complex arithmetic

Because the current version of the STx EVAL command does not support a complex data type, a package of functions is used to implement arithmetic and special handling for complex numbers.

The package consists of the following functions:

cr2p, cp2r, conj, cr2len, cr2phi, cget, cset, conj, cmul, cdiv, ctrn, cdot, cmulv

complex numerical objects

• A complex number or complex scalar is a numerical object v with 2 rows and 1 column (a vector):

v[0] =	re (cartesian: real part) or len (polar: length)
v[1] =	im (cartesian: imaginary part) or phi (polar: phase)

• A complex vector with N elements is a numerical object v with 2N rows and 1 column (a vector):

v[2*i] =	rei or leni
v[2*i+1] =	imi or phii

• A complex matrix with MxN elements is a numerical object v with 2N rows and M columns (a matrix):

v[2*i,j] =	rei,j or leni,j
v[2*i+1,j] =	imi,j or phii,j

• In general a numerical object containing N x M complex numbers (N≥1, M≥1), consists of 2N rows and M columns, because each complex number uses two cells of a row.
• If a numerical object containing N x M complex numbers, is converted element-wise to real, the resulting object consists of N x M real numbers.

complex → complex

xc	... any complex type
rc ..	... same complex type as xc

rc = cr2p(xc)

Convert xc from cartesian (real, imaginary) to polar (length, phase) format.

rc = cp2r(xc)

Convert xc from polar (length, phase) to cartesian (real, imaginary) format.

rc = conj(xc)

Conjugate xc; xc must be in cartesian format.

complex → real

xc	... any complex type
r	... same real type as xc

r = cr2len(xc)

Compute length of xc; xc is stored in cartesian format.

r = cr2phi(xc)

Compute phase of xc; xc is stored in cartesian format.

r = cget(xc, 0)

Get real part or length of xc (depends on format of xc).

r = cget(xc, 1)

Get imaginary part or phase of xc (depends on format of xc).

real → complex

x	... any real type
y	... same type as x
rc	... same complex type as x

rc = cset(x, y)

Combine elements of x (real part or length) and y (imaginary part or phase) to complex numbers.

multiplication and division (element-wise)

xc	... any complex type (re,im)
yc	... same complex type as 'xc'
n	... real or complex number (re,im)
result rc	... same complex type as xc

rc = cmul(xc, n)

rc = cmul(n, xc)

Multiply each element of xc with the real or complex number n.

rci,j = xci,j * n

rc = cmul(xc, yc)

Multiply xc and yc element by element.

rci,j = xci,j * yci,j

rc = cdiv(xc)

Compute the inverse of each element of xc.

rci,j = 1 / xci,j

rc = cdiv(xc, n)

Divide each element of xc by the complex number n.

rci,j = xci,j / n

rc = cdiv(xc, yc)

Divide xc by yc element by element.

rci,j = xci,j / yci,j

special functions

rcmatrix = ctrn(xcmatrix)

Transposed the complex matrix xc.

rci,j = xcj,i

rcscalar = cdot(xcvector, ycvector)

Compute the dot product (inner product) of the two complex vectors xc and yc (both with N elements).

rc = sumi=0..N-1 (xci * yci) , i=0..N-1

rcmatrix = cmulv(xcvector, ycvector)

Compute the tensor (or dyadic) product of the two complex vectors xc and yc.

rci,j = xci * ycj

rcvector = cmulv(xcvector, ycmatrix)

Compute the product of the complex vector xc (N elements) and the complex matrix yc (N rows, M columns).

rcj = sumi=0..N-1 (xci * yci,j) , j=0..M-1

rcvector = cmulv(xcmatrix, ycvector)

Compute the product of the complex matrix xc (N rows, M columns) and the complex vector yc (M elements).

rci = sumj=0..M-1 (xci,j * ycj) , i=0..N-1

rcmatrix = cmulv(xcmatrix, ycmatrix)

Compute the product of the complex NxM matrix xc and the complex MxL matrix yc. The result is the complex NxL matrix rc.

rci,k = sumj=0..M-1 (xci,j * ycj,i) , i=0..N-1 and k=0..L-1

See also

complex numbers, vvset, vvget, vv, fft, dft

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