Index |
This Keisαn service aims to achieve the high precision of calculation by applying Arithmetic Operation System with variable number of digits.
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| Feature of Keisαn service |
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| 1. The number of operation digits can be specified. |
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Example) |
digit number 6 |
√5=sqrt (5)=2.23607 |
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digit number 14 |
√5=sqrt(5)=2.2360679774998 |
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| 2. All operations inside CPU are carried out in the decimal system. |
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Example) |
Keisαn |
(0.9-0.3*3)= 0 |
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Excel |
(0.9-0.3*3)= 1.11022E-16 |
This error is caused by the conversion from binary to decimal. |
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| 3. The number of correct digits in the operation result is automatically judged and the precise operation result is returned. |
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Example) |
digit number 6 |
1-0.999876= 1.24000E-4 |
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digit number 10 |
gamma(3.5)=3.323350970 |
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| 4. Complex number, probability function, special function, and repeated calculation can be handled. |
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Example) |
Complex number calculation
digit number 6
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(1+i)^(1+i) = 0.273957+0.583701i |
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Error function
digit number 6
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x=0.1 |
erf(x)= 0.112463 |
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Repeated calculation |
x=2,1,5 |
sqrt(x)= |
| x |
sqrt(x) |
| 2 |
1.41421 |
| 3 |
1.73205 |
| 4 |
2 |
| 5 |
2.23607 |
| 6 |
2.44949 |
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| 5. The library of expressions can be utilized. |
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Example) |
Area of triangle |
s=(a+b+c)/2;a=3;b=4;c=5;sqrt(s*(s-a)*(s-b)*(s-c)) |
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| Description rule of expression
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| 1. Numerical value |
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The numerical value is displayed with combination of 0,1,2,3,4,5,6,7,8,9,
.(decimal point), E(exponent), i (imaginary number), + and - . |
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Example) |
12340.56789 |
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E2 |
100 |
(1* 102) |
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1.23E+4 |
12300 |
(1.23* 104 note: + after E is the sign of exponent followed by integer.) |
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1.23E-4 |
0.000123 |
(1.23*10-4 note: - after E is the sign of exponent followed by integer.) |
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i |
0+1i |
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E+2i |
0+100i |
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1.23+4.56i |
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(Note: In order to handle the expression as complex number, put it in the parenthesis.) |
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1.23-4.56i |
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(Note: In order to handle the expression as complex number, put it in the parenthesis.) |
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1.23E3+4.56E2i |
1230+456i |
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1.23E-3-4.56E+2i |
0.00123-456i |
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Note) |
A capital letter E is used for an exponent. (A small letter e is used for Napier's constant.) |
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| 2. Variable |
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The variable begins with an English letter and is composed of letters, words and/or numbers.
However, it is not possible to use the same variable identifier as special sign and reserved word.
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Valid variable identifier: |
a sinx a3 interest rate and the like |
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Invalid variable identifier: |
reserved word such as e, i, inf, sin and #abc |
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| 3. Function |
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The function consists of the reserved word, and the one or more of arguments which are surrounded by the parenthesis. |
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If there are more than two arguments, they have to be divided by comma. |
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The numerical value, variable, function and expression can be described as the argument. |
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Example) |
sin(x) |
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normalcd(x,μ,2) |
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normalcd(x+y,ln(12),σ) |
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pi, inf, e |
The parenthesis is not reuired for constant. |
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| 4. Operator |
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The operators (+, -, *, /, ^, ! and = ) execute operations between each element of numerical value, variable and function. |
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Element 1+Element 2 |
Add Element 1 to Element 2. |
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Element 1-Element 2 |
Subtract Element 2 from Element 1. |
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Element 1*Element 2 |
Multiply Element 1 by Element 2.(* it is possible also to abbreviate) |
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Element 1/Element 2 |
Divide Element 1 by Element 2. |
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Element 1^Element 2 |
Calculate Element 1 to the power of Element 2. |
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Element 1! |
Factorial of Element 1. (Note that the factorial operation is performed only for one element.) |
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Element 1=Element 2 |
Substitute the operation result of Element 2 into Element 1. Therefore Element 1 must be variable. |
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| 5. Expression |
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The expression is composed of numerical value, variable and function, each element of which is connected by the operator. Further, the expressions can be connected by the operator as one element.
