Âñòðîåííûå ôóíêöèè
Mathcad 8 Pro
3.1. Âñòðîåííûå ôóíêöèè Mathcad ïî
ãðóïïàì:
1. Âîñåìíàäöàòü ôóíêöèé Áåññåëÿ
(Bessel):
2. Ïÿòü ôóíêöèé ðàáîòû ñ
êîìïëåêñíûìè ÷èñëàìè (Complex Numbers):
4. Òðè ôóíêöèè òèïà âûðàæåíèÿ
(Expression Type):
5. Äâàäöàòü ïÿòü ôóíêöèé ðàáîòû ñ
ôàéëàìè (File Access):
6. Øåñòüäåñÿò ÷åòûðå ôóíêöèè
îáðàáîòêè ñèãíàëîâ (Signal Processing):
7. Äåâÿíîñòî ïÿòü ôóíêöèé îáðàáîòêè
îáðàçîâ (Image Processing):
8. Âîñåìü ôóíêöèé ïðåîáðàçîâàíèé
Ôóðüå (Fourier Transform):
9. Äâåíàäöàòü ãèïåðáîëè÷åñêèõ
ôóíêöèé (Hyperbolic):
10. Äåâÿíîñòî ïÿòü ôóíêöèé
îáðàáîòêè îáðàçîâ (Image Processing):
11. Îäèííàäöàòü ôóíêöèé
èíòåðïîëÿöèè è ýêñòðàïîëÿöèè (Interpolation and Prediction):
12. Òðè ëîãàðèôìè÷åñêèå è
ýêñïîíåíöèàëüíûå ôóíêöèè (Log and Exponential):
33. Ïÿòü ôóíêöèé òåîðèè ÷èñåë è
êîìáèíàòîðèêè (Numbers Theory/Combinatorics):
14. Ïÿòü ôóíêöèé ñòóïåíåê è óñëîâèé
(Piecewise Continuous):
15. Øåñòíàäöàòü ôóíêöèé ïëîòíîñòè
âåðîÿòíîñòè (Probably Density):
16. Òðèäöàòü øåñòü ôóíêöèé
ðàñïðåäåëåíèÿ âåðîÿòíîñòè (Probably Distribution):
17. Âîñåìíàäöàòü ôóíêöèé ñëó÷àéíûõ
÷èñåë (Random Numbers):
18. Äåñÿòü ôóíêöèé ðåãðåññèè è
ñãëàæèâàíèÿ (Regression and Smoothing):
19. Ñåìü ôóíêöèé ðåøåíèÿ
àëãåáðàè÷åñêèõ óðàâíåíèé è ñèñòåì, à òàêæå îïòèìèçàöèè (Solving):
20. ×åòûðå ôóíêöèè ñîðòèðîâêè
ìàññèâîâ (Sorting):
21. Äâåíàäöàòü ñïåöèàëüíûõ ôóíêöèé
(Special):
22. Øåñòüäåñÿò äâå ñòàòèñòè÷åñêèå
ôóíêöèè (Statistics):
23. Âîñåìü òåêñòîâûõ ôóíêöèé
(String):
24. ×åòûðå ôóíêöèè îêðóãëåíèÿ è
ðàáîòû ñ ÷àñòüþ ÷èñëà (Truncation and Round-Off):
25. Ôóíêöèè, îïðåäåëåííûå
ïîëüçîâàòåëåì (User function):
26. Òðèäöàòü ÷åòûðå ôóíêöèè ðàáîòû
ñ âåêòîðàìè è ìàòðèöàìè (Vector and Matrix):
27. Äâå ôóíêöèè âîëíîâîãî
ïðåîáðàçîâàíèÿ (Wavelet Transform):
3.2. Âñòðîåííûå ôóíêöèè Mathcad ïî
àëôàâèòó:
3.3. Ôóíêöèè ÷èñëåííûõ ìåòîäîâ
(Numerical Recipes):
Îáîçíà÷åíèÿ:
x è y – âåùåñòâåííûå ÷èñëà;
S – öåïî÷êà ñèìâîëîâ (òåêñò);
z – âåùåñòâåííîå ëèáî êîìïëåêñíîå ÷èñëî;
m, n, i, j è k – öåëûå ÷èñëà;
v, u è âñå èìåíà, íà÷èíàþùèåñÿ ñ
v, – âåêòîðû;
A è B – ìàòðèöû ëèáî âåêòîðû;
M è N – êâàäðàòíûå ìàòðèöû;
F – âåêòîð-ôóíêöèÿ;
file – ëèáî èìÿ ôàéëà, ëèáî ôàéëîâàÿ ïåðåìåííàÿ, ïðèñîåäèíåííàÿ ê èìåíè
ôàéëà.
Âñå óãëû èçìåðÿþòñÿ â ðàäèàíàõ. Ìíîãîçíà÷íûå ôóíêöèè (àðêñèíóñ, íàïðèìåð) è ôóíêöèè ñ êîìïëåêñíûì àðãóìåíòîì âñåãäà âîçâðàùàþò ãëàâíîå çíà÷åíèå.
