Variable-precision arithmetic (arbitrary-precision arithmetic)
Support for character vectors that do not define a number has been removed. Instead, first
create symbolic numbers and variables using sym and
syms, and then use operations on them. For example, use vpa((1
+ sqrt(sym(5)))/2) instead of vpa('(1 + sqrt(5))/2').
vpa( uses variable-precision
floating-point arithmetic (VPA) to evaluate each element of the symbolic
input x)x to at least d significant
digits, where d is the value of the digits function.
The default value of digits is 32.
Evaluate symbolic inputs with variable-precision
floating-point arithmetic. By default, vpa calculates
values to 32 significant digits.
syms x p = sym(pi); piVpa = vpa(p)
piVpa = 3.1415926535897932384626433832795
a = sym(1/3); f = a*sin(2*p*x); fVpa = vpa(f)
fVpa = 0.33333333333333333333333333333333*sin(6.283185307179586476925286766559*x)
Evaluate elements of vectors or matrices with variable-precision arithmetic.
V = [x/p a^3]; M = [sin(p) cos(p/5); exp(p*x) x/log(p)]; vpa(V) vpa(M)
ans = [ 0.31830988618379067153776752674503*x, 0.037037037037037037037037037037037] ans = [ 0, 0.80901699437494742410229341718282] [ exp(3.1415926535897932384626433832795*x), 0.87356852683023186835397746476334*x]
Note
You must wrap all inner inputs with vpa,
such as exp(vpa(200)). Otherwise the inputs are
automatically converted to double by MATLAB®.
vpaBy default, vpa evaluates
inputs to 32 significant digits. You can change the number of significant
digits by using the digits function.
Approximate the expression 100001/10001 with
seven significant digits using digits. Save the
old value of digits returned by digits(7).
The vpa function returns only five significant
digits, which can mean the remaining digits are zeros.
digitsOld = digits(7); y = sym(100001)/10001; vpa(y)
ans = 9.9991
Check if the remaining digits are zeros by using a higher precision
value of 25. The result shows that the remaining
digits are in fact a repeating decimal.
digits(25) vpa(y)
ans = 9.999100089991000899910009
Alternatively, to override digits for a
single vpa call, change the precision by specifying
the second argument.
Find π to 100 significant digits by specifying the second argument.
vpa(pi,100)
ans = 3.141592653589793238462643383279502884197169... 39937510582097494459230781640628620899862803... 4825342117068
Restore the original precision value in digitsOld for
further calculations.
digits(digitsOld)
While symbolic results are exact, they might
not be in a convenient form. You can use vpa to
numerically approximate exact symbolic results.
Solve a high-degree polynomial for its roots using solve.
The solve function cannot symbolically solve
the high-degree polynomial and represents the roots using root.
syms x y = solve(x^4 - x + 1, x)
y = root(z^4 - z + 1, z, 1) root(z^4 - z + 1, z, 2) root(z^4 - z + 1, z, 3) root(z^4 - z + 1, z, 4)
Use vpa to numerically approximate the
roots.
yVpa = vpa(y)
yVpa = 0.72713608449119683997667565867496 - 0.43001428832971577641651985839602i 0.72713608449119683997667565867496 + 0.43001428832971577641651985839602i - 0.72713608449119683997667565867496 - 0.93409928946052943963903028710582i - 0.72713608449119683997667565867496 + 0.93409928946052943963903028710582i
vpa Uses Guard Digits to Maintain PrecisionThe value of the digits function
specifies the minimum number of significant digits used. Internally, vpa can
use more digits than digits specifies. These
additional digits are called guard digits because they guard against
round-off errors in subsequent calculations.
Numerically approximate 1/3 using four significant
digits.
a = vpa(1/3, 4)
a = 0.3333
Approximate the result a using 20 digits.
The result shows that the toolbox internally used more than four digits
when computing a. The last digits in the result
are incorrect because of the round-off error.
vpa(a, 20)
ans = 0.33333333333303016843
Hidden round-off errors can cause unexpected results.
Evaluate 1/10 with the default 32-digit precision,
and then with the 10 digits precision.
a = vpa(1/10, 32) b = vpa(1/10, 10)
a = 0.1 b = 0.1
Superficially, a and b look
equal. Check their equality by finding a - b.
a - b
ans = 0.000000000000000000086736173798840354720600815844403
The difference is not equal to zero because b was
calculated with only 10 digits of precision and
contains a larger round-off error than a. When
you find a - b, vpa approximates b with
32 digits. Demonstrate this behavior.
a - vpa(b, 32)
ans = 0.000000000000000000086736173798840354720600815844403
vpa Restores Precision of Common Double-Precision InputsUnlike exact symbolic values, double-precision
values inherently contain round-off errors. When you call vpa on
a double-precision input, vpa cannot restore
the lost precision, even though it returns more digits than the double-precision
value. However, vpa can recognize and restore
the precision of expressions of the form p/q, pπ/q, (p/q)1/2, 2q,
and 10q,
where p and q are modest-sized
integers.
First, demonstrate that vpa cannot restore
precision for a double-precision input. Call vpa on
a double-precision result and the same symbolic result.
dp = log(3); s = log(sym(3)); dpVpa = vpa(dp) sVpa = vpa(s) d = sVpa - dpVpa
dpVpa = 1.0986122886681095600636126619065 sVpa = 1.0986122886681096913952452369225 d = 0.00000000000000013133163257501600766255995767652
As expected, the double-precision result differs from the exact result at the 16th decimal place.
Demonstrate that vpa restores precision
for expressions of the form p/q, pπ/q, (p/q)1/2, 2q,
and 10q,
where p and q are modest sized
integers, by finding the difference between the vpa call
on the double-precision result and on the exact symbolic result. The
differences are 0.0 showing that vpa restores
lost precision in the double-precision input.
vpa(1/3) - vpa(1/sym(3)) vpa(pi) - vpa(sym(pi)) vpa(1/sqrt(2)) - vpa(1/sqrt(sym(2))) vpa(2^66) - vpa(2^sym(66)) vpa(10^25) - vpa(10^sym(25))
ans = 0.0 ans = 0.0 ans = 0.0 ans = 0.0 ans = 0.0
vpa does not convert fractions
in the exponent to floating point. For example, vpa(a^sym(2/5)) returns a^(2/5).
vpa uses more digits than the
number of digits specified by digits. These extra
digits guard against round-off errors in subsequent calculations and
are called guard digits.
When you call vpa on a numeric
input, such as 1/3, 2^(-5),
or sin(pi/4), the numeric expression is evaluated
to a double-precision number that contains round-off errors. Then, vpa is
called on that double-precision number. For accurate results, convert
numeric expressions to symbolic expressions with sym.
For example, to approximate exp(1), use vpa(exp(sym(1))).
If the second argument d is not
an integer, vpa rounds it to the nearest integer
with round.
vpa restores precision for numeric
inputs that match the forms p/q, pπ/q, (p/q)1/2, 2q,
and 10q,
where p and q are modest-sized
integers.
Atomic operations using variable-precision arithmetic round to nearest.
The differences between variable-precision arithmetic and IEEE Floating-Point Standard 754 are
Inside computations, division by zero throws an error.
The exponent range is larger than in any predefined
IEEE mode. vpa underflows below approximately 10^(-323228496).
Denormalized numbers are not implemented.
Zeroes are not signed.
The number of binary digits in the mantissa of a result may differ between variable-precision arithmetic and IEEE predefined types.
There is only one NaN representation.
No distinction is made between quiet and signaling NaN.
No floating-point number exceptions are available.
digits | double | root | vpaintegral