rouben@math13.math.umbc.edu (Rouben Rostamian) (03/11/91)
I had trouble with posting this porgram earlier today. This is a repost.
I apologize if you received duplicate copies.
--
Rouben Rostamian Telephone: (301) 455-2458
Department of Mathematics and Statistics e-mail:
University of Maryland Baltimore County bitnet: rostamian@umbc.bitnet
Baltimore, MD 21228, U.S.A. internet: rouben@math9.math.umbc.edu
---------------------------------------------------------------------------
This is version 2.0 of SPLINE. SPLINE generates a piecewise cubic and twice
continuously differentiable interpolation y(x) of a set of points (x_i,y_i),
i=1,2,...,n. It is assumed throughout that x_1 < x_2 < ... < x_n.
SPLINE's default action is to generate a _natural_ cubic spline, i.e., the
second derivative y'' vanishes at the end points x_1 and x_n. The default
action of SPLINE may be modified by specifying optional switches which are
described later.
------- INPUT --------------------------------------------------------
SPLINE reads its input from the stack. The n coordinate points may be
specified in two different formats:
THE COORDINATE-PAIRS INPUT FORMAT:
n+1: (x_1,y_1)
n: (x_2,y_2)
...
2: (x_n,y_n)
1: n
THE ARRAY INPUT FORMAT:
2: [ y_1 y_2 ... y_n ]
1: [ x_1 x_n ]
In the ARRAY input format the interval [x_1 x_n] is automatically divided
into n equally spaced nodes. The COORDINATE-PAIRS format is useful if
the nodes are not equally spaced.
----------- OUTPUT ----------------------------------------------------
The output of SPLINE is a *program* which can be used as a user-defined
function. The program, which is placed on level 1 of the stack, has the
following general format:
1: << -> X
<<
Description of the spline curve here
>>
>>
This program may be stored in a variable, say TRY, and may be
evaluated as "TRY(X)" (algebraic mode) or as "X TRY" (RPN mode.)
TRY(X) may be plotted with the usual plotting commands.
-------- OPTIONAL SWITCHES ---------------------------------------------
o SPLINE by default imposes the natural boundary conditions
y''(x_1) = y''(x_n) = 0. It is possible to specify instead the first
derivatives a := y'(x_1) and b := y'(x_n) as boundary conditions.
For this, enter the data in one of the two formats described
before, then push the list { a b } into the level 1 of stack.
o SPLINE can also return the first derivative y'(x) and the second
derivative y''(x) of the cubic spline interpolant. To get these, enter
data as before, optionally enter the list { a b } from the
previous paragraph, then push the characters 'D1' into level 1
to compute y'(x), or 'D2' to compute y''(x).
---------- EXAMPLES --------------------------------------------------
Example 1:
5: (0,0)
4: (1,1)
3: (2,4)
2: (4,16)
1: 4
SPLINE returns the program:
1: << -> X
<<
CASE 'X' 2 >=
THEN '2.60869565217*(4-X)^3/2/6+6.86956521739*(X-2)+2.26086956522'
END 'X' 1 >=
THEN '(2.34782608696*(2-X)^3+2.60869565217*(X-1)^3)/6
+2.95652173913*(X-1)+.608695652173'
END '2.34782608696X^3/6+.608695652173*X'
END EVAL
>>
>>
Example 2a:
2: [ 0 1 0 1 0 1 0 ]
1: [ 0 9 ]
The second derivative is zero at the end points.
Example 2b:
3: [ 0 1 0 1 0 1 0 ]
2: [ 0 9 ]
1: { 0 0 }
The first derivative is zero at the ends. Replacing { 0 0 } by { 1 -1 }
makes the first derivative equal 1 and -1 at the ends.
Example 2c:
4: [ 0 1 0 1 0 1 0 ]
3: [ 0 9 ]
2: { 0 0 }
1: 'D1'
Computes the first derivative y'(x) of the spline y(x) of example 2b.
Replacing 'D1' with 'D2' will computes the second derivative. It is
instructive to save y(x), y'(x), and y''(x) into variables Y0, Y1, and Y2,
and PLOT { 'Y2(X)' 'Y1(X)' Y0(X)' } with XRANGE set to 0,9 and AUTO.
-------- REFERENCE -----------------------------------------------------
Stoer and Bulirsch, Numerical Analysis
-------- REMARKS -------------------------------------------------------
SPLINE does not use, change, create or modify any global variables.
It does not modify parts of the stack it does not own and does not
alter any system flags, although the calculator has to be in the
symbolic mode for SPLINE to operate. It clears _user_ flags 6,7,8,9.
-------- NOTES ---------------------------------------------------------
SPLINE V2.0 is completely different from an earlier version (no version
number) that I posted to comp.sys.handhelds a few weeks ago. This new
version generates code which executes about 4 times faster than the
previous version. It also has many additional features. I will not
describe the differences here because because I consider the previous
version obsolete.