Èìåíà ïðèâåäåííûõ ôóíêöèé
íå÷óâñòâèòåëüíû ê øðèôòó, íî ÷óâñòâèòåëüíû ê ðåãèñòðó – èõ ñëåäóåò ïå÷àòàòü â
òî÷íîñòè òàê, êàê îíè ïðèâåäåíû. Ðåêîìåíäóåòñÿ íå íàáèðàòü èìåíà ôóíêöèé ÷åðåç
êëàâèàòóðó, à ïîëüçîâàòüñÿ êíîïêîé (êîìàíäîé) Insert
Function... (Âñòàâèòü ôóíêöèþ...) èç ìåíþ Insert. Ýòî ïîçâîëèò ðàáîòàòü ñ
Ìàñòåðîì ôóíêöèé – ñì. ðèñ. 1.28 â ýòþäå 1. Ïîñëå èìåíè ôóíêöèè ñëåäóåò ÷èòàòü
«âîçâðàùàåò[1]» è äàëåå ïî
òåêñòó.
Àâòîð äîëæåí ïðèçíàòüñÿ
÷èòàòåëþ, ÷òî îí íå ñîâñåì óâåðåí â àáñîëáþòíîé ïðàâèëüíîñòè ïåðåâîäà ñóòè
íåêîòîðûõ ôóíêöèé, îñîáåííî òåõ, ñ êàêèìè àâòîð íå ðàáîòàë. Áûëà äàæå èäåÿ
îñòàâèòü òóò àíãëèéñêèé òåêñò. Îíà îñóùåñòâëåíà íàïîëîâèíó – ñïèñîê âñòðîåííûõ
ôóíêöèé ýëåêòðîííîãî ó÷åáíèêà Numerical Recieptes
(ñì. íèæå) íå ïåðåâåäåí. Â êîíöå îïèñàíèÿ íåêîòîðûõ ôóíêöèé óêàçàíû íîìåðà
ðèñóíêîâ, ãäå îíè çàäåéñòâîâàíû.
Ai(x), bei(n, x), ber(n, x), Bi(x), I0(x), I1(x), In(m, x), J0(x), J1(x), Jn(m, x), js(n, x), K0(x), K1(x), Kn(m, x), Y0(x), Y1(x), Yn(m, x) è ys(n, x).
arg(z), csgn(z), Im(z), Re(z) è signum(z).
Bulstoer(y, x1, x2, npts, D),
bulstoer(y, x1, x2, acc, D, kmax, save), bvalfit(v1,
v2, x1, x2, xf, D, load1, load2, score), multigrid(M,
ncycle), relax(A, B, C, D, E, F, U, rjac), Rkadapt(y, x1, x2, npts, D),
rkadapt(y, x1, x2, acc, D, kmax, save), rkfixed(y,
x1, x2, npts, D), sbval(v, x1, x2, D, load, score), Stiffb(y, x1, x2, npts, D, J),
stiffb(y, x1, x2, acc, D, J, kmax, save), Stiffr(y, x1,
x2, npts, D, J)
è stiffr(y, x1, x2, acc, D, J,
kmax, save).
IsArray(x), IsScalar(x) è IsString(x).
APPENDPRN(file), LoadColormap(file), READ(file), READ_BLUE(file), READBMP(file), READ_GREEN(file), READ_HLS(file), READ_HLS_HUE(file), READ_HLS_LIGHT(file), READ_HLS_SAT(file), READ_HSV(file), READ_HSV_HUE(file), READ_HSV_SAT(file), READ_HSV_VALUE(file), READ_IMAGE(file), READPRN(file), READ_RED(file), READRGB(file), READRGB(file), SaveColormap(file, M), WRITE(file), WRITE_HLS(file), WRITE_HSV(file), WRITEBMP(file), WRITEPRN(file) è WRITERGB(file).
(ñì.
äîêóìåíòàöèþ è help ïàêåòà è îäíîèìåííîãî ýëåêòðîííîãî ó÷åáíèêà).
(ñì.
äîêóìåíòàöèþ è help ïàêåòà).
CFFT(A), cfft(A), FFT(v), fft(v), ICFFT(A), icfft(A), IFFT(v) è ifft(v).
sinh(z), cosh(z), tanh(z), csch(z), sech(z), coth(z), asinh(z), acosh(z), atanh(z), acoth(z), asech(z) è acsch(z).
(ñì.
äîêóìåíòàöèþ è help ïàêåòà).
linterp(vx, vy, x), cspline(vx,
vy), pspline(vx, vy), lspline(vx,
vy), interp(vs, vx, vy, x), cspline(Mxy,
Mz), pspline(Mxy, Mz), lspline(Mxy,
Mz), interp(vs, Mxy, Mz, v) è predict(v, m,
n) è bspline(vx, vy, u, n).
exp(z) (èëè ez), ln(z) è log(z, b).
combin(n, k), gcd(A), lcm(A), mod(n, k) è premut(n, k).