The operational order between each element follows "the operational order of expressions".
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Example) |
Element 1 + Element 2 - Element 3 * Element 4/Element 5 ^ Element 6! +
Element 7… |
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Note) Note that Element 6! is immediately followed by the operator. |
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| 6. Miscellaneous |
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Minus "-", excluding the exponential part of expression, is regarded as a binary operator. |
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Minus "-" at the beginning of the expression or immediately after the open parenthesis or immediately after the semicolon, is considered as that the beginning of zero is abbreviated. |
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Example) |
- a+b → (0-a) +b |
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- a^b → 0- (a^b) |
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a^-b → Error |
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a^ (- b) → a^ (0-b) |
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a*b; - a*b → a*b; 0- (a*b) |
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a*-b → Error |
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In order to make expression easy to see, insert the space between words. |
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| Order of operations in expression |
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| 1. |
The expression is operated in the higher priority order of the operator. |
| 2. |
The operation is performed from the left side if there is more than one operator of the same priority. |
| 3. |
The operation inside the parenthesis is performed with high priority. |
| 4. |
Further, the parenthesis can be described inside the parenthesis. In that case, the priority is given to the inside parenthesis. |
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| Type and priority of operator |
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| Priority |
Operator |
Meaning |
high
｜
｜
｜
｜
↓
low |
Function |
Intrinsic function |
| !!, !, ^ |
Double factorial, Factorial and Power |
| *, / |
Multiplication and Division |
| +, - |
Addition and Subtraction |
| = |
Substitution |
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| Mistakable description example of expression. |
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| 1. The order of operations without parenthesis. |
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2+3*4 |
→ |
2+(3*4) |
The multiplication is calculated first than the addition. |
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3*4^5 |
→ |
3*(4^5) |
The power is calculated first than the multiplication. |
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sin(2)^3 |
→ |
(sin(2))^3 |
The function is calculated first than the power. |
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1/2i |
→ |
1/(2i) |
The imaginary number sign takes precedence. |
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3!!! |
→ |
(3!!)! |
The double factorial takes precedence from the factorial. |
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| 2. The expression without multiplication sign. |
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3a |
○ |
It is calculated as 3*a. |
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a3 |
× |
a3 is handled as variable identifier. |
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a(b+c) |
× |
It is not calculated as a* (b+c). "a" is handled as user defined function. |
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(a+b)(b+c) |
○ |
It is calculated as (a+b)*(b+c). |
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| Numerical value, operator and constant |
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| Classification |
Functional name |
Reserved word |
| Numerical value |
Numerical value |
1234567890 |
0 ~9 |
| Decimal point |
0.123 |
. |
| Sign |
-123, +456 |
+,- |
| Exponent |
1.23E-45 |
E |
| Imaginary number |
2+3i |
i |
| Operator |
Addition |
x+y |
+ |
| Subtraction |
x-y |
- |
| Multiplication |
x*y |
* |
| Division |
x/y |
/ |
| Power |
x^y |
^ |
| Factorial |
n! |
! |
| Double factorial |
n!! |
!! |
| Substitution |
a=123 |
