Lecture 15

In this lecture we will see more examples on establishing and proving loop invariants.

Dot_Product.

double dot_product ( int nelt, double *v1, double *v2 ) {

int i;

double t = 0.0;

for ( i= 0; i < nelt; i++ ) {

t += v1[i] * v2[i];

}

return t;

}

To establish the invariant first look into the specifications of the dot_product routine, i.e. the pre-conditions and the post-conditions.

Pre-conditions: nelt > 0;

v1, v2 are vectors of length nelt.

Post-conditions: The routine must return å^{nelt - 1} v1[i] * v2[i]

ⁱ⁼⁰

Hence, the loop invariant is:

t = sum so far

= å ^{i - 1} v1[j] * v2[j]

^j=0

Initially, i = 0 and t = 0.0.

Simulating the execution,

Assume t = å ^{i - 1} v1[j] * v2[j]

^j=0

t' = t + v1[i] * v2[i]

i' = i + 1

Now, t' = å ^{j = i - 1} v1[j] * v2[j] + v1[i] * v2[i]

^j=0

= å ^{j = i} v1[j] * v2[j]

^j=0

But i = i' - 1

Therefore, t' = å^{i’ - 1} v1[j] * v2[j]

^j=0

The loop terminates when i = nelt and hence:

t = å ^{j = nelt - 1} v1[j] * v2[j]

^j=0

which satisfies the post-condition.

Loop Invariant for an if statement:

double exp ( double b, int e ) {

double x = 1.0, y = b;

// Imagine i=0

while ( 0 != e ) {

if ( e odd ) {

x = x * y;

}

y = y * y;

e = e div 2; // imagine i = i+ 1

}

return x;

}

Explanation of strategy:

The binary expansion of e is e = e31 e30…e1 e0 = å³¹ 2ⁱ * ei

^{i = 0}

ð b^e = b raised to ( å³¹ 2ⁱ * ei )

^{i = 0}

= Õ³¹ ( ( b raised to 2ⁱ ) raised to ei )

i=0

= ( b raised to e0 ) * ( ( b² ) raised to e1) * ( ( b⁴ ) raised to e2 )…

Loop Invariant:

For all i from 0 to 31, x = Õⁱ ( ( b raised to 2^j ) raised to ej ) y = b raised to 2^j

^{j = 0}

At every iteration, y’ = y * y = y² = ( (b raised to 2ⁱ) raised to 2 ) = b raised to 2^{i + 1}

The only values of ei are 0 or 1.

if ei = 0,

x’ = x

When, i’ = i +1

x’ = Õ^i-1 ( ( b raised to 2^j ) raised to ej ) * y

^j=0

= ( Õ^i-1 ( ( b raised to 2^j ) raised to ej ) ) * ( ( b raised to 2ⁱ ) raised to ei )

^j=0

= Õⁱ ( ( b raised to 2^j ) raised to ej )

^j=0

= Õ^i’-1 ( ( b raised to 2^j ) raised to ej )

^j=0

if ei = 1

x’ = x * y

= Õ^i-1( ( b raised to 2^j ) raised to ej ) * ( ( b raised to 2ⁱ ) raised to ei )

^j=0

= Õⁱ ( ( b raised 2^j ) raised to ej )

j=0