ε(i, j, k), F(x), if(cond, x, y), δ(x, y) è sign(x).
dbeta(x, s1, s2),
dbinom(k, n, p), dchisq(x, d), dexp(x, r), dF(x, d1, d2), dgamma(x, s), dgeom(k, p), dhypergeom(m, a, b, n), dlnorm(x, mu, sigma),
dlogis(x, l, s), dnbinom(k, n, p), dnorm(x, mu, sigma), dpois(k, l) dt(x, d),
dunif(x, a, b)
è dweibull(x, s).
cnorm(x),
pbeta(x, s1, s2), pbinom(k, n, p), pcauchy(x, l, s), pchisq(x, d), pexp(x, r), pF(x, d1, d2), pgamma(x, s), pgeom(k, p), phypergeom(m, a, b, n), plnorm(x, mu, sigma),
plogis(x, l, s), pnbinom(k, n, p), pnorm(x, mu, sigma), ppois(k, l),
pt(x, d), punif(x, a, b), pweibull(x, s), qbeta(p, s1, s2), qbinom(p, n, q), qcauchy(p, l, s), qchisq(p, d), qexp(p, r), qF(p, d1, d2), qgamma(p, s), qgeom(p, q), qhypergeom(p, n, M, N), qlnorm(p, mu, sigma),
qlogis(p, l, s), qnbinom(p, n, r), qnorm(p, mu, sigma), qpois(p, l),
qt(p, d) è qunif(p, a, b).
rbeta(m, s1, s2), rbinom(m, n, p), rcauchy(m, l, s), rchisq(m, d), rexp(m, r), rF(m, d1, d2), rgamma(m, s), rgeom(m, p), rhypergeom(m, a, b, n), rlnorm(m, mu, sigma), rlogis(m, l, s), rnbinom(m, n, p), rnd(x), rnorm(m, mu, sigma), rpois(m, l), rt(m, d), runif(m, a, b) è rweibull(m, s).
genfit(vx,
vy, vg, F), intercept(vx, vy), ksmooth(vx, vy, b), linfit(vx, vy, F),
loess(Mx,
My, span), medsmooth(vy, n), regress(Mx, vy, n), slope(vx, vy),
stderr(vx,
vy) è supsmooth(vx, vy).
Find(var1, var2,...), lsolve(M, v), Maximize(f, var1,
var2,...), MinErr(var1, var2,...), Minimize(f,
var1, var2,...), polyroots(v) è root(f(var),
var).
csort(A, j), reverse(A), reverse(v), rsort(A, j) è sort(v).
erf(z), erfc(x), fhyper(a, b, c, x), Gamma(a, z), Her(n, x), ibeta(a, x, y), Jac(n, a, b, x), Lag(n, x), Leg(n, x), mhyper(a, b, x), Tcheb(n, x) è Ucheb(n, x).
mean(A), median(A), var(A), Var(A), cvar(A,B), stdev(A), Stdev(A), corr(A, B), dbeta(x, s1, s2), dbinom(k, n, p), dcauchy(x, l,
s), dchisq(x, d), dexp(x, r), dF(x, d1, d2), dgamma(x, s), dgeom(k, p), dhypergeom(m, a, b,
n), dlnorm(x, m, s), dlogis(x, l, s), dnbinom(k, n, p), dnorm(x, m, s), dpois(k, l), dt(x, d), dunif(x, a, b), dweibull(x, s), cnorm(x), fhyper(a, b, c, x), mhyper(a, b, x), pbeta(x, s1,
s2), pbinom(k, n, p), pcauchy(x, l,
s), pchisq(x, d), pexp(x, r), pF(x, d1, d2), pgamma(x, s), pgeom(k, p), phypergeom(m, a, b,
n), plnorm(x, m, s), plogis(x, l, s), pnbinom(k, n, p), pnorm(x, m, s), ppois(k, l), pt(x, d), punif(x, a, b), pweibull(x, s), qbeta(x, s1, s2), qbinom(p, n,
q), qcauchy(p, l, s), qchisq(p, d), qexp(p, r), qF(p, d1, d2), qgamma(p, s), qgeom(p, q), qhypergeom(p, a, b, n), qlnorm(p, m, s), qlogis(p, l, s), qnbinom(p, n, q), qnorm(p, m, s), qpois(p, l), qt(p, d), qunif(p, a, b) è qweibull(p,
s).
concat(S1, S2,
S3,...), strlen(S), search(S,
SubS, m), substr(S, m, n), str2num(S), num2str(z), str2vec(S) è vec2str(v).
ceil(x), floor(x), round(x, n) è trunc(x).
kronecker(m, n) è Psi(z).
augment(A, B), cholesky(M), cols(A), cond1(M), cond2(M), conde(M), condi(M), diag(v), eigenvals(M), eigenvec(M, z), eigenvecs(M), geninv(A), genvals(M, N), genvecs(M, N), identity(n), last(v), lenght(v), lu(M), matrix(m, n, f), max(A), min(A), norm1(M), norm2(M), norme(M), normi(M), qr(A), rank(A), rows(A), rref(A), stack(A, B), submatrix(A, ir, jr, ic, jc), svd(A), svds(A) è tr(M).
iwave(v) è wave(v).
