LinearSystemImpl.mesa
Copyright © 1985 by Xerox Corporation. All rights reserved.
Last edited by Maureen Stone 27-Oct-81 11:16:26
Written by Michael Plass, 8-Oct-81
Tim Diebert May 21, 1985 5:51:30 pm PDT
Stone, October 16, 1985 11:22:42 am PDT
DIRECTORY LinearSystem;
LinearSystemImpl: CEDAR PROGRAM EXPORTS LinearSystem = BEGIN OPEN LinearSystem;
SolveN:
PUBLIC
PROCEDURE [
A:MatrixN, b:ColumnN, n:
INTEGER]
RETURNS [x:ColumnN] =
BEGIN
-- solve Ax=b by Gaussian Elimination
FOR i:
INTEGER
IN [0..n)
DO
bestk:INTEGER ← i;
FOR k:
INTEGER
IN [i..n)
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:RowN ← A[i]; A[i] ← A[bestk]; A[bestk] ← t};
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:
INTEGER
IN (i..n)
DO
IF
A[k][i]#0
THEN
BEGIN
r:REAL = A[k][i]/A[i][i]; -- Singular A causes Real.RealException = divide by zero
FOR j:
INTEGER
IN [i..n)
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
END;
ENDLOOP;
ENDLOOP;
Now A is upper-triangular, so back substitute
x ← NEW[VecSeq[n]];
FOR i:INTEGER IN [0..n) DO x[i] ← 0; ENDLOOP;
FOR i:
INTEGER
DECREASING
IN [0..n)
DO
xi:REAL ← b[i];
FOR j:
INTEGER
IN (i..n)
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END;
Solve2:
PUBLIC
PROCEDURE [
A:Matrix2, b:Column2]
RETURNS [x:Column2] =
BEGIN
n:
NAT = 2;
BEGIN -- solve Ax=b by Gaussian Elimination
FOR i:[1..n]
IN [1..n]
DO
bestk:[1..n] ← i;
FOR k:[1..n]
IN [i..n]
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:Row2 ← A[i]; A[i] ← A[bestk]; A[bestk] ← t}; -- sorry about the dependence on n
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:(1..n]
IN (i..n]
DO
r:REAL = A[k][i]/A[i][i]; -- Singular A causes divide by zero
FOR j:[1..n]
IN [i..n]
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
ENDLOOP
ENDLOOP;
Now A is upper-triangular, so back substitute
FOR i:[1..n]
DECREASING
IN [1..n]
DO
xi:REAL ← b[i];
FOR j:[1..n]
IN (i..n]
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END
END;
Solve3:
PUBLIC
PROCEDURE [
A:Matrix3, b:Column3]
RETURNS [x:Column3] =
BEGIN
n:
NAT = 3;
BEGIN -- solve Ax=b by Gaussian Elimination
FOR i:[1..n]
IN [1..n]
DO
bestk:[1..n] ← i;
FOR k:[1..n]
IN [i..n]
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:Row3 ← A[i]; A[i] ← A[bestk]; A[bestk] ← t}; -- sorry about the dependence on n
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:(1..n]
IN (i..n]
DO
r:REAL = A[k][i]/A[i][i]; -- Singular A causes divide by zero
FOR j:[1..n]
IN [i..n]
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
ENDLOOP
ENDLOOP;
Now A is upper-triangular, so back substitute
FOR i:[1..n]
DECREASING
IN [1..n]
DO
xi:REAL ← b[i];
FOR j:[1..n]
IN (i..n]
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END
END;
Solve4:
PUBLIC
PROCEDURE [
A:Matrix4, b:Column4]
RETURNS [x:Column4] =
BEGIN