= |
| Multiple expression |
Expression separator |
a=2+3; b=4/7 |
; |
| Constant |
Napier's constant |
e= 2.71828… |
e |
| Infinity |
∞ |
inf |
| Pi |
π= 3.14159… |
pi |
| Euler's constant |
γ= 0.5772… |
euler |
| Logical value |
true = 1, false = 0 |
true,false |
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| Elementary function |
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| Classification |
Functional name |
Reserved word |
| Logarithm and exponential function |
Square root |
√ |
sqrt(x) |
| Common logarithm |
log |
log(x) |
| Natural logarithm |
ln |
ln(x) |
| Exponential |
ex |
e^x |
| Power |
xy |
x^y |
| Trigonometric function |
Sine |
sin |
sin(x) |
| Cosine |
cos |
cos(x) |
| Tangent |
tan |
tan(x) |
| Invese sine |
sin-1 |
asin(x) |
| Invese cosine |
cos-1 |
acos(x) |
| Invese tangent |
tan-1 |
atan(x) |
| Hyperbolic function |
Hyperbolic sine |
sinh |
sinh(x) |
| Hyperbolic cosine |
cosh |
cosh(x) |
| Hyperbolic tangent |
tanh |
tanh(x) |
| Invese hyperbolic sine |
sinh-1 |
asinh(x) |
| Invese hyperbolic cosine |
cosh-1 |
acosh(x) |
| Invese hyperbolic tangent |
tanh-1 |
atanh(x) |
| Combination and Permutation |
Combination |
nCr |
combination(n,r) |
| Permutation |
nPr |
permutation(n,r) |
| Factorial |
x! |
x! |
| Double factorial |
x!! |
x!! |
Numerical function
(real number) |
Floor function |
int(4.2)=4, int(-4.2)=-5 |
int (x) |
| Ceiling function |
ceiling(4.2)=5, ceiling(-4.2)=-4 |
ceiling (x) |
| Integer part |
IP(4.2)=4, IP(-4.2)=-4 |
IP (x) |
| Fractional part |
FP(4.2)=0.2, FP(-4.2)=-0.2 |
FP (x) |
| Remainder on division of x by y |
mod(9,5)=4, mod(-9,5)=1 |
mod (x, y) |
| y*FP(x/y) |
remainder(9,5)=4, remainder(-9,5)=-4 |
remainder (x, y) |
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Rounding x to n decimal place(s)
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round(-2.34,1)=-2.3 |
round (x, n) |
| Round down x to n decimal place(s) |
roundup(-2.34,1)=-2.3 |
rounddown (x, n) |
| Round up x to n decimal place(s) |
rounddown(-2.34,1)=-2.4 |
roundup (x, n) |
| IP(x*10n)/10n |
truncate(-2.34,1)=-2.3 |
truncate (x, n) |
| Numerical function (complex number) |
Sign |
sign(z)=1,0,-1 |
sign(z) |
| Complex conjugate number |
x+iy→ x-iy |
conjugate(z) |
| Real part |
x+iy→ x |
Re(z) |
| Imaginary part |
x+iy→ y |
lm(z) |
| Absolute value |
|z| = |reiθ| = r |
abs(z) |
| Complex argument |
reiθ→θ |
argument(z) |
| Cartesian coordinate conversion |
reiθ→x+iy |
cartesian(z) |
| Cartesian coordinate conversion |
(r,θ) → x+iy |
cartesian(r,θ) |
| Polar coordinate conversion |
x+iy → reiθ |
polar(z) |
| Polar coordinate conversion |
(x, y) → reiθ |
polar(x, y) |
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| Probability function |
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| Classification |
Functional name |
Reserved word |
| Upper cumulative distribution Q |
Probability density f |
Percentage point x |
Non centrality λ |
| Continuous distribution function |
Standard
normal distribution |
normalcd(x) |
normalpd(x) |
normalicd(Q) |
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| Normal
distribution |
normalcd(x,μ,σ) |
normalpd(x,μ,σ) |
normalicd(Q,μ,σ) |
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| Logarithmic normal distribution |
lognormalcd(x,μ,σ) |
lognormalpd(x,μ,σ) |
lognormalicd(Q,μ,σ) |
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| Chi-square distribution |
chi2cd(x,ν) |
chi2pd(x,ν) |
chi2icd(Q,ν) |
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| t-distribution of student |
tcd(x,ν) |
tpd(x,ν) |
ticd(Q,ν) |
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| F-distribution |
fcd(x,ν1,ν2) |
fpd(x,ν1,ν2) |
ficd(Q,ν1,ν2) |
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| Noncentral chi-square distribution |
ncchi2cd(x,ν,λ) |
ncchi2pd(x,ν,λ) |
ncchi2icd(Q,ν,λ) |
ncchi2il(P,x,ν) |
| Noncentral t-distribution |
nctcd(x,ν,λ) |
nctpd(x,ν,λ) |
ncticd(Q,ν,λ) |
nctil(Q,x,ν) |
| Noncentral F-distribution |
ncfcd(x,ν1,ν2,λ) |
ncfpd(x,ν1,ν2,λ) |
ncficd(Q,ν1,ν2,λ) |
ncfil(Q,x,ν1,ν2) |
| Uniform
distribution |
uniformcd(x,a,b) |
uniformpd(x,a,b) |
uniformicd(Q,a,b) |
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| Exponential
distribution |
exponentialcd(x,b) |
exponentialpd(x,b) |
exponentialicd(Q,b) |
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| Weibull