1.
acos(z) – àðêêîñèíóñ z.
2.
acosh(z) – ãèïåðáîëè÷åñêèé
àðêêîñèíóñ – îáðàòíàÿ ôóíêöèÿ ê ãèïåðáîëè÷åñêîìó êîñèíóñó.
3.
acot(z) – àðêêîòàíãåíñ z (â ðàäèàíàõ). Ðåçóëüòàò – ìåæäó 0 è p, åñëè z âåùåñòâåííîå. Ðåçóëüòàò –
äåéñòâèòåëüíàÿ ÷àñòü, åñëè z – êîìïëåêñíîå ÷èñëî.
4.
acoth(z) – îáðàòíûé ãèïåðáîëè÷åñêèé
êîòàíãåíñ z. Ðåçóëüòàò – äåéñòâèòåëüíàÿ
÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
5.
acsc(z) – àðêêîñåêàíñ z (â ðàäèàíàõ). Ðåçóëüòàò – äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
6.
acsch(z) – ãèïåðáîëè÷åñêèé àðêêîñåêàíñ.
Ðåçóëüòàò – äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
7.
Ai(x) – çíà÷åíèå ôóíêöèè Ýéðè
ïåðâîãî âèäà.
8.
angle(x, y) – óãîë (â ðàäèàíàõ) ìåæäó
ïîëîæèòåëüíûì íàïðàâëåíèåì îñè x è ðàäèóñ-âåêòîðîì òî÷êè (x, y). Ðåçóëüòàò îò 0 äî 2p.
9.
APPENDPRN(file) – äîáàâëåíèå ìàòðèöû A ê ñóùåñòâóþùåìó ôàéëó äàííûõ (îáëàäàþùåìó ñòðóêòóðîé ASCII) file.prn íà äèñêå. Êàæäàÿ ñòðîêà
ìàòðèöû ñòàíîâèòñÿ íîâîé ñòðîêîé â äàííîì ôàéëå. Ñóùåñòâóþùèå äàííûå äîëæíû
èìåòü ñòîëüêî æå ñòîëáöîâ, ñêîëüêî A.
10.
arg(z) – óãîë (â ðàäèàíàõ) ìåæäó
ïîëîæèòåëüíûì íàïðàâëåíèåì îñè x è êîìïëåêñíîãî ÷èñëà z. Ðåçóëüòàò îò -p äî p. Âîçâðàùàåò q, êîãäà z ïðåäñòàâëåíî â âèäå r×ei×q.
11.
asec(z) – êîñåêàíñ z (â ðàäèàíàõ). Ðåçóëüòàò – äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
12.
asech(z) – ãèïåðáîëè÷åñêèé êîñåêàíñ z. Ðåçóëüòàò – äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
13.
asin(z) – óãîë (â ðàäèàíàõ), ñèíóñ
êîòîðîãî ðàâåí z; ðåçóëüòàò îò -p/2 äî p/2, åñëè z âåùåñòâåííîå.
Äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî z.
14.
asinh(z) – àðåàñèíóñ: îáðàòíàÿ
ôóíêöèÿ ê ãèïåðáîëè÷åñêîìó ñèíóñó z. Ðåçóëüòàò – äåéñòâèòåëüíàÿ
÷àñòü äëÿ êîìïëåêñíîãî ÷èñëà z.
15.
atan(z) – àðêòàíãåíñ z. Ðåçóëüòàò îò -p/2 äî p/2, åñëè z âåùåñòâåííîå.
Äåéñòâèòåëüíàÿ ÷àñòü äëÿ êîìïëåêñíîãî z.
16.
atan2(x, y) – óãîë (â ðàäèàíàõ) îò ïîëîæèòåëüíîãî
íàïðàâëåíèÿ îñè àáñöèññ äî ïðÿìîé, ñîäåðæàùåé íà÷àëî êîîðäèíàò (0, 0) è òî÷êó ñ
êîîðäèíàòàìè (x, y). È x è y äîëæíû áûòü âåùåñòâåííûìè
÷èñëàìè. Ðåçóëüòàò îò -p äî p.
17.
atanh(z) – àðåàòàíãåíñ: îáðàòíàÿ
ôóíêöèÿ ê ãèïåðáîëè÷åñêîìó òàíãåíñó. Ðåçóëüòàò – äåéñòâèòåëüíàÿ ÷àñòü äëÿ
êîìïëåêñíîãî ÷èñëà z.
