#phase_velocity(kdim, fdim, kconv, fconv, kf0_is_c0 = true, no_kfreorder = false) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 757

def phase_velocity(kdim,fdim,kconv,fconv,kf0_is_c0=true,no_kfreorder=false)
  kax = self.axis(kdim)
  fax = self.axis(fdim)
  kax.pos = kax.pos*kconv if kconv
  fax.pos = fax.pos*fconv if fconv
  cunits = fax.pos.units / kax.pos.units

  f = fax.pos.val
  k = kax.pos.val
  nk = k.length
  nf = f.length
  if no_kfreorder
    k[nk/2+1..-1] = -k[nk/2+1..-1][-1..0]+k[nk/2]
    f[nf/2+1..-1] = -f[nf/2+1..-1][-1..0]+f[nf/2]
  end
  f = -f
  cp = f.newdim(0) / k.newdim(1) #cp[kdim,fdim]
  jf0 = f.eq(0).where[0]  # where f==0
  jk0 = k.eq(0).where[0]  # where k==0
  if kf0_is_c0
    cp[jk0,jf0] = 0.0       # treat k=f=0 as stationary (c=0)
  else
    cp[jk0,jf0] = 1.0/0.0   # not to count k=f=0 component at all (c=infty)
  end

  [cp, cunits]
end

#phase_velocity_binning(kdim, fdim, cbins, kconv = nil, fconv = nil) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 673

def phase_velocity_binning(kdim, fdim, cbins, kconv=nil, fconv=nil)

  # < process arguments >

  case cbins
  when Hash 
    min = cbins["min"] ||raise(ArgumentError,"a Hash cbins must have 'min'")
    max = cbins["max"] ||raise(ArgumentError,"a Hash cbins must have 'max'")
    int = cbins["int"] ||raise(ArgumentError,"a Hash cbins must have 'int'")
    cbins = Array.new
    eps = int.abs*1e-6   # epsilon to deal with float steps
    (min.to_f..(max.to_f+eps)).step(int){|c| cbins.push(c)}
    cbins = NArray.to_na(cbins)
  when Array
    cbins = NArray.to_na(cbins)
  when NArray
  else
    raise ArgumentError, "cbins must be a Hash or Array or NArray"
  end

  kdim = dim_index(kdim) if kdim.is_a?(String)
  kdim += rank if kdim < 0
  fdim = dim_index(fdim) if fdim.is_a?(String)
  fdim += rank if fdim < 0

  # < sort along wavenumber/freuqency axis >

  pw = self.spect_zero_centering(kdim).spect_one_sided(fdim)

  # < process axes >

  cp, cunits = pw.phase_velocity(kdim,fdim,kconv,fconv,false)

  vcbins = VArray.new(cbins, {"units"=>cunits.to_s, 
                "long_name"=>"phase velocity bounds"}, "cbounds")
  vccent = VArray.new( (cbins[0..-2] + cbins[1..-1])/2, 
                {"units"=>cunits.to_s, "long_name"=>"phase velocity"}, "c")
  axc = Axis.new(true).set_cell(vccent, vcbins).set_pos_to_center
  axes = [axc]   # the first dimension will be "c"
  gr = pw.grid
  (0...pw.rank).each do |d|
    if d!=kdim && d!=fdim
      axes.push(gr.axis(d))
    end
  end
  newgrid = Grid.new(*axes)

  nk = pw.shape[kdim]
  nf = pw.shape[fdim]
  cp.reshape!(nk*nf)

  # < reorder input data >

  dimorder = (0...pw.rank).collect{|i| i}
  dimorder.delete(fdim)
  dimorder.unshift(fdim)
  dimorder.delete(kdim)
  dimorder.unshift(kdim)   # --> [kdim, fdim, the other dims...]
  sh = pw.shape
  reshape = [nk*nf]
  (0...rank).each{|i| reshape.push(sh[i]) if i!=fdim && i!=kdim}
  pwv = pw.val.transpose(*dimorder).reshape(*reshape)  
                           # --> [ combined k&fdim, the other dims...]