n:
NAT = 4;
BEGIN -- solve Ax=b by Gaussian Elimination
FOR i:[1..n]
IN [1..n]
DO
bestk:[1..n] ← i;
FOR k:[1..n]
IN [i..n]
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:Row4 ← A[i]; A[i] ← A[bestk]; A[bestk] ← t}; -- sorry about the dependence on n
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:(1..n]
IN (i..n]
DO
r:REAL = A[k][i]/A[i][i]; -- Singular A causes divide by zero
FOR j:[1..n]
IN [i..n]
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
ENDLOOP
ENDLOOP;
Now A is upper-triangular, so back substitute
FOR i:[1..n]
DECREASING
IN [1..n]
DO
xi:REAL ← b[i];
FOR j:[1..n]
IN (i..n]
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END
END;
Solve5:
PUBLIC
PROCEDURE [
A:Matrix5, b:Column5]
RETURNS [x:Column5] =
BEGIN
n:
NAT = 5;
BEGIN -- solve Ax=b by Gaussian Elimination
FOR i:[1..n]
IN [1..n]
DO
bestk:[1..n] ← i;
FOR k:[1..n]
IN [i..n]
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:Row5 ← A[i]; A[i] ← A[bestk]; A[bestk] ← t}; -- sorry about the dependence on n
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:(1..n]
IN (i..n]
DO
r:REAL = A[k][i]/A[i][i]; -- Singular A causes divide by zero
FOR j:[1..n]
IN [i..n]
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
ENDLOOP
ENDLOOP;
Now A is upper-triangular, so back substitute
FOR i:[1..n]
DECREASING
IN [1..n]
DO
xi:REAL ← b[i];
FOR j:[1..n]
IN (i..n]
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END
END;
Solve6:
PUBLIC
PROCEDURE [
A:Matrix6, b:Column6]
RETURNS [x:Column6] =
BEGIN
n:
NAT = 6;
BEGIN -- solve Ax=b by Gaussian Elimination
FOR i:[1..n]
IN [1..n]
DO
bestk:[1..n] ← i;
FOR k:[1..n]
IN [i..n]
DO
IF ABS[A[k][i]] > ABS[A[bestk][i]] THEN bestk ← k;
ENDLOOP;
{t:Row6 ← A[i]; A[i] ← A[bestk]; A[bestk] ← t}; -- sorry about the dependence on n
{t:REAL ← b[i]; b[i] ← b[bestk]; b[bestk] ← t};
FOR k:(1..n]
IN (i..n]
DO
r:REAL = A[k][i]/A[i][i]; -- Singular A causes divide by zero
FOR j:[1..n]
IN [i..n]
DO
A[k][j] ← A[k][j] - A[i][j]*r
ENDLOOP;
b[k] ← b[k] - b[i]*r
ENDLOOP
ENDLOOP;
Now A is upper-triangular, so back substitute
FOR i:[1..n]
DECREASING
IN [1..n]
DO
xi:REAL ← b[i];
FOR j:[1..n]
IN (i..n]
DO
xi ← xi - A[i][j]*x[j];
ENDLOOP;
x[i] ← xi / A[i][i]
ENDLOOP
END
END;
InvalidMatrix: PUBLIC SIGNAL = CODE;
InvalidOperation: PUBLIC SIGNAL = CODE;
ValidMatrix:
PROCEDURE [a: MatrixN]
RETURNS [nrows,ncols:
INTEGER] = {
IF a=NIL OR a.nrows=0 THEN SIGNAL InvalidMatrix;
nrows ← a.nrows;
ncols ← a[0].ncols;
FOR i:
INTEGER
IN [0..nrows)
DO
IF a[i].ncols#ncols THEN SIGNAL InvalidMatrix;
ENDLOOP;
RETURN[nrows,ncols];
};
Sign: PROC[i,j: INTEGER] RETURNS[REAL] = {RETURN[IF (i+j) MOD 2 = 0 THEN 1 ELSE -1]};
MakeAij:
PROC[a, Aij: MatrixN, i,j:
INTEGER] = {
--assume Aij is well formed
n,m: INTEGER ← 0; --row index, column index for new matrix