18.
augment(A, B) – ãîðèçîíòàëüíîå ñëèÿíèå
äâóõ ìàòðèö (âåêòîðîâ); îáå ìàòðèöû äîëæíû èìåòü îäèíàêîâûé ðàçìåð.
19.
bei(n, x) – çíà÷åíèå ìíèìîé ôóíêöèè
Áåññåëÿ-Êåëüâèíà.
20.
ber(n, x) – çíà÷åíèå âåùåñòâåííîé
ôóíêöèè Áåññåëÿ-Êåëüâèíà ïîðÿäêà n.
21.
Bi(x) – çíà÷åíèå
ôóíêöèè Ýéðè âòîðîãî âèäà.
22.
bspline(vx, vy, u, n) – âåêòîð êîýôôèöèåíòîâ B-ñïëàéíà ñòåïåíè n. Ïîëó÷åííûé âåêòîð
ñòàíîâèòñÿ ïåðâûì àðãóìåíòîì ôóíêöèè interp.
23.
bulstoer(y, x1, x2,
acc, D, kmax, save) – ðåøåíèå ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ
èñïîëüçîâàíèåì ìåòîäà Áóëèðøà ¾ Øò¸ðà (Bulirsh ¾ Stoer). y – âåêòîð íà÷àëüíûõ çíà÷åíèé ïî èíòåðâàëó (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. Kmax – ìàêñèìàëüíîå ÷èñëî
ïðîìåæóòî÷íûõ òî÷åê, ïî êîòîðûì äîëæíî áûòü àïïðîêñèìèðîâàíî ðåøåíèå. Save – íàèìåíüøåå äîïóñòèìîå ïðîñòðàíñòâî ìåæäó âåëè÷èíàìè, ïî êîòîðûì
àïïðîêñèìèðîâàíî ðåøåíèå.
24.
Bulstoer(v, x1, x2,
npts, D) –
ðåøåíèå ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ èñïîëüçîâàíèåì ìåòîäà
Áóëèðøà ¾ Øò¸ðà. y – âåêòîð íà÷àëüíûõ çíà÷åíèé ïî èíòåðâàëó (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. Npts – êîëè÷åñòâî òî÷åê, íå ÿâëÿþùèõñÿ
èçíà÷àëüíûìè, ïî êîòîðûì àïïðîêñèìèðîâàíî ðåøåíèå.
25.
bvalfit(v1, v2, x1,
x2, xf, D, load1, load2, score) – íà÷àëüíûå óñëîâèÿ äëÿ êðàåâîé çàäà÷è. v1,
v2 – âåêòîðû
íà÷àëüíûõ çíà÷åíèé íåîïðåäåëåííûõ ñëåâà îò x1 è x2 ñîîòâåòñòâåííî. D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. Load(x1, v1), load(x1, v2) – âåùåñòâåííûå âåêòîðû
çíà÷åíèé ôóíêöèè, n ýëåìåíòîâ êîòîðûõ
àíàëîãè÷íû çíà÷åíèÿì n íåèçâåñòíûõ ôóíêöèé â
òî÷êàõ x1, x2 ñîîòâåòñòâåííî. Score(xf, y) – âåùåñòâåííûé n-ýëåìåíòíûé âåêòîð çíà÷åíèé
ôóíêöèè, îïðåäåëÿþùèé, íàñêîëüêî, ïî âàøåìó æåëàíèþ, ðåøåíèå äîëæíî
ñîîòâåòñòâîâàòü çíà÷åíèþ â òî÷êå xf (ðåøåíèå èçâåñòíî â
íåêîòîðîé ïðîìåæóòî÷íîé òî÷êå xf).
26.
ceil(x) – íàèìåíüøåå öåëîå, íå
ïðåâûøàþùåå x.
27.
cfft(A) – áûñòðîå äèñêðåòíîå
ïðåîáðàçîâàíèå Ôóðüå ìàññèâà êîìïëåêñíûõ ÷èñåë A. Âîçâðàùàåò ìàññèâ òàêîãî
æå ðàçìåðà, êàê è åãî àðãóìåíò. A – âåùåñòâåííàÿ èëè
êîìïëåêñíàÿ ìàòðèöà èëè âåêòîð.
28.
CFFT(A) – òî æå, ÷òî è â ïóíêòå 27,
íî èñïîëüçóåò äðóãèå íîðìó è çíàê.
29.
cholesky(M) – òðåóãîëüíîå ðàçëîæåíèå
ìàòðèöû M ìåòîäîì Õîëåöêîãî. M = L×LT, ãäå M – ñèììåòðè÷íàÿ ìàòðèöà, L – òðåóãîëüíàÿ ìàòðèöà.
Âîçâðàùàåò L.
30.
cnorm(x) – ñóììàðíîå ñòàíäàðòíîå
íîðìàëüíîå ðàñïðåäåëåíèå.
31.
cols(A) – ÷èñëî ñòîëáöîâ â ìàññèâå A. A – ìàòðèöà èëè âåêòîð (ðèñ. 4.10).