-------- PROGRAM OBJECT CHECKSUMS --------------------------------------
Checksum: #3B69h
Bytes: 2212
-------- COMMENTED PROGRAM (Uncommented program follows) --------------
%%HP: T(3)A(D)F(.);
\<<
@ Display version while working
" SPLINE V2.0
" 1 DISP
6 CF 7 CF 8 CF 9 CF @ Prepare flags 6-9
IF DUP TYPE 6 SAME @ Check if 'D1' or 'D2' are specified
THEN
CASE DUP 'D1' SAME
THEN 6 SF @ Set flag 6 if 'D1' specified
END DUP 'D2' SAME
THEN 7 SF @ Set flag 7 if 'D2' specified
END DUP \->STR ": Unknown flag" + DOERR
END
DROP @ Drop 'D1' or 'D2' from stack
END
IF DUP TYPE 5 \=/ @ See if the end-derivatives are specified
THEN { 0 0 } @ If not, insert {0 0} (as a place holder)
ELSE 8 SF @ If yes, then set flag 8
END
IF OVER TYPE 0 SAME @ If we have a number n in level 2 then we
THEN OVER 2 + ROLLD @ expect n pairs of coordinates above it
ELSE SWAP ROT @ Otherwise we expect two arrays in levels 2
END @ and 3. In either case, we move the { s1 s2 }
@ list form level 1 to the top of the stack.
IF DUP TYPE 0 \=/ @ If we don't have a number in level 1
THEN 9 SF @ Then the coordinates are given as arrays
DUP SIZE EVAL @ Determine the size of array
END
@ Set up the local variables
DUP 1 - { } { } 0 0 0 0 0 0 0 0 0 0 0 0
\-> n k x y h \Gl \Gm s d m a b \Gl0 \Gmn s11 s1n
@ Begin the main program
\<<
IF 9 FC?C
THEN 1 n @ Flag 9 is clear so data is in coord. pairs
FOR j @ Convert coordinate pairs into lists x and y
C\->R 'y' STO+ 'x' STO+
NEXT
ELSE
OBJ\-> EVAL \->LIST 'y' STO @ Store array of y_i into the list y
OBJ\-> DROP OVER - k / @ Compute the mesh size
0 k
FOR j
DUP j * 3 PICK + 3 ROLLD @ Generate the x mesh
NEXT
DROP2 n \->LIST 'x' STO @ Store the x values into the list x
END
EVAL 's1n' STO 's11' STO @ Read the end-derivative values
x EVAL @ Compute the list of
1 k @ interval lengths h_j = x_{j+1} - x_j
FOR j OVER - n ROLLD
NEXT
DROP
k \->LIST 'h' STO
1 k @ Compute the list of slopes s_j
FOR j @ s_j = ( y_{j+1} - y_j ) / h_j
y j GETI 3 ROLLD GET - NEG h j GET /
NEXT
k \->LIST 's' STO
@ Compute the elements d_j of the list d:
IF 8 FS?
THEN @ End-derivatives are specified
1 '\Gl0' STO
1 '\Gmn' STO
s 1 GET s11 - h 1 GET / 6 *
ELSE 0
END
@ Still computing d:
1 n 2 -
FOR j
s j GETI 3 ROLLD GET - NEG h j GETI 3 ROLLD GET + / 6 *
NEXT
@ End of computation of d:
IF 8 FS?C
THEN
s1n s k GET - h k GET / 6 *
ELSE 0
END
n \->LIST 'd' STO
@ Compute lambda_j:
h OBJ\-> 1 - 1 SWAP
FOR j
DUP 3 PICK + / k ROLLD
NEXT
DROP
n 2 - \->LIST '\Gl' STO
@ Compute gamma_j:
\Gl OBJ\-> 1 SWAP
FOR j
NEG 1 +
n 2 - ROLL
NEXT
n 2 - \->LIST '\Gm' STO
@ Compute the moments m_j:
n IDN 2 *
2 k
FOR j
j DUP 1 - 2 \->LIST
\Gm j 1 - GET
PUT
j DUP 1 + 2 \->LIST
\Gl j 1 - GET
PUT
NEXT
2 \Gl0 PUT
n SQ 1 - \Gmn PUT
INV
d OBJ\-> \->ARRY *
'm' STO
@ Compute a_j:
1 k
FOR j
m j GETI 3 ROLLD GET - h j GET * 6 / s j GET +
NEXT
k \->LIST 'a' STO