  # < binning >

  shc = newgrid.shape
  pwc = NArray.new(pwv.typecode, *shc)    # will have no missing data
  nc = axc.length
  for jc in 0...nc
    w = (cp.gt(cbins[jc]) & cp.lt(cbins[jc+1])).where
    pwc[jc,false] += pwv[w,false].sum(0) if w.length>0
    w = (cp.eq(cbins[jc])).where
    pwc[jc,false] += pwv[w,false].sum(0)/2 if w.length>0  # half from bdry
    w = (cp.eq(cbins[jc+1])).where
    pwc[jc,false] += pwv[w,false].sum(0)/2 if w.length>0  # half from bdry
  end

  vpwc = VArray.new(pwc,pw.data,pw.name)
  gpwc = GPhys.new(newgrid,vpwc)

  gpwc
end

#phase_velocity_binning_iso_norml(kdim, fdim, cmin, cmax, cint, kconv = nil, fconv = nil) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 665

def phase_velocity_binning_iso_norml(kdim, fdim, cmin, cmax, cint, 
                               kconv=nil, fconv=nil)
  cbins = {"min"=>cmin,"max"=>cmax,"int"=>cint}
  pwc = phase_velocity_binning(kdim, fdim, cbins, kconv, fconv)
  fact = UNumeric[int, pwc.coord(0).units]
  pwc/fact
end

#phase_velocity_filter(xdim, tdim, cmin = nil, cmax = nil, xconv = nil, tconv = nil, remove_xtmean = false) ⇒ Object

Raises:

  • (ArgumentError) 


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# File 'lib/numru/gphys/gphys_fft.rb', line 628

def phase_velocity_filter(xdim, tdim, cmin=nil, cmax=nil, xconv=nil, tconv=nil, remove_xtmean=false)
  raise(ArgumentError,"need at least cmin or cmax") if !(cmin || cmax)


  xdim = dim_index(xdim) if xdim.is_a?(String)
  xdim += rank if xdim < 0
  tdim = dim_index(tdim) if tdim.is_a?(String)
  tdim += rank if tdim < 0
  fc = self.fft(nil,xdim,tdim)
  
  kdim = xdim
  fdim = tdim
  kconv = ( xconv ? 1.0/xconv : nil )
  fconv = ( tconv ? 1.0/tconv : nil )
  cp, = fc.phase_velocity(kdim,fdim,kconv,fconv,!remove_xtmean,true)

  fcv = fc.val
  nk = fc.shape[kdim]
  nf = fc.shape[fdim]
  sel = [true]*fc.rank
  for jf in 0...nf
    for jk in 0...nk
      c = cp[jk,jf]
      if ( cmin && c<cmin or cmax && c>cmax)
        sel[kdim]=jk
        sel[fdim]=jf
        fcv[*sel] = 0.0
      end
    end
  end
  fc.replace_val(fcv)
  gp = fc.fft(true,xdim,tdim)
  gp = gp.real if (self.typecode <= NArray::FLOAT)
  GPhys.new(self.grid_copy, gp.data)
            #^ use the original grid, since units may have changed
end

#rawspect2powerspect(*dims) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 612

def rawspect2powerspect(*dims)
  # developpers memo: Needs Units conversion.
  factor = nil
  dims.each{|dim|
  ax = self.coord(dim)
  dwn = UNumeric.new( ((ax[-1].val - ax[0].val)/(ax.length - 1)).abs,
   ax.units )
    if !factor
    factor = dwn**(-1)
  else
    factor = factor / dwn.to_f
  end
  }
  self * factor
end

#regrid(to) ⇒ Object

Interpolate to conform the grid to a target GPhys object

ARGUMENTS

  • to [GPhys] : the target gphys

RETURN VALUE

  • a GPhys



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# File 'lib/numru/gphys/interpolate.rb', line 208

def regrid(to)
  coords = to.axnames.collect{|nm| to.coord(nm)}
  interpolate(*coords)
end

#running_mean(dim, len_or_wgt = nil, bc = BC_SIMPLE, nminvalid = 1) ⇒ Object

Running mean along a dimension (with optional weight specification).