32.
combin(n, k) – ÷èñëî ïîäíàáîðîâ (êàæäûé
ðàçìåðîì k), êîòîðîå ìîæåò áûòü
ñôîðìèðîâàíî èç îáúåêòîâ n. Ýòî ÷èñëî ïîäíàáîðîâ
èçâåñòíî êàê êîìáèíàöèÿ; n è k – öåëûå ÷èñëà, áîëüøå ëèáî ðàâíûå íóëþ (0£k£n).
33.
concat(S1, S2,...) – «ñêëåèâàíèå» äâóõ (èëè
áîëåå) òåêñòîâûõ ïåðåìåííûõ S1 è S2. Ñòàâèò S2 â êîíåö S1 è òàê äàëåå (ðèñ. 1.30).
34.
cond1(M) – ÷èñëî îáóñëîâëåííîñòè
ìàòðèöû, âû÷èñëåííîå â íîðìå L1.
35.
cond2(M) – ÷èñëî îáóñëîâëåííîñòè
ìàòðèöû, âû÷èñëåííîå â íîðìå L2.
36.
condå(M) – ÷èñëî îáóñëîâëåííîñòè
ìàòðèöû, âû÷èñëåííîå â íîðìå åâêëèäîâîãî ïðîñòðàíñòâà.
37.
condi(M) – ÷èñëî îáóñëîâëåííîñòè
ìàòðèöû, îñíîâàííîå íà ðàâíîìåðíîé íîðìå.
38.
corr(A, B) – êîýôôèöèåíò êîððåëÿöèè äëÿ
äâóõ ìàññèâîâ A è B, ãäå A è B – ìàòðèöû ðàçìåðà m´n, èëè âåêòîðû òàêîãî æå ðàçìåðà.
39.
cos(z) – êîñèíóñ z (ðèñ 1.15, 1,16).
40.
cosh(z) – ãèïåðáîëè÷åñêèé êîñèíóñ z.
41.
cot(z) – êîòàíãåíñ z.
42.
coth(z) – ãèïåðáîëè÷åñêèé êîòàíãåíñ z.
43.
csc(z) – êîñåêàíñ z.
44.
csch(z) – ãèïåðáîëè÷åñêèé êîñåêàíñ z.
45.
csgn(z) – âîçâðàùàåò 0, åñëè z=0, 1 – åñëè Re(z)>0 èëè (Re(z)=0 è Im(z)>0), -1 – â äðóãèõ ñëó÷àÿõ (ðèñ.
7.13).
46.
csort(A, j) – ñîðòèðîâêà ìàòðèöû A ïî ñòîëáöó j (ïåðåñòàíîâêà ñòðîê ïî
âîçðàñòàíèþ çíà÷åíèé ýëåìåíòîâ â ñòîëáöå j). Ðåçóëüòàò – ìàòðèöà
òàêîãî æå ðàçìåðà, êàê A.
47.
cspline(vx, vy) – âåêòîð êîýôôèöèåíòîâ
êóáè÷åñêîãî ñïëàéíà, ïîñòðîåííîãî ïî âåêòîðàì
vx è vy. Ïîëó÷åííûé âåêòîð ñòàíîâèòñÿ ïåðâûì àðãóìåíòîì ôóíêöèè interp (ðèñ. 4.8, 4.10).
48.
cvar(A, B) – êîâàðèàöèÿ ýëåìåíòîâ äâóõ
ìàññèâîâ A è B. A è B – âåùåñòâåííûå èëè êîìïëåêñíûå ìàòðèöû èëè âåêòîðû ðàçìåðîì m´n.
49.
dbeta(x, s1, s2) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
b-ðàñïðåäåëåíèÿ.
50.
dbinom(k, n, p) – çíà÷åíèå âåðîÿòíîñòè Pr(X=k), ãäå X – ñëó÷àéíàÿ âåëè÷èíà,
èìåþùàÿ áèíîìèàëüíîå ðàñïðåäåëåíèå.
51.
dcauchy(x, l, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ðàñïðåäåëåíèÿ Êîøè.
52.
dchisq(x, d) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
c-êâàäðàò-ðàñïðåäåëåíèÿ.
53.
dexp(x, r) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ýêñïîíåíöèàëüíîãî ðàñïðåäåëåíèÿ.
54.
dF(x, d1, d2) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ðàñïðåäåëåíèÿ Ôèøåðà.
55.
dgamma(x, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
g-ðàñïðåäåëåíèÿ.
56.
dgeom(k, p) – òî æå, ÷òî è ïóíêòå 55, íî
äëÿ ãåîìåòðè÷åñêîãî ðàñïðåäåëåíèÿ.
57.
dhypergeom(m, a, b,
n) – òî æå, ÷òî
è ïóíêòå 55, íî äëÿ ãèïåðãåîìåòðè÷åñêîãî ðàñïðåäåëåíèÿ. Âîçâðàùàåò çíà÷åíèå
âåðîÿòíîñòè Pr(x=m).