distribution |
weibullcd(x,a,b) |
weibullpd(x,a,b) |
weibullicd(Q,a,b) |
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| Gamma
distribution |
gammacd(x,a,b) |
gammapd(x,a,b) |
gammaicd(Q,a,b) |
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| Beta
distribution |
betacd(x,a,b) |
betapd(x,a,b) |
betaicd(Q,a,b) |
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| Laplace
distribution |
laplacecd(x,a,b) |
laplacepd(x,a,b) |
laplaceicd(Q,a,b) |
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| Cauchy
distribution |
cauchycd(x,a,b) |
cauchypd(x,a,b) |
cauchyicd(Q,a,b) |
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| Logistic
distribution |
logisticcd(x,α,β) |
logisticpd(x,α,β) |
logisticicd(Q,α,β) |
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| Discrete distribution function |
Poisson
distribution |
poissoncd(x,λ) |
poissonpd(x,λ) |
poissonicd(Q,λ) |
poissonil(Q,x) |
| Binomial
distribution |
binomialcd(x,n,p) |
binomialpd(x,n,p) |
binomialicd(Q,n,p) |
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| Negative
binomial distribution |
negbinomialcd(x,k,p) |
negbinomialpd(x,k,p) |
negbinomialicd(Q,k,p) |
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| Geometric
distribution |
geometriccd(x,p) |
geometricpd(x,p) |
geometricicd(Q,p) |
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| Hypergeometric
distribution |
hypgeometriccd(x,n,M,N) |
hypgeometricpd(x,n,M,N) |
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< Note > For the reserved word which uses lower cumulative probability
P, add lower to the above-mentioned each reserved word end.
Example) P=poissoncdlower (x,λ), x=poissonicdlower (P,λ), λ=poissonillower
(P, x) |
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| Bessel function |
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| Classification |
Functional name |
Reserved word |
| f (x) |
f '(x) |
f-1 (0), f '-1 (0),
s=1,2. |
| Bessel function |
Bessel function of the 1st kind |
Jν(x) |
besselj(ν,x) |
besseljdf(ν,x) |
besseljzeros(ν,s), besseljdfzeros(ν,s) |
| Bessel function of the 2nd kind |
Yν(x) |
bessely(ν,x) |
besselydf(ν,x) |
besselyzeros(ν,s), besselydfseros(ν,s) |
| Modified bessel function of the 1st kind |
Iν(x) |
besseli(ν,x) |
besselidf(ν,x) |
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| Modified bessel function of the 2nd kind |
Kν(x) |
besselk(ν,x) |
besselkdf(ν,x) |
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| Hankel function of the 1st kind |
H(1)ν(x) |
hankelH1(ν,x) |
hankelH1df(ν,x) |
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| Hankel function of the 2nd kind |
H(2)ν(x) |
hankelH2(ν,x) |
hankelH2df(ν,x) |
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| Spherical Bessel function |
Spherical bessel function of the 1st kind |
jν(x) |
sphericalbesselj(ν,x) |
sphericalbesseljdf(ν,x) |
sphericalbesseljzeros(ν,s), sphericalbesseljdfzeros(ν,s) |
| Spherical bessel function of the 2nd kind |
yν(x) |
sphericalbessely(ν,x) |
sphericalbesselydf(ν,x) |
sphericalbesselyzeros(ν,s), sphericalbesselydfzeros(ν,s) |
| Modified spherical bessel function of the 1st kind |
iν(x) |
sphericalbesseli(ν,x) |
sphericalbesselidf(ν,x) |
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| Modified spherical bessel function of the 2nd kind |
kν(x) |
sphericalbesselk(ν,x) |
sphericalbesselkdf(ν,x) |
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| Spherical hankel function of the 1st kind |
h(1)ν(x) |
sphericalhankelH1(ν,x) |
sphericalhankelH1df(ν,x) |
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| Spherical hankel function of the 2nd kind |
h(2)ν(x) |
sphericalhankelH2(ν,x) |
sphericalhankelH2df(ν,x) |
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| Airy function |
Airy
function |
Ai(x) |
airyai(x) |
airyaidf(x) |
airyaizeros(s), airyaidfzeros(s) |
| Airy function |
Bi(x) |
airybi(x) |
airybidf(x) |
airybizeros(s), airybidfzeros(s) |
| Kelvin function |
Kelvin function of the 1st kind, ber |
berν(x) |
ber(ν,x) |
berdf(ν,x) |
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| Kelvin function of the 1st kind, bei |
beiν(x) |
bei(ν,x) |
beidf(ν,x) |
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| Kelvin function of the 2nd kind, ker |
kerν(x) |
ker(ν,x) |
kerdf(ν,x) |
