# Module: NumRu::Derivative

Defined in:
lib/numru/derivative.rb

## Constant Summary collapse

LINEAR_EXT =

<<module constant>>

`1`
CYCLIC_EXT =
`2`
MIRROR_A =
`3`
MIRROR_B =
`4`

## Class Method Details

### .b_expand(z, dim, bc) ⇒ Object

 ``` 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230``` ```# File 'lib/numru/derivative.rb', line 216 def b_expand(z,dim,bc) case bc when LINEAR_EXT ze = b_expand_linear_ext(z,dim) # linear extention when CYCLIC_EXT ze = b_expand_cyclic(z,dim) when MIRROR_A ze = b_expand_mirror_A(z,dim) when MIRROR_B ze = b_expand_mirror_B(z,dim) else raise ArgumentError,"unsupported boundary condition: #{bc}." end ze end```

### .b_expand_cyclic(z, dim) ⇒ Object

 ``` 247 248 249 250``` ```# File 'lib/numru/derivative.rb', line 247 def b_expand_cyclic(z,dim) k = z.shape[dim]-1 z[*([true]*dim + [[k,0..k,0]] + [false])] end```

### .b_expand_linear_ext(z, dim) ⇒ Object

Raises:

• (ArgumentError)
 ``` 232 233 234 235 236 237 238 239 240 241 242 243 244 245``` ```# File 'lib/numru/derivative.rb', line 232 def b_expand_linear_ext(z,dim) raise ArgumentError,"Len of #{dim}th dim (#{z.shape[dim]}) must be >= 2" if z.shape[dim] < 2 val0 = z[*([true]*dim + [0] + [false])] # first val1 = z[*([true]*dim + [1] + [false])] # second valm1 = z[*([true]*dim + [-1] + [false])] # last valm2 = z[*([true]*dim + [-2] + [false])] # one before last # expand boundary ze = z[*([true]*dim + [[0,0..(z.shape[dim]-1),0]] + [false])] ze[*([true]*dim + [0] + [false])] = 2*val0-val1 ze[*([true]*dim + [-1] + [false])] = 2*valm1-valm2 return ze end```

### .b_expand_mirror_A(z, dim) ⇒ Object

 ``` 252 253 254 255``` ```# File 'lib/numru/derivative.rb', line 252 def b_expand_mirror_A(z,dim) k = z.shape[dim]-1 z[*([true]*dim + [[0,0..k,k]] + [false])] end```

### .b_expand_mirror_B(z, dim) ⇒ Object

Raises:

• (ArgumentError)
 ``` 257 258 259 260 261``` ```# File 'lib/numru/derivative.rb', line 257 def b_expand_mirror_B(z,dim) raise ArgumentError,"Len of #{dim}th dim (#{z.shape[dim]}) must be >= 2" if z.shape[dim] < 2 k = z.shape[dim]-1 z[*([true]*dim + [[1,0..k,k-1]] + [false])] end```

### .cderiv(z, x, dim, bc = LINEAR_EXT) ⇒ Object

Raises:

• (ArgumentError)
 ``` 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189``` ```# File 'lib/numru/derivative.rb', line 174 def cderiv(z, x, dim, bc=LINEAR_EXT) dim += z.rank if dim<0 raise ArgumentError,"dim value (#{dim}) must be smaller than z.rank and >= 0" if dim >= z.rank || dim<0 raise ArgumentError,"rank of x (#{x.rank}) must be 1" if x.rank != 1 # <> ze = b_expand(z,dim,bc) xe = b_expand_linear_ext(x,0) # expand boundary of axis. # <> dz = cdiff(ze,dim) dx = cdiff(xe,0) if dx.rank != dz.rank # make dx.rank == dz.rank dx = dx.reshape(*([1]*dim + [true] + [1]*(dz.rank-1-dim))) end dzdx = dz/dx return dzdx end```

### .cdiff(z, dim) ⇒ Object

 ``` 263 264 265 266 267 268``` ```# File 'lib/numru/derivative.rb', line 263 def cdiff(z,dim) z1 = z[*([true]*dim + [2..-1] + [false])] z2 = z[*([true]*dim + [0..-3] + [false])] cz = z1-z2 # cz[i] = cz[n+1] - cz[n-1] return cz end```

### .deriv2nd(z, x, dim, bc = LINEAR_EXT) ⇒ Object

2nd derivative covering uniform grids

Raises:

• (ArgumentError)
 ``` 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213``` ```# File 'lib/numru/derivative.rb', line 192 def deriv2nd(z, x, dim, bc=LINEAR_EXT) dim += z.rank if dim<0 if dim < 0 || dim >= z.rank raise ArgumentError,"dim value(#{dim}) must be between 0 and (#{z.rank-1}" end raise ArgumentError,"rank of x (#{x.rank}) must be 1" if x.rank != 1 # <> ze = b_expand(z,dim,bc) xe = b_expand_linear_ext(x,0) # always linear extention # <> to_rankD = [1]*dim + [true] + [1]*(ze.rank-1-dim) # to exand 1D to rank D dx20 = xe[2..-1] - xe[0..-3] # x_{i+1} - x_{i-1} (for i=1..-2) dx21 = xe[2..-1] - xe[1..-2] # x_{i+1} - x_{i} (for i=1..-2) dx10 = xe[1..-2] - xe[0..-3] # x_{i} - x_{i-1} (for i=1..-2) a2 = 2/(dx21*dx20).reshape(*to_rankD) a1 = (-2)/(dx21*dx10).reshape(*to_rankD) a0 = 2/(dx10*dx20).reshape(*to_rankD) d2zdx2 = ze[ *([true]*dim+[2..-1,false]) ] * a2 \ + ze[ *([true]*dim+[1..-2,false]) ] * a1 \ + ze[ *([true]*dim+[0..-3,false]) ] * a0 return d2zdx2 end```

### .threepoint_O2nd_deriv(z, x, dim, bc = LINEAR_EXT) ⇒ Object

Raises:

• (ArgumentError)
 ``` 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171``` ```# File 'lib/numru/derivative.rb', line 148 def threepoint_O2nd_deriv(z, x, dim, bc=LINEAR_EXT) dim += z.rank if dim<0 if dim < 0 || dim >= z.rank raise ArgumentError,"dim value(#{dim}) must be between 0 and (#{z.rank-1}" end raise ArgumentError,"rank of x (#{x.rank}) must be 1" if x.rank != 1 # <> ze = b_expand(z,dim,bc) xe = b_expand_linear_ext(x,0) # always linear extention # <> to_rankD = [1]*dim + [true] + [1]*(ze.rank-1-dim) # to exand 1D to rank D dx = xe[1..-1] - xe[0..-2] # x_{i} - x_{i-1} (for i=0..n-2) dx2 = dx**2 s = dx[0..-2] # x_{i} - x_{i-1} (for i=0..n-3) t = dx[1..-1] # x_{i+1} - x_{i} (for i=0..n-3) s2 = dx2[0..-2].reshape(*to_rankD) # s**2 t2 = dx2[1..-1].reshape(*to_rankD) # t**2 numerator = ze[ *([true]*dim+[2..-1,false]) ] * s2\ + ze[ *([true]*dim+[1..-2,false]) ] * (t2-s2) \ - ze[ *([true]*dim+[0..-3,false]) ] * t2 denominator = (s*t*(s+t)).reshape(*to_rankD) dzdx = numerator / denominator return dzdx end```