58.
diag(v) – äèàãîíàëüíàÿ ìàòðèöà,
ýëåìåíòû ãëàâíîé äèàãîíàëè êîòîðîé – ýëåìåíòû âåêòîðà v.
59.
dlnorm(x, m, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ ëîãíîðìàëüíîãî ðàñïðåäåëåíèÿ.
60.
dlogis(x, l, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ïîñëåäîâàòåëüíîãî ðàñïðåäåëåíèÿ.
61.
dnbinom(k, n, p) – òî æå, ÷òî è ïóíêòå 55, íî
äëÿ îòðèöàòåëüíîãî áèíîìèàëüíîãî ðàñïðåäåëåíèÿ.
62.
dnorm(x, m, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ íîðìàëüíîãî ðàñïðåäåëåíèÿ.
63.
dpois(k, l) – òî æå, ÷òî è ïóíêòå 55, íî äëÿ ðàñïðåäåëåíèÿ Ïóàññîíà.
64.
dt(x, d) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ðàñïðåäåëåíèÿ Ñòüþäåíòà.
65.
dunif(x, a, b) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ðàâíîìåðíîãî ðàñïðåäåëåíèÿ.
66.
dweibull(x, s) – ïëîòíîñòü âåðîÿòíîñòè äëÿ
ðàñïðåäåëåíèÿ Âåéáóëëà.
67.
eigenvals(M) – âåêòîð ñîáñòâåííûõ
çíà÷åíèé ìàòðèöû M.
68.
eigenvec(M, z) – íîðìèðîâàííûé ñîáñòâåííûé
âåêòîð êâàäðàòíîé ìàòðèöû M, ñîîòâåòñòâóþùèé åå
ñîáñòâåííîìó çíà÷åíèþ z.
69.
eigenvecs(M) – ìàòðèöà, ñòîëáöàìè êîòîðîé
ÿâëÿþòñÿ ñîáñòâåííûå âåêòîðû ìàòðèöû M. Ïîðÿäîê ðàñïîëîæåíèÿ
ñîáñòâåííûõ âåêòîðîâ ñîîòâåòñòâóåò ïîðÿäêó ñîáñòâåííûõ çíà÷åíèé, âîçâðàùàåìûõ
ôóíêöèåé eigenvals.
70.
erf(x) – ôóíêöèÿ îøèáîê.
71.
erfc(x) – äîïîëíèòåëüíàÿ ôóíêöèÿ
îøèáêè. x – âåùåñòâåííîå ïîëîæèòåëüíîå
ñêàëÿðíîå ÷èñëî.
72.
error(S) – ñîîáùåíèå îá îøèáêå äëÿ
òåêñòîâîé ïåðåìåííîé S (ðèñ. 4.10, 6.4).
73.
exp(z) – çíà÷åíèå ýêñïîíåíöèàëüíîé
ôóíêöèè ez.
74.
fft(v) – áûñòðîå äèñêðåòíîå
ïðåîáðàçîâàíèå Ôóðüå äëÿ âåùåñòâåííûõ ÷èñåë. v – âåùåñòâåííûé âåêòîð ñ 2n ýëåìåíòàìè, ãäå n
– öåëîå ÷èñëî.
Ïîëó÷èì âåêòîð ðàçìåðà 2n-1+1.
75.
FFT(v) – òî æå, ÷òî è fft(v), íî èñïîëüçóåò äðóãóþ íîðìó è çíàê. Âîçâðàùàåò âåêòîð ðàçìåðà 2n-1+1.
76.
fhyper(a, b, c, x) – çíà÷åíèå
ãèïåðãåîìåòðè÷åñêîé ôóíêöèè Ãàóññà â òî÷êå x;
a, b, c – äàííûå
ïàðàìåòðû.
77.
Find(var1, var2,...) – çíà÷åíèÿ var1, var2 ,... , ïðåäñòàâëÿþùèå ðåøåíèå ñèñòåìû óðàâíåíèé. ×èñëî
âîçâðàùàåìûõ çíà÷åíèé ðàâíî ÷èñëó àðãóìåíòîâ. Âîçâðàùàåò ñêàëÿð, åñëè òîëüêî
îäèí àðãóìåíò, â äðóãîì ñëó÷àå – âåêòîð îòâåòà (ðèñ. 1.6, 1.8, 1.9, 1.14, 1.15,
3.3, 3.4).
78.
floor(x) – íàèáîëüøåå öåëîå ÷èñëî,
ìåíüøåå èëè ðàâíîå x. x äîëæíî áûòü äåéñòâèòåëüíûì ÷èñëîì (ðèñ. 6.9).
79.
gcd(A) – íàèáîëüøåå öåëîå ÷èñëî, íà
êîòîðîå äåëÿòñÿ âñå ýëåìåíòû ìàññèâà A. Ýòî öåëîå ÷èñëî –
íàèáîëüøèé îáùèé äåëèòåëü ýëåìåíòîâ â A. A – ìàòðèöà èëè âåêòîð, âñå ýëåìåíòû – öåëûå ÷èñëà áîëüøå íóëÿ.
