Learn more about Symbolic Math Toolbox

Variable-Precision Arithmetic

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Overview Example: Using the Different Kinds of Arithmetic Another Example Using Different Kinds of Arithmetic

Overview

There are three different kinds of arithmetic operations in this toolbox:

Numeric	MATLAB floating-point arithmetic
Rational	MuPAD® software exact symbolic arithmetic
VPA	MuPAD software variable-precision arithmetic

For example, the MATLAB statements

format long
1/2 + 1/3

use numeric computation to produce

  ans =
   0.833333333333333

With Symbolic Math Toolbox software, the statement

sym(1/2) + 1/3

uses symbolic computation to yield

ans =
5/6

And, also with the toolbox, the statements

digits(25)
vpa('1/2 + 1/3')

use variable-precision arithmetic to return

ans =
0.8333333333333333333333333

The floating-point operations used by numeric arithmetic are the fastest of the three, and require the least computer memory, but the results are not exact. The number of digits in the printed output of MATLAB double quantities is controlled by the format statement, but the internal representation is always the eight-byte floating-point representation provided by the particular computer hardware.

In the computation of the numeric result above, there are actually three roundoff errors, one in the division of 1 by 3, one in the addition of 1/2 to the result of the division, and one in the binary to decimal conversion for the printed output. On computers that use IEEE^® floating-point standard arithmetic, the resulting internal value is the binary expansion of 5/6, truncated to 53 bits. This is approximately 16 decimal digits. But, in this particular case, the printed output shows only 15 digits.

The symbolic operations used by rational arithmetic are potentially the most expensive of the three, in terms of both computer time and memory. The results are exact, as long as enough time and memory are available to complete the computations.

Variable-precision arithmetic falls in between the other two in terms of both cost and accuracy. A global parameter, set by the function digits, controls the number of significant decimal digits. Increasing the number of digits increases the accuracy, but also increases both the time and memory requirements. The default value of digits is 32, corresponding roughly to floating-point accuracy.

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Example: Using the Different Kinds of Arithmetic

Rational Arithmetic

By default, Symbolic Math Toolbox software uses rational arithmetic operations, i.e., MuPAD software's exact symbolic arithmetic. Rational arithmetic is invoked when you create symbolic variables using the sym function.

The sym function converts a double matrix to its symbolic form. For example, if the double matrix is

format short;
A = [1.1,1.2,1.3;2.1,2.2,2.3;3.1,3.2,3.3]

A =
    1.1000    1.2000    1.3000
    2.1000    2.2000    2.3000
    3.1000    3.2000    3.3000

its symbolic form is:

S = sym(A)

S =
[ 11/10,   6/5, 13/10]
[ 21/10,  11/5, 23/10]
[ 31/10,  16/5, 33/10]

For this matrix A, it is possible to discover that the elements are the ratios of small integers, so the symbolic representation is formed from those integers. On the other hand, the statement

E = [exp(1) (1 + sqrt(5))/2; log(3) rand]

returns a matrix

E =
    2.7183    1.6180
    1.0986    0.6324

whose elements are not the ratios of small integers, so

sym(E)

reproduces the floating-point representation in a symbolic form:

ans =
[ 3060513257434037/1125899906842624,   910872158600853/562949953421312]
[ 2473854946935173/2251799813685248, 1423946432832521/2251799813685248]

Variable-Precision Numbers

Variable-precision numbers are distinguished from the exact rational representation by the presence of a decimal point. A power of 10 scale factor, denoted by 'e', is allowed. To use variable-precision instead of rational arithmetic, create your variables using the vpa function.

For matrices with purely double entries, the vpa function generates the representation that is used with variable-precision arithmetic. For example, if you apply vpa to the matrix S defined in the preceding section, with digits(4), by entering

vpa(S)

MATLAB returns the output

ans = 
[ 1.1, 1.2, 1.3]
[ 2.1, 2.2, 2.3]
[ 3.1, 3.2, 3.3]

Applying vpa to the matrix E defined in the preceding section, with digits(25), by entering

digits(25)
F = vpa(E)

returns

F = 
[  2.718281828459045534884808,  1.618033988749894902525739]
[  1.098612288668109560063613, 0.6323592462254095103446616]

Converting to Floating-Point

To convert a rational or variable-precision number to its MATLAB floating-point representation, use the double function.

In the example, both double(sym(E)) and double(vpa(E)) return E.

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Another Example Using Different Kinds of Arithmetic

The next example is perhaps more interesting. Start with the symbolic expression

f = sym('exp(pi*sqrt(163))')

The statement

format long;
double(f)

produces the printed floating-point value

ans =
    2.625374126407687e+017

Using the second argument of vpa to specify the number of digits,

vpa(f,18)

returns

ans =
262537412640768744.0

and, too,

vpa(f,25)

returns

ans =
262537412640768744.0

You might suspect that f actually has an integer value. This suspicion is reinforced by the 30 digit value:

vpa(f,30)

ans =
262537412640768743.999999999999

Finally, the 40–digit value:

vpa(f,40)

ans =
262537412640768743.9999999999992500725972

shows that f is very close to, but not exactly equal to, an integer.

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Simplifications and Substitutions

Linear Algebra

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