{smcl}
{bf:Mata course:  Accuracy 1:  multiplication and addition}
{hline}


{title:1.  Base 10 vs. base 2 (binary)}

	0
	1/8  = .001
	2/8  = .01
	3/8  = .011
	4/8  = .1
	5/8  = .101
	6/8  = .11
	7/8  = .111

	.1 = 0.00011001100110011001...
	.2 = 0.0011001100110011001...
	.3 = 0.010011001100110011...
	.4 = 0.011001100110011001...
	.5 = 0.1
	.6 = 0.10011001100110011...
	.7 = 0.10110011001100110...
	.8 = 0.11001100110011001...
	.9 = 0.11000110011001100...


{title:2.  Floating-point arithmetic (decimal)}

{p 4 4 2}
    Write number as 

		{it:z}  =  ({it:s}, {it:x}, {it:y}) =  {it:s}*{it:x}*10^{it:y}

{p 4 4 2}
    where            
		{it:s}  =  1  or  -1

		0 <= {it:x} < 10

{p 4 4 2}
    Example:

		.5  =  (1, 5, -1)  =  5*10^(-1)


{title:3.  Fixed-precision floating-point arithmetic (decimal)}

{p 4 4 2}
    {it:x} is recorded as {it:d} digits, say 5.

{p 4 4 2}
    Example:

		.5  =  (1, 5.0000, -1) = 5*10(^-1)

		1/3 =  (1, 3.3333, -1) = 3.3333*10^(-1)


{title:4.  Fixed-precision floating-point arithmetic (binary)}

{p 4 4 2}
    Write number as 

		{it:z}  =  ({it:s}, {it:x}, {it:y}) =  {it:s}*{it:x}*2^{it:y}

{p 4 4 2}
    where             
		{it:s}  =  1  or  -1

		0 <= {it:x} < 2

{p 4 4 2}
    {it:x} is recorded as {it:d} binary digits, say 24 (float) or 52 (double)


{title:5.  Multiplication}

{p 4 4 2}
    Method:

		{it:z1}  =  {it:s1}*{it:x1}*{it:b}^{it:y1}
		{it:z2}  =  {it:s2}*{it:x2}*{it:b}^{it:y2}

             {it:z1}*{it:z2}  =  {it:s1}*{it:s2} * {it:x1}*{it:x2}*{it:b}^({it:y1}+{it:y2})


{p 8 8 2}
    The {it:x1}*{it:x2} part will have more than {it:d} digits.{break}
    Truncate or round or {it:d} digits.

{p 4 4 2}
Result:

{p 8 8 2}
    In the worst case, the error is one binary digit.  

{p 8 8 2}
    If our precision is {it:d}, we have {it:d}-1 or {it:d} precision result.



{title:6.  Addition}

{p 4 4 2}
    Method:

		{it:z1}  =  {it:s1}*{it:x1}*10^{it:y1}
		{it:z2}  =  {it:s2}*{it:x2}*10^{it:y2}

{p 8 8 2}
      Add the usual way.{break}
      Truncate or round to {it:d} digits.

{p 4 4 2}
      Example:

         12.345 + .12345:

                         12.345
                           .12345
			----------
                         12.46845
			 -- ---

{p 4 4 2}
    Result:

{p 8 8 2}
      Given {it:d} digits of precision, our result will have between 
      0 and {it:d} digits of precision.


{title:7.  Summary}

{p 4 8 2}
1.
    Multiplication (and division) introduce little (next to none) inaccuracy.

{p 4 8 2}
2.
    Addition (and subtraction) have the potential to introduce lots of
    inaccuracy.

{p 4 8 2}
{it:Example:}

{p 8 8 2}
Do not code

			{cmd:1 - normal(}{it:z}{cmd:)}

{p 8 8 2}
Code 

			{cmd:normal(-}{it:z}{cmd:)}



{title:8.  Implication for complex operators}

{p 4 8 2}
1.  What is said above about addition applies complex addition.

{p 4 8 2}
2.  What is said above about addition applies to complex multiplication, 
    because complex multiplication involves addition:

	    ({it:a} + {it:b}{it:i})(c + d{it:i})  =  ({it:ac}-{it:bd}) + ({it:ad}+{it:bc}){it:i}


{title:9.  Implication for matrix operators}


{p 4 8 2}
1.  What is said above about addition applies to matrices.

{p 4 8 2}
2.  What is said above about addition applies to matrix multiplication
    because matrix multiplication involves addition.
   


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{bf:{view talk.smcl:Top}}