FOR row:
INTEGER
IN [0..a.nrows)
DO
IF row=i THEN LOOP; --row index for original matrix
m ← 0;
FOR col:
INTEGER
IN [0..a[0].ncols)
DO
--column index for new matrix
IF col=j THEN LOOP;
Aij[n][m] ← a[row][col]; --column 0 of old matrix supplies the aij values
m ← m+1;
ENDLOOP;
n ← n+1;
ENDLOOP;
};
Invert:
PUBLIC PROCEDURE [a: MatrixN]
RETURNS [ai: MatrixN] = {
nrows,ncols: INTEGER;
det: REAL;
Aij: MatrixN;
[nrows,ncols] ← ValidMatrix[a];
IF nrows#ncols THEN SIGNAL InvalidOperation;
det ← Determinant[a];
ai ← Create[nrows,ncols];
Aij ← Create[nrows-1,ncols-1];
FOR i:
INTEGER
IN [0..nrows)
DO
FOR j:
INTEGER
IN [0..ncols)
DO
MakeAij[a,Aij,i,j];
ai[j][i] ← Sign[i,j]*Determinant[Aij]/det;
ENDLOOP;
ENDLOOP;
RETURN[ai];
};
Determinant:
PUBLIC PROC[a: MatrixN]
RETURNS [det:
REAL] = {
nrows,ncols: INTEGER;
[nrows,ncols] ← ValidMatrix[a];
IF nrows#ncols THEN SIGNAL InvalidOperation;
IF nrows=1 THEN det ← a[0][0]
ELSE IF nrows=2 THEN det ← a[0][0]*a[1][1]-a[0][1]*a[1][0]
ELSE {
i,j: INTEGER;
Aij: MatrixN ← Create[nrows-1,ncols-1];
det ← 0;
j ← 0; --always use column 0 for now
FOR i
IN [0..nrows)
DO
MakeAij[a,Aij,i,j];
det ← det + a[i][j]*Sign[i,j]*Determinant[Aij];
ENDLOOP;
};
RETURN[det];
};
Create:
PUBLIC
PROC [nrows, ncols:
INTEGER]
RETURNS [a: MatrixN] = {
a ← NEW[MatrixSeq[nrows]];
FOR i: INTEGER IN [0..nrows) DO a[i] ← NEW[VecSeq[ncols]]; ENDLOOP;
};
Copy:
PUBLIC
PROC [a: MatrixN]
RETURNS[MatrixN] = {
nrows,ncols: INTEGER;
new: MatrixN;
[nrows,ncols] ← ValidMatrix[a];
new ← Create[nrows, ncols];
FOR i:
INTEGER
IN [0..nrows)
DO
FOR j: INTEGER IN [0..ncols) DO new[i][j] ← a[i][j]; ENDLOOP;
ENDLOOP;
RETURN[new];
};
Transpose:
PUBLIC
PROCEDURE [a: MatrixN]
RETURNS [transpose: MatrixN] = {
nrows,ncols: INTEGER;
[nrows,ncols] ← ValidMatrix[a];
transpose ← Create[ncols,nrows];
FOR i:
INTEGER
IN [0..nrows)
DO
FOR j:
INTEGER
IN [0..ncols)
DO
transpose[j][i] ← a[i][j];
ENDLOOP;
ENDLOOP;
RETURN[transpose];
};
Multiply:
PUBLIC PROCEDURE [a: MatrixN, b: MatrixN]
RETURNS [c: MatrixN] = {
nra,nca: INTEGER;
nrb,ncb: INTEGER;
[nra,nca] ← ValidMatrix[a];
[nrb,ncb] ← ValidMatrix[b];
IF nca#nrb THEN SIGNAL InvalidOperation;
c ← Create[nra,ncb];
FOR j:
INTEGER
IN [0..nra)
DO
FOR i:
INTEGER
IN [0..ncb)
DO
c[j][i] ← 0;
FOR k:
INTEGER
IN [0..nca)
DO
c[j][i] ← c[j][i] + a[j][k]*b[k][i];
ENDLOOP;
ENDLOOP;
ENDLOOP;
};
MultiplyVec:
PUBLIC
PROC[a: MatrixN, v: ColumnN]
RETURNS [c: RowN] = {
nrows,ncols,nels: INTEGER;
[nrows,ncols] ← ValidMatrix[a];
IF v=NIL THEN SIGNAL InvalidMatrix;
nels ← v.ncols; --actually nrows for this case. historical
IF nels#ncols THEN SIGNAL InvalidOperation;
c ← NEW[VecSeq[nrows]];
FOR i:
INTEGER
IN [0..nrows)
DO
c[i] ← 0;
FOR j:
INTEGER
IN [0..nels)
DO
c[i] ← c[i]+a[i][j]*v[j];
ENDLOOP;
ENDLOOP;
};
END.