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File: LinearEquationSystems.sm

Introduction

This worksheet describes the solution of both fully-determined and overdetermined systems of linear equations using matrices. A simple "shopping basket" problem is used for illustration.

Definitions

Set a parameter to identify fruit and a unit to mimic currency

Apple 1 :

p 1 :

Orange 1 :

Banana 1 :

Fully Determined System of Linear Equations

For clarity, the Basket and Bill arrays are assembled individually then stacked.

Bill details

Basket details

Bill_1 15 p * :

Basket_1 1 Apple * 1 Orange * 0 Banana * 13 mat :

Bill_2 22 p * :

Basket_2 1 Apple * 1 Orange * 1 Banana * 13 mat :

Bill_3 25 p * :

Basket_3 1 Apple * 2 Orange * 0 Banana * 13 mat :

Bills_fd Bill_1 Bill_2 Bill_3 stack : p 15222531 mat

Baskets_fd Basket_1 Basket_2 Basket_3 stack : 11011112033 mat

Describing the problem using arrays…

Bills Baskets ItemCost * ≡

…allows simple solution…

ItemCost Baskets 1 -^Bills * ≡

Calculate item costs for the examples:

ItemCost_fd Baskets_fd 1 -^Bills_fd * : p 510731 mat

Check the results:

Baskets_fd ItemCost_fd * p 15222531 mat

Bills_fd p 15222531 mat

Overdetermined System of Linear Equations

With more equations than there are unknowns, the set of linear equations are Overdetermined. The minimum least squares item costs can be calculated using the algorithm in the reference. Reference: https://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)

β X transpose X * (1 -^X transpose y * (* ≡

This algorithm can also be used to solve a fully determined system:

Baskets_fd transpose Baskets_fd * (1 -^Baskets_fd transpose Bills_fd * (* p 510731 mat

Overdetermined system definition: • add two baskets and bills to the previous examples • values have been chosen to avoid the item costs previously calculated

Bill_4 22 p * :

Basket_4 1 Apple * 1 Orange * 1 Banana * 13 mat :

Bill_5 31 p * :

Basket_5 2 Apple * 2 Orange * 013 mat :

Bills_od Bills_fd Bill_4 Bill_5 stack : p 152225223151 mat

Baskets_od Baskets_fd Basket_4 Basket_5 stack : 11011112011122053 mat

Solve this overdetermined system:

Set optimization to Numeric or None ItemCost_od Baskets_od transpose Baskets_od * (1 -^Baskets_od transpose Bills_od * (* : p 5.8 9.6 6.6 31 mat

Check the results:

Baskets_od ItemCost_od * p 15.4 222522 30.8 51 mat

Bills_od p 152225223151 mat

As expected, there are small discepancies due to the use of different item costs in the additional bills.