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| Kelvin function of the 2nd kind, kei |
keiν(x) |
kei(ν,x) |
keidf(ν,x) |
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| Kelvin function of the 3rd kind, her |
herν(x) |
her(ν,x) |
herdf(ν,x) |
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| Kelvin function of the 3rd kind, hei |
heiν(x) |
hei(ν,x) |
heidf(ν,x) |
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| Special function |
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| Classification |
Functional name |
Reserved word |
Extended reserved word |
| Gamma function |
Gamma
function |
Γ(a) |
gamma(a) |
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| Lower incomplete gamma function |
γ(a,x) |
igamma1(a,x) |
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| Upper incomplete gamma function |
Γ(a,x) |
igamma2(a,x), gamma(a,x) |
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| Lower regularized incomplete gamma function |
P(a,x) |
gammap(a,x) |
gammapdf(a,x), gammapix(y,a), gammapia(y,x) |
| Upper regularized incomplete gamma function |
Q(a,x) |
gammaq(a,x) |
gammaqdf(a,x), gammaqix(y,a), gammaqia(y,x) |
| Logarithm of gamma function |
lnΓ(a) |
lngamma(a) |
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| Digamma
function |
ψ(a) |
polygamma(a) |
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| Polygamma
function |
ψ(n)(a) |
polygamma(n,a) |
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| Double factorial |
x!! |
x!! |
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| Pochhammer function |
(x)n |
pochhammer(x,n) |
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| Beta function |
Beta
function |
B(a,b) |
beta(a,b) |
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| Regularized incomplete beta function |
Ix(a,b) |
ibeta(x,a,b) |
ibetadf(x,a,b), ibetaix(y,a,b) |
| Regularized incomplete beta function |
I1-x(a,b) |
ibetac(x,a,b) |
ibetacdf(x,a,b), ibetacix(y,a,b) |
| Error function |
Error
function |
erf(x) |
erf(x) |
erfdf(x), erfix(y) |
| Complementary error function |
erfc(x) |
erfc(x) |
erfcdf(x), erfcix(y) |
| Imaginary error function |
erfi(x) |
erfi(x) |
erfidf(x) |
| Elementary function integral |
Exponential integral |
Ei(x) |
Ei(x) |
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| Exponential integral |
En(x) |
Ei(n,x) |
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| Logarithmic integral |
li(x) |
li(x) |
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| Sine integral |
Si(x) |
Si(x) |
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| Cosine integral |
Ci(x) |
Ci(x) |
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| Hyperbolic sine integral |
Shi(x) |
Shi(x) |
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| Hyperbolic cosine integral |
Chi(x) |
Chi(x) |
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| Fresnel integral |
Fresnel sine integral |
S(x) |
fresnelS(x) |
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| Fresnel cosine integral |
C(x) |
fresnelC(x) |
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| Elliptic integral |
Complete elliptic integral of the 1st kind |
K(k) |
ellipticK(k) |
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| Complete elliptic integral of the 2nd kind |
E(k) |
ellipticE(k) |
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| Complete elliptic integral of the 3rd kind |
Π(n,k) |
ellipticPi(n,k) |
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| Incomplete elliptic integral of the 1st kind |
F(x,k) |
ellipticF(x,k) |
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| Incomplete elliptic integral of the 2nd kind |
E(x,k) |
ellipticE(x,k) |
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| Incomplete elliptic integral of the 3rd kind |
Π(x,n,k) |
ellipticPi(x,n,k) |
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| Elliptic function |
Jacobi elliptic function sn |
sn(u) |
jacobiSN(u,k) |
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| Jacobi elliptic function cn |
cn(u) |
jacobiCN(u,k) |
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| Jacobi elliptic function dn |
dn(u) |
jacobiDN(u,k) |
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| Jacobi amplitude function |
am(u) |
jacobiAM(u,k) |
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| Orthogonal polynomial |
