Complex Numbers
This chapter describes the complex numbers as supported by MAD-NG. The module for Complex numbers is not exposed, only the contructors are visible from the MAD environment and thus, complex numbers are handled directly by their methods or by the generic functions of the same name from the module MAD.gmath. Note that complex have value semantic like a pair of number equivalent to a C structure or an array const num_t[2] for direct compliance with the C API.
Types promotion
The operations on complex numbers may involve other data types like real numbers leading to few combinations of types. In order to simplify the descriptions, the generic names num and cpx are used for real and complex numbers respectively. The following table summarizes all valid combinations of types for binary operations involving at least one complex type:
Left Operand Type |
Right Operand Type |
Result Type |
|---|---|---|
number |
complex |
complex |
complex |
number |
complex |
complex |
complex |
complex |
Constructors
The constructors for complex numbers are directly available from the MAD environment, except for the special case of the imaginary postfix, which is part of the language definition.
- i
The imaginary postfix that qualifies literal numbers as imaginary numbers, i.e.
1iis the imaginary unit, and1+2iis the complex number \(1+2i\).
- complex(re_, im_)
Return the complex number equivalent to
re + im * 1i. Default:re_ = 0,im_ = 0.
- tocomplex(str)
Return the complex number decoded from the string
strcontaining the literal complex number"a+bi"(with no spaces) whereaandbare literal numbers, i.e. the strings"1","2i"and"1+2i"will give respectively the complex numbers \(1+0i\), \(0+2i\) and \(1+2i\).
Attributes
- cpx.re
The real part of the complex number
cpx.
- cpx.im
The imaginary part of the complex number
cpx.
Functions
- is_complex(a)
Return
trueifais a complex number,falseotherwise. This function is only available from the moduleMAD.typeid.
- is_scalar(a)
Return
trueifais a number or a complex number,falseotherwise. This function is only available from the moduleMAD.typeid.
Methods
Operator-like Methods
Functions |
Return values |
Metamethods |
C functions |
|---|---|---|---|
|
\(-z\) |
|
|
|
\(z + z_2\) |
|
|
|
\(z - z_2\) |
|
|
|
\(z \cdot z_2\) |
|
|
|
\(z / z_2\) |
|
|
|
\(z \mod z_2\) |
|
|
|
\(z ^ {z_2}\) |
|
|
|
\(z = z_2\) |
|
Real-like Methods
Functions |
Return values |
C functions |
|---|---|---|
|
\(|z|\) |
|
|
\(\cos^{-1} z\) |
|
|
\(\cosh^{-1} z\) |
|
|
\(\cot^{-1} z\) |
|
|
\(\coth^{-1} z\) |
|
|
\(\sin^{-1} z\) |
|
|
\(\frac{\sin^{-1} z}{z}\) |
|
|
\(\sinh^{-1} x\) |
|
|
\(\frac{\sinh^{-1} z}{z}\) |
|
|
\(\tan^{-1} z\) |
|
|
\(\tanh^{-1} z\) |
|
|
\(\lceil\Re(z)\rceil+i\,\lceil\Im(z)\rceil\) |
|
|
\(\cos z\) |
|
|
\(\cosh z\) |
|
|
\(\cot z\) |
|
|
\(\coth z\) |
|
|
\(\exp z\) |
|
|
\(\lfloor\Re(z)\rfloor+i\,\lfloor\Im(z)\rfloor\) |
|
|
\(z - \operatorname{trunc}(z)\) |
|
|
\(\sqrt{z^2+z_2^2}\) |
|
|
\(\sqrt{z^2+z_2^2+z_3^2}\) |
|
|
\(\frac{v}{z}\) |
|
|
\(\frac{v}{\sqrt z}\) |
|
|
\(\log z\) |
|
|
\(\log_{10} z\) |
|
|
\(z^n\) |
|
|
\(\lfloor\Re(z)\rceil+i\,\lfloor\Im(z)\rceil\) |
|
|
\(\sin z\) |
|
|
\(\frac{\sin z}{z}\) |
|
|
\(\sinh z\) |
|
|
\(\frac{\sinh z}{z}\) |
|
|
\(z \cdot z\) |
|
|
\(\sqrt{z}\) |
|
|
\(\tan z\) |
|
|
\(\tanh z\) |
|
|
\(\operatorname{trunc} \Re(z)+i\,\operatorname{trunc} \Im(z)\) |
|
|
\(\frac{z}{|z|}\) |
In methods inv() and invsqrt(), default is v_ = 1.