ARGUMENTS

  • dim (Integer or String) : the dimension

  • len_or_wgt : If Integer, specifies the length; if 1D NArray, specifies the weight (e.g., NArray[1.0, 2.0, 1.0] for the 1-2-1 smooting)

  • bc (Integer; optional) : Speficy one of the folloing:

    • GPhys::BC_SIMPLE (default) : Averaging is trucated at the boundaries (the number of grid points used is reduced near the boundaries). The shape of the object is conserved.

    • GPhys::BC_CYCLIC : Cyclic boundary condition. Shape conserved.

    • GPhys::BC_TRIM : Grids near the boundaries are trimmed to secure the number of grid points to average. Shape not conserved; length along the dim is reduced by (len-1).

  • nminvalid (Integer; optional; defualt=1): This parameter is used only when the data have missing. Minimum number of grid points needed for averaging. Must be from 1 to len.

RETURN VALUE

  • a GPhys

REMARK AND LIMITATION

  • If the length of the running mean is even number, fewer grid points are used from the “left” side; e.g., If len is 6, result is a mean over self..self.

  • Regardless the na_type of self, double is used for avaraging, so:

  • This method does not support complex numbers.



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# File 'lib/numru/gphys/gphys_dim_op.rb', line 46

def running_mean(dim, len_or_wgt=nil, bc=BC_SIMPLE, nminvalid=1)

  #< process arguments >

  dim = dim_index(dim)  # to handle String or negative specification

  case len_or_wgt
  when nil
    raise ArgumentError, "You need to specify the length (Integer) or the weight (1D NArray) as the 2nd argument"
  when Integer
    # len_or_wgt is a length
    len = len_or_wgt  
    wgt = NArray.float(len).fill!(1.0)
  else
    # len_or_wgt is a weight
    wgt = len_or_wgt
    if (!wgt.respond_to?(:rank) || wgt.rank != 1)
      raise ArgumentError, "wgt: expect a 1D NArray(-like obj)"
    end
    len = wgt.length
  end

  #< calc running mean >

  vi = self.val
  if (vi.typecode > NArray::DFLOAT)
    raise("This method supports only real or integer data")
  end 
  if vi.is_a?(NArrayMiss)
    vi, missval = nam2na_missval(vi)
    vo = c_running_mean(vi,dim,wgt,bc,missval,nminvalid)
    vo = NArrayMiss.to_nam(vo, vo.ne(missval) )
  else
    vo = c_running_mean(vi,dim,wgt,bc)
  end

  #< grid >

  if (bc ==  BC_TRIM)
    fst = (len-1)/2    # if odd len/2, if even len/2-1
    lst = -(len/2) - 1 
    grid = self.grid[ *([true]*dim + [fst..lst, false]) ]
  else
    grid = self.grid
  end

  #< result >
  vvo = VArray.new( vo, self.data, self.name )  # Inherit name & attrs
  GPhys.new( grid, vvo )

end

#spect_one_sided(dim) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 601

def spect_one_sided(dim)
  dim = dim + self.rank if dim<0
  len = self.shape[dim]
  b = self[ *([true]*dim + [0..len/2,false]) ] * 2
  b[*([true]*dim + [0,false])] = b[*([true]*dim + [0,false])] / 2
  if (self.shape[dim] % 2) == 0  # --> even number
    b[*([true]*dim + [-1,false])] = b[*([true]*dim + [-1,false])] / 2
  end
  b
end

#spect_zero_centering(dim) ⇒ Object 



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# File 'lib/numru/gphys/gphys_fft.rb', line 587

def spect_zero_centering(dim)
  dim = dim + self.rank if dim<0
  len = self.shape[dim]
  b = self[ *( [true]*dim + [[(len+1)/2..len-1,0..len/2],false] ) ].copy
  s1 = [true]*dim + [0, false]
  s2 = [true]*dim + [-1, false]
  if (len % 2) == 0   #--> even number
    b[*s1] = b[*s1]/2      # the ends are duplicated --> halved
    b[*s2] = b[*s1]
  end
  b.coord(dim)[0..len/2-1] = -b.coord(dim)[len/2+1..-1].val[-1..0]
  b
end

#threepoint_O2nd_deriv(*args) ⇒ Object 



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# File 'lib/numru/gphys/derivative.rb', line 94

def threepoint_O2nd_deriv(*args)
  Derivative::threepoint_O2nd_deriv(self,*args)
end