@ Compute b_j:
1 k
FOR j
y j GET m j GET h j GET SQ * 6 / -
NEXT
k \->LIST 'b' STO
@ Now we compute the individual arcs of the spline:
CASE 6 FS?C
THEN @ Will compute y'(x)
1 k
FOR j
m j 1 + GET 'X' x j GET - SQ *
m j GET x j 1 + GET 'X' - SQ * -
h j GET / 2 / a j GET +
NEXT
END 7 FS?C
THEN @ Will compute y''(x)
1 k
FOR j
m j GET x j 1 + GET 'X' - *
m j 1 + GET 'X' x j GET - * +
h j GET / NEXT
END
1 k
FOR j @ Will compute y(x)
m j GET x j 1 + GET 'X' - 3 ^ *
m j 1 + GET 'X' x j GET - 3 ^ * +
h j GET / 6 /
a j GET 'X' x j GET - * +
b j GET +
NEXT
END
@ Create the output program:
"\<<\-> X\<<CASE " n ROLLD
k 2
FOR j
'X' " " + x j GET + " \>= THEN " + SWAP + " END " +
j ROLLD
-1
STEP
" END EVAL\>>\>>"
@ Concatenate all parts:
1 n
FOR j +
NEXT
OBJ\->
\>>
\>>
----------- UNCOMMENTED PROGRAM -------------------------------------------
%%HP: T(3)A(D)F(.);
\<<
" SPLINE V2.0
"
1 DISP 6 CF 7 CF 8
CF 9 CF
IF DUP TYPE 6
SAME
THEN
CASE DUP 'D1'
SAME
THEN 6 SF
END DUP 'D2'
SAME
THEN 7 SF
END DUP \->STR
": Unknown flag" +
DOERR
END DROP
END
IF DUP TYPE 5 \=/
THEN { 0 0 }
ELSE 8 SF
END
IF OVER TYPE 0
SAME
THEN OVER 2 +
ROLLD
ELSE SWAP ROT
END
IF DUP TYPE 0 \=/
THEN 9 SF DUP
SIZE EVAL
END DUP 1 - { } {
} 0 0 0 0 0 0 0 0 0
0 0 0 \-> n k x y h \Gl
\Gm s d m a b \Gl0 \Gmn
s11 s1n
\<<
IF 9 FC?C
THEN 1 n
FOR j C\->R 'y'
STO+ 'x' STO+
NEXT
ELSE OBJ\-> EVAL
\->LIST 'y' STO OBJ\->
DROP OVER - k / 0 k
FOR j DUP j *
3 PICK + 3 ROLLD
NEXT DROP2 n
\->LIST 'x' STO
END EVAL 's1n'
STO 's11' STO x
EVAL 1 k
FOR j OVER - n
ROLLD
NEXT DROP k
\->LIST 'h' STO 1 k
FOR j y j GETI
3 ROLLD GET - NEG h
j GET /
NEXT k \->LIST
's' STO
IF 8 FS?
THEN 1 '\Gl0' STO
1 '\Gmn' STO s 1 GET
s11 - h 1 GET / 6 *
ELSE 0
END 1 n 2 -
FOR j s j GETI
3 ROLLD GET - NEG h
j GETI 3 ROLLD GET
+ / 6 *
NEXT
IF 8 FS?C
THEN s1n s k
GET - h k GET / 6 *
ELSE 0
END n \->LIST 'd'
STO h OBJ\-> 1 - 1
SWAP
FOR j DUP 3
PICK + / k ROLLD
NEXT DROP n 2 -
\->LIST '\Gl' STO \Gl
OBJ\-> 1 SWAP
FOR j NEG 1 + n
2 - ROLL
NEXT n 2 -
\->LIST '\Gm' STO n IDN
2 * 2 k
FOR j j DUP 1 -
2 \->LIST \Gm j 1 - GET
PUT j DUP 1 + 2
\->LIST \Gl j 1 - GET
PUT
NEXT 2 \Gl0 PUT n
SQ 1 - \Gmn PUT INV d
OBJ\-> \->ARRY * 'm'
STO 1 k
FOR j m j GETI
3 ROLLD GET - h j
GET * 6 / s j GET +
NEXT k \->LIST
'a' STO 1 k
FOR j y j GET m
j GET h j GET SQ *
6 / -
NEXT k \->LIST
'b' STO
CASE 6 FS?C
THEN 1 k
FOR j m j 1
+ GET 'X' x j GET -
SQ * m j GET x j 1
+ GET 'X' - SQ * -
h j GET / 2 / a j
GET +
NEXT
END 7 FS?C
THEN 1 k
FOR j m j
GET x j 1 + GET 'X'
- * m j 1 + GET 'X'
x j GET - * + h j
GET /
NEXT
END 1 k
FOR j m j GET
x j 1 + GET 'X' - 3
^ * m j 1 + GET 'X'
x j GET - 3 ^ * + h
j GET / 6 / a j GET
'X' x j GET - * + b
j GET +
NEXT
END
"\<<\-> X\<<CASE " n
ROLLD k 2
FOR j 'X' " " +
x j GET +
" \>= THEN " + SWAP +
" END " + j ROLLD
-1
STEP
" END EVAL\>>\>>" 1 n
FOR j +
NEXT OBJ\->
\>>
\>>