Complex-like Methods
Functions |
Return values |
C functions |
|---|---|---|
|
\(|z|\) |
|
|
\(\arg z\) |
|
|
\(z^*\) |
|
|
\(|\Re(z)|+i\,|\Im(z)|\) |
|
|
\(\Im(z)\) |
|
|
\(|z|\,e^{i \arg z}\) |
|
|
\(\operatorname{proj}(z)\) |
|
|
\(\Re(z)\) |
|
|
\(\Re(z)\cos \Im(z)+i\,\Re(z)\sin \Im(z)\) |
|
|
\(\Re(z), \Im(z)\) |
Error-like Methods
Error-like methods call C wrappers to the corresponding functions from the Faddeeva library from the MIT, considered as one of the most accurate and fast implementation over the complex plane [FADDEEVA] (see mad_num.c).
Functions |
Return values |
C functions |
|---|---|---|
|
\(\frac{2}{\sqrt\pi}\int_0^z e^{-t^2} dt\) |
|
|
\(1-\operatorname{erf}(z)\) |
|
|
\(-i\operatorname{erf}(i z)\) |
|
|
\(e^{z^2}\operatorname{erfc}(z)\) |
|
|
\(e^{-z^2}\operatorname{erfc}(-i z)\) |
|
|
\(\frac{-i\sqrt\pi}{2}e^{-z^2}\operatorname{erf}(iz)\) |
Operators
The operators on complex follow the conventional mathematical operations of Complex numbers.
- -cpx
Return a complex resulting from the negation of the operand as computed by
cpx:unm().
- num + cpx
- cpx + num
- cpx + cpx
Return a complex resulting from the sum of the left and right operands as computed by
cpx:add().
- num - cpx
- cpx - num
- cpx - cpx
Return a complex resulting from the difference of the left and right operands as computed by
cpx:sub().
- num * cpx
- cpx * num
- cpx * cpx
Return a complex resulting from the product of the left and right operands as computed by
cpx:mul().
- num / cpx
- cpx / num
- cpx / cpx
Return a complex resulting from the division of the left and right operands as computed by
cpx:div(). If the right operand is a complex number, the division uses a robut and fast algorithm implemented inmad_cpx_div_r()[1].
- num % cpx
- cpx % num
- cpx % cpx
Return a complex resulting from the rest of the division of the left and right operands, i.e. \(x - y \lfloor \frac{x}{y} \rfloor\), as computed by
cpx:mod(). If the right operand is a complex number, the division uses a robut and fast algorithm implemented inmad_cpx_div_r()[1].
- num ^ cpx
- cpx ^ num
- cpx ^ cpx
Return a complex resulting from the left operand raised to the power of the right operand as computed by
cpx:pow().
- num == cpx
- cpx == num
- cpx == cpx
Return
falseif the real or the imaginary part differ between the left and right operands,trueotherwise. A numberawill be interpreted as \(a+i0\) for the comparison.
C API
These functions are provided for performance reason and compliance (i.e. branch cut) with the C API of other modules dealing with complex numbers like the linear and the differential algebra. For the same reason, some functions hereafter refer to the section 7.3 of the C Programming Language Standard [ISOC99CPX].
-
num_t mad_cpx_abs_r(num_t x_re, num_t x_im)
Return the modulus of the complex
xas computed bycabs().
-
num_t mad_cpx_arg_r(num_t x_re, num_t x_im)
Return the argument in \([-\pi, +\pi]\) of the complex
xas computed bycarg().
-
void mad_cpx_unit_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe complexxdivided by its modulus as computed bycabs().
-
void mad_cpx_proj_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe projection of the complexxon the Riemann sphere as computed bycproj().
-
void mad_cpx_rect_r(num_t rho, num_t ang, cpx_t *r)
Put in
rthe rectangular form of the complexrho * exp(i*ang).
-
void mad_cpx_inv_r(num_t x_re, num_t x_im, cpx_t *r)
-
cpx_t mad_cpx_inv(cpx_t x)
Put in
ror return the inverse of the complexx.
-
void mad_cpx_invsqrt_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe square root of the inverse of the complexx.
-
void mad_cpx_div_r(num_t x_re, num_t x_im, num_t y_re, num_t y_im, cpx_t *r)
-
cpx_t mad_cpx_div(cpx_t x, cpx_t y)
Put in
ror return the complexxdivided by the complexy.
