How to Convert symbolic variables to numeric

Greetings!

How to convert a symbolic expression into a numerical expression?

best regards,

Walid.

[Dyuman Joshi]

The general way is to use a numeric data type

%random example
y = [(sqrt(sym(5))+1)/2 sym(sin(pi/3)*cos(pi/4)) exp(sym(pi))]

y = 

z = double(y)

z = 1×3

1.6180 0.6124 23.1407

[Guendouz walid]

Thank you.

[Star Strider]

Reference: How to calculate the derivative of vector and specifically my Answer.

Answer 1

[Steven Lord]

If it contains no symbolic variable use double.

two = sym(2);
sqrt2 = sqrt(two)

sqrt2 = 

x = double(sqrt2)

x = 1.4142

If it does contain a symbolic variable you can approximate the numeric parts with vpa. You could also use vpa even if it doesn't contain a symbolic variable.

z = vpa(sqrt2)

z = 

1.4142135623730950488016887242097

syms y
polynomialInY = y.^2 + sqrt2*y - 1

polynomialInY = 

vpa(polynomialInY)

ans = 

In this case trying to convert polynomialInY to a double array will error. You could subs a value for y into that expression (to eliminate the symbolic variable) then convert the result.

double(polynomialInY)

Error using symengine
Unable to convert expression containing symbolic variables into double array. Apply 'subs' function first to substitute values for variables.

Error in sym/double (line 872)
Xstr = mupadmex('symobj::double', S.s, 0);

[Dyuman Joshi]

@Steven Lord, what would you call sym(sqrt(2)) as? Symbolic constant?

[Steven Lord]

I don't know if there's a standard term for that, I'd probably just call them a number stored as a symbolic expression. But symbolic constant seems like a reasonable description as well.

But I'd be careful with expressions of that type. The sym function by default will "figure out" if you have sqrt(n) for small n that it should represent it as the square root of n. But for larger n it doesn't.

x1 = sym(sqrt(123456783))

x1 = 

But if you swap the order we're taking the square root of a symbolic integer value.

x2 = sqrt(sym(123456783))

x2 = 

[Dyuman Joshi]

Thanks for the info, Steven!

I found more useful info regarding this from documentation of sym -

"Use sym on subexpressions instead of the entire expression for better accuracy. Using sym on entire expressions is inaccurate because MATLAB® first converts the expression to a floating-point number, which loses accuracy. sym cannot always recover this lost accuracy."

Meaning, x2 should be preferred/used over x1.

[Walter Roberson]

@Dyuman Joshi

It is safest to convert every number to sym() inside a symbolic expression.

In the case of a symbolic expression raised to an integer power, MATLAB can be counted on to work out itself that the power needs to be converted to symbolic... Likewise, symbolic expression divided by an integer is reliably converted. But in both cases, only if the expression is already symbolic.

sym((sym(138)/sym(490))^sym(10))    %safest

ans = 

sym((sym(138)/sym(490))^10)         %pretty safe

ans = 

sym((sym(138)/490)^10)              %pretty safe

ans = 

sym((138/490)^10)                   %unsafe !

ans = 

You have to be careful in converting scientific notation

sym(6.02139280e-23)               %unsafe !

ans = 

sym(6.02139280)*10^-23            %less unsafe but still inaccurate

ans = 

sym(602139280)*sym(10)^-31       %safe

ans = 

sym(602139280)*10^-23           %looks safe but is not really

ans = 

The 10^-23 in the last example is not internally being handled by 10^-23 being exactly represented as would be the case for sym(10)^-23 . Instead, the 10^-23 is being handled by sym() examining the number being passed in and doing a continued fraction analysis to determine whether it is "sufficiently close to" a "nice number".

[Steven Lord]

The description for the 'r' value for the flag input argument to the sym function lists the forms that it recognizes for "modest-sized integers p and q". This value for the flag input is the default which is why:

x = sym(sqrt(2))

x = 

works to give . 2 counts as "modest-sized".