Hermite polynomial |
Hn(x) |
hermiteH(n,x) |
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| Chebyshev polynomial of the 1st kind |
Tn(x) |
chebyshevT(n,x) |
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| Chebyshev polynomial of the 2nd kind |
Un(x) |
chebyshevU(n,x) |
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| Legendre polynomial |
Pn(x) |
legendreP(n,x) |
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| Associated Legendre polynomial |
Pnm(x) |
legendreP(n,,x) |
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| Laguerre polynomial |
Ln(x) |
laguerreL(n,x) |
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| Associated Laguerre polynomial |
Lnα(x) |
laguerreL(n,α,x) |
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| Gegenbauer polynomial |
Cnλ(x) |
gegenbauerC(n,λ,x) |
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| Jacobi polynomial |
Pnα,β(x) |
jacobiP(n,α,β,x) |
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| Spherical harmonics |
Ynm(θ,φ) |
sphericalharmonicY(n,m,θ,φ) |
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| Hypergeometric function |
Confluent hypergeometric function of the 1st kind |
1F1(a;b;x) |
F11(a,b,x) |
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| Confluent hypergeometric function of the 2nd kind |
U(a;b;x) |
FU11(a,b,x) |
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| Gauss hypergeometric function |
2F1(a,b;c;x) |
F21(a,b,c,x) |
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| Miscellaneous function |
Bernoulli number |
Bn |
bernoulli(n) |
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| Riemann zeta function |
ζ(x) |
zeta(x) |
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| Programming |
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This programming language is a "C-language-like" simple one and specialized for numerical computation.
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| Classification |
Reserved word |
Quick instruction |
| Flow control |
if(expression) {statement;...} elseif(expression) {statement;...} else{statement;...} |
If the conditional expression of "if" is true, the statement is executed. If false, the statement is executed by the conditional expression of "elseif". The if statement ends after the execution of one of statements. |
| while(expression) {statement;...} |
If the conditional expression is true, the statement is executed. The statement is repeatedly executed and ends when the conditional expression becomes false. |
| do{statement;...} while(expression); |
After the statement is unconditionally executed, if the conditional expression is true, the statement is again executed. If false, the statement ends. |
| for(initialize; check; update) {statement;...} |
At first the variable is initialized, and if the conditional expression is true, the statement is executed. After the execution of the statement, the variable is incremented, the conditional expression is again evaluated, and then the same procedure described above is repeatedly executed. When the conditional expression becomes false, the statement ends. |
| break |
It escapes from a loop unconditionally. |
| continue |
It returns to the beginning of a loop unconditionally. |
| Display control |
print(expression1,expression2,...) |
The value of expression is outputted. |
| println(expression1,expression2,...) |
After outputting the value of expression, a new line is started. |
| Comment |
/*comment*/ |
It participates in neither expression nor control. |
Logical operator
If a conditional expression is truth, the value is true.
If not, the value is false. |
not |
Priority order: not＞and＞or |
| and |
| or |
| == |
The left expression is equal to the right expression. ＝ |
| <> |
The left expression is not equal to the right expression. ≠ |
| =<, <= |
The left expression is less than or equal to the right expression. ≦ |
| =>, >= |
The left expression is greater than or equal to the right expression. ≧ |
| < |
The left expression is less than the right expression. ＜ |
| > |
The left expression is greater than the right expression. ＞ |
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Logical value
|
true |
true=1（≠0） |
| false |
false=0 |
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Service 