-
void mad_cpx_mod_r(num_t x_re, num_t x_im, num_t y_re, num_t y_im, cpx_t *r)
Put in
rthe remainder of the division of the complexxby the complexy.
-
void mad_cpx_pow_r(num_t x_re, num_t x_im, num_t y_re, num_t y_im, cpx_t *r)
Put in
rthe complexxraised to the power of complexyusingcpow().
-
void mad_cpx_powi_r(num_t x_re, num_t x_im, int n, cpx_t *r)
-
cpx_t mad_cpx_powi(cpx_t x, int n)
Put in
ror return the complexxraised to the power of the integernusing a fast algorithm.
-
void mad_cpx_sqrt_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe square root of the complexxas computed bycsqrt().
-
void mad_cpx_exp_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe exponential of the complexxas computed bycexp().
-
void mad_cpx_log_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe natural logarithm of the complexxas computed byclog().
-
void mad_cpx_sin_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe sine of the complexxas computed bycsin().
-
void mad_cpx_cos_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe cosine of the complexxas computed byccos().
-
void mad_cpx_tan_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe tangent of the complexxas computed byctan().
-
void mad_cpx_sinh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic sine of the complexxas computed bycsinh().
-
void mad_cpx_cosh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic cosine of the complexxas computed byccosh().
-
void mad_cpx_tanh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic tangent of the complexxas computed byctanh().
-
void mad_cpx_asin_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe arc sine of the complexxas computed bycasin().
-
void mad_cpx_acos_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe arc cosine of the complexxas computed bycacos().
-
void mad_cpx_atan_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe arc tangent of the complexxas computed bycatan().
-
void mad_cpx_asinh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic arc sine of the complexxas computed bycasinh().
-
void mad_cpx_acosh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic arc cosine of the complexxas computed bycacosh().
-
void mad_cpx_atanh_r(num_t x_re, num_t x_im, cpx_t *r)
Put in
rthe hyperbolic arc tangent of the complexxas computed bycatanh().
-
void mad_cpx_sinc_r(num_t x_re, num_t x_im, cpx_t *r)
-
cpx_t mad_cpx_sinc(cpx_t x)
Put in
ror return the sine cardinal of the complexx.
-
void mad_cpx_sinhc_r(num_t x_re, num_t x_im, cpx_t *r)
-
cpx_t mad_cpx_sinhc(cpx_t x)
Put in
ror return the hyperbolic sine cardinal of the complexx.
-
void mad_cpx_asinc_r(num_t x_re, num_t x_im, cpx_t *r)
-
cpx_t mad_cpx_asinc(cpx_t x)
Put in
ror return the arc sine cardinal of the complexx.
-
void mad_cpx_asinhc_r(num_t x_re, num_t x_im, cpx_t *r)
-
cpx_t mad_cpx_asinhc(cpx_t x)
Put in
ror return the hyperbolic arc sine cardinal of the complexx.
-
void mad_cpx_wf_r(num_t x_re, num_t x_im, num_t relerr, cpx_t *r)
-
cpx_t mad_cpx_wf(cpx_t x, num_t relerr)
Put in
ror return the Faddeeva function of the complexx.
-
void mad_cpx_erf_r(num_t x_re, num_t x_im, num_t relerr, cpx_t *r)
-
cpx_t mad_cpx_erf(cpx_t x, num_t relerr)
Put in
ror return the error function of the complexx.
-
void mad_cpx_erfc_r(num_t x_re, num_t x_im, num_t relerr, cpx_t *r)
-
cpx_t mad_cpx_erfc(cpx_t x, num_t relerr)
Put in
ror return the complementary error function of the complexx.
-
void mad_cpx_erfcx_r(num_t x_re, num_t x_im, num_t relerr, cpx_t *r)
-
cpx_t mad_cpx_erfcx(cpx_t x, num_t relerr)
Put in
ror return the scaled complementary error function of the complexx.
References
Smith, “Algorithm 116: Complex division”, Commun. ACM, 5(8):435, 1962.
Baudin and R. L. Smith, “A robust complex division in Scilab”, October 2012. http://arxiv.org/abs/1210.4539.
Oeftiger, R. De Maria, L. Deniau et al, “Review of CPU and GPU Faddeeva Implementations”, IPAC2016. https://cds.cern.ch/record/2207430/files/wepoy044.pdf.
ISO/IEC 9899:1999 Programming Languages - C. https://www.iso.org/standard/29237.html.
Footnotes