I don't understand the problem this solves. At least in my experience each transaction on the bank statement has a reference to a business transaction attached (usually an invoice number). The amount of money that just lands on the account without a reference is negligible in comparison and usually easily manually associated.
Often banks will batch up transactions into a single one, and especially in a real-time market, that may not match what you expect.
For example lets say I ask you to sell 10 shares of Google if the price goes over $140 (this is typically called a limit order). Now your bank comes back and says the sold 2 shares at $140.02, 7 shares at $140.03, and 1 share at $139.77. Did they satisfy their obligation?
The answer is yes, but it's difficult to determine that, and you can't use exact math to do it easily. You expected $1400 from that sale, but you got $1400.02. Now do it again, but you have half a dozen orders at different prices. That's where it turns into the knapsack problem.
The problem is severely compounded when you look at why you're reconciling (it's to make sure your assets changed the way you expected, and fix things when it didn't). Often banks will drop a transaction, or add an extra one (these systems are annoyingly manual, and subject to error). How do you find the exact error and track it down? Especially when the trade happened, but you don't have the actual record of it, and your records show that it didn't.