80.
genfit(vx, vy, vg, F) – âåêòîð, ñîäåðæàùèé
ïàðàìåòðû, êîòîðûå íàèëó÷øèì îáðàçîì àïïðîêñèìèðóþò ôóíêöèþ F îò x è n ïàðàìåòðîâ u0,
u1,... , un -1 ê äàííûì â vx è vy. F – ôóíêöèÿ, êîòîðàÿ
âîçâðàùàåò n+1-âåêòîð, ñîäåðæàùèé f è åå ÷àñòíûå ïðîèçâîäíûå ïî n ïàðàìåòðàì. Âåêòîðû vx è vy äîëæíû áûòü îäèíàêîâîãî
ðàçìåðà. vg – n ýëåìåíòíûé âåêòîð ïðèáëèçèòåëüíûõ çíà÷åíèé äëÿ n ïàðàìåòðîâ (ðèñ. 4.6).
81.
geninv(A) – ìàòðèöà, ëåâàÿ îáðàòíàÿ
ìàòðèöå A, L×A=E, ãäå E – åäèíè÷íàÿ ìàòðèöà ðàçìåðîì n
íà n, L – ïðÿìîóãîëüíàÿ ìàòðèöà
ðàçìåðîì n íà m, A – ïðÿìîóãîëüíàÿ ìàòðèöà
ðàçìåðîì m íà n.
82.
genvals(M, N) – âåêòîð îáîáùåííûõ
ñîáñòâåííûõ çíà÷åíèé vj ìàòðèöû M: M×x=vj×N×x. M è N – ìàòðèöû ñ äåéñòâèòåëüíûìè
ýëåìåíòàìè, x – íåíóëåâîé ñîáñòâåííûé
âåêòîð.
83.
genvecs(M, N) – ìàòðèöà, ñîäåðæàùàÿ
íîðìèðîâàííûå ñîáñòâåííûå âåêòîðû, îòâå÷àþùèå ñîáñòâåííûì çíà÷åíèÿì â v (âåêòîð, êîòîðûé âîçâðàùåí ôóíêöèåé genvals).
j-é ñòîëáåö
ýòîé ìàòðèöû ÿâëÿåòñÿ ñîáñòâåííûì âåêòîðîì
x,
óäîâëåòâîðÿþùèì ñîáñòâåííîìó çíà÷åíèþ óðàâíåíèÿ M×x=vj×N×x. Êâàäðàòíûå ìàòðèöû M è N ñîäåðæàò äåéñòâèòåëüíûå çíà÷åíèÿ.
84.
gmean(A) – ãåîìåòðè÷åñêîå ñðåäíåå
ýëåìåíòîâ ìàññèâà A, A – âåùåñòâåííàÿ ìàòðèöà èëè âåêòîð ðàçìåðà m´n, âñå ýëåìåíòû A äîëæíû áûòü áîëüøå íóëÿ
(ðèñ. 3.14).
85.
Her(n, x) – ïîëèíîì Ýðìèòà ñòåïåíè n â òî÷êå x.
86.
hist(intervals, A) – ãèñòîãðàììà. Âåêòîð
intervals çàäàåò ãðàíèöû èíòåðâàëîâ â
ïîðÿäêå âîçðàñòàíèÿ. A – ìàññèâ äàííûõ. Âîçâðàùàåò âåêòîð òîé æå ðàçìåðíîñòè,
÷òî è âåêòîð intervals, è ñîäåðæèò ÷èñëî òî÷åê èç
A, ïîïàâøèõ â ñîîòâåòñòâóþùèé èíòåðâàë.
87.
hmean(A) – ãàðìîíè÷åñêîå ñðåäíåå
ýëåìåíòîâ ìàññèâà A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðîì m´n, âñå ýëåìåíòû A äîëæíû áûòü íåíóëåâûìè.
88.
I0(x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ ïåðâîãî ðîäà íóëåâîãî ïîðÿäêà.
89.
I1(x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ ïåðâîãî ðîäà ïåðâîãî ïîðÿäêà.
90.
ibeta(a, x, y) – íåïîëíàÿ b-ôóíêöèÿ ñ ïàðàìåòðîì a â òî÷êå (x, y).
91.
icfft(A) – îáðàòíîå ïðåîáðàçîâàíèå
Ôóðüå, ñîîòâåòñòâóþùåå cfft. Âîçâðàùàåò ìàññèâ òàêîãî
æå ðàçìåðà, êàê è åãî àðãóìåíò.
92.
ICFFT(A) – îáðàòíîå ïðåîáðàçîâàíèå
Ôóðüå, ñîîòâåòñòâóþùåå CFFT. Âîçâðàùàåò ìàññèâ òàêîãî
æå ðàçìåðà, êàê è åãî àðãóìåíò.
93.
identity(n) – åäèíè÷íàÿ êâàäