A recreational programming exercise:

Multiplication of a Hilbert matrix with its inverse using
Bignum.Rationals as a test case for rational arithmetic.

R=r
OCL=18706
CL=18706

test/hilbert.go

@@ -0,0 +1,167 @@
+// $G $D/$F.go && $L $F.$A && ./$A.out
+
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// A little test program for rational arithmetics.
+// Computes a Hilbert matrix, its inverse, multiplies them
+// and verifies that the product is the identity matrix.
+
+package main
+
+import Big "bignum"
+import Fmt "fmt"
+
+
+func assert(p bool) {
+	if !p {
+		panic("assert failed");
+	}
+}
+
+
+var (
+	Zero = Big.Rat(0, 1);
+	One = Big.Rat(1, 1);
+)
+
+
+type Matrix struct {
+	n, m int;
+	a *[]*Big.Rational;
+}
+
+
+func (a *Matrix) at(i, j int) *Big.Rational {
+	assert(0 <= i && i < a.n && 0 <= j && j < a.m);
+	return a.a[i*a.m + j];
+}
+
+
+func (a *Matrix) set(i, j int, x *Big.Rational) {
+	assert(0 <= i && i < a.n && 0 <= j && j < a.m);
+	a.a[i*a.m + j] = x;
+}
+
+
+func NewMatrix(n, m int) *Matrix {
+	assert(0 <= n && 0 <= m);
+	a := new(Matrix);
+	a.n = n;
+	a.m = m;
+	a.a = new([]*Big.Rational, n*m);
+	return a;
+}
+
+
+func NewUnit(n int) *Matrix {
+	a := NewMatrix(n, n);
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			x := Zero;
+			if i == j {
+				x = One;
+			}
+			a.set(i, j, x);
+		}
+	}
+	return a;
+}
+
+
+func NewHilbert(n int) *Matrix {
+	a := NewMatrix(n, n);
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			x := Big.Rat(1, i + j + 1);
+			a.set(i, j, x);
+		}
+	}
+	return a;
+}
+
+
+func MakeRat(x *Big.Natural) *Big.Rational {
+	return Big.MakeRat(Big.MakeInt(false, x), Big.Nat(1));
+}
+
+
+func NewInverseHilbert(n int) *Matrix {
+	a := NewMatrix(n, n);
+	for i := 0; i < n; i++ {
+		for j := 0; j < n; j++ {
+			x0 := One;
+			if (i+j)&1 != 0 {
+				x0 = x0.Neg();
+			}
+			x1 := Big.Rat(i + j + 1, 1);
+			x2 := MakeRat(Big.Binomial(uint(n+i), uint(n-j-1)));
+			x3 := MakeRat(Big.Binomial(uint(n+j), uint(n-i-1)));
+			x4 := MakeRat(Big.Binomial(uint(i+j), uint(i)));
+			x4 = x4.Mul(x4);
+			a.set(i, j, x0.Mul(x1).Mul(x2).Mul(x3).Mul(x4));
+		}
+	}
+	return a;
+}
+
+
+func (a *Matrix) Mul(b *Matrix) *Matrix {
+	assert(a.m == b.n);
+	c := NewMatrix(a.n, b.m);
+	for i := 0; i < c.n; i++ {
+		for j := 0; j < c.m; j++ {
+			x := Zero;
+			for k := 0; k < a.m; k++ {
+				x = x.Add(a.at(i, k).Mul(b.at(k, j)));
+			}
+			c.set(i, j, x);
+		}
+	}
+	return c;
+}
+
+
+func (a *Matrix) Eql(b *Matrix) bool {
+	if a.n != b.n || a.m != b.m {
+		return false;
+	}
+	for i := 0; i < a.n; i++ {
+		for j := 0; j < a.m; j++ {
+			if a.at(i, j).Cmp(b.at(i,j)) != 0 {
+				return false;
+			}
+		}
+	}
+	return true;
+}
+
+
+func (a *Matrix) String() string {
+	s := "";
+	for i := 0; i < a.n; i++ {
+		for j := 0; j < a.m; j++ {
+			x := a.at(i, j);  // BUG 6g bug
+			s += Fmt.sprintf("\t%s", x);
+		}
+		s += "\n";
+	}
+	return s;
+}
+
+
+func main() {
+	n := 10;
+	a := NewHilbert(n);
+	b := NewInverseHilbert(n);
+	I := NewUnit(n);
+	ab := a.Mul(b);
+	if !ab.Eql(I) {
+		Fmt.println("a =", a);
+		Fmt.println("b =", b);
+		Fmt.println("a*b =", ab);
+		Fmt.println("I =", I);
+		panic("FAILED");
+	}
+}