NB. Matrix to find spline 2nd derivatives (natural splines): 
NB. (SpMD x) on vector x yeilds matrix B, where (B mm y)
NB. gives the spline 2nd derivatives, i.e.,  D = x spline y 
NB. 
SpMD=: monad define
k1=. 6% h1=. }: h=. DEL y 
k3=. 6% h3=. }. h
h2=. +: h3+h1
h1=. 0, (}. h1)
h3=. (}: h3),0
AI=. trivert h1;h2;h3        NB. invert (N-2)x(N-2) 
AI=. 0,(|: 0,(|: AI),0),0    NB. border with zeros
k1=. 0,k1,0
k3=. 0,k3,0
k2=. - (k1 + k3)
AI mm full_matrix k1;k2;k3
)

NB. Invert a tridiagonal matrix by direct substitution.
NB. Rows are  a_i*x_(i-1), b_i*x_i, c_i*x_(i+1)  
NB. a,b,c have same length, i.e., a[0]=0 and c[n]=0. 
NB. Usage: inverse(full nxn matrix)<-- tridgl a;b;c 
NB. 
trivert=: monad define
'a b c'=. y 
n1=. <: n=. <: nn=. #b
x=. %{.b
inv=. x (<0 0)}(=@i. nn)  
v=. (x*{.c) 0}v=. nn#0
j=. 0
while. n1>:j=.>:j do.
  b1=. j{b
  a1=. j{a
  v0=. (<:j){v
  x=. %b1-a1*v0
  c1=. j{c
  v=. (x*c1)j}v
  inv1=. j{inv
  inv0=. (<:j){inv
  inv=. (x*inv1-a1*inv0)j}inv
end.
x=. %(n{b)-(n{a)*(n1{v)
inv=. (x*(n{inv)-(n{a)*(n1{inv)) n}inv
j=. n
while. 0<:j=.<:j do.
  inv1=. j{inv
  v1=. j{v
  inv2=. (>:j){inv
  inv=. (inv1-v1*inv2)j}inv
end.
)

NB. Expand the three diagonal bands of a tridiagonal 
NB. matrix into the full matrix. 
NB. Rows are  a_i, b_i, c_i
NB. a,b,c have same length, i.e., a[0]=0 and c[n]=0.
NB. Usage: full_matrix a;b;c --> (full nxn matrix)
NB.
full_matrix=: monad define
'a b c'=. y
abc=. b* I=. =@i. (#b)   NB. I is the identity matrix
abc=. abc+ 1|."1 (a* I)
abc+ _1|."1 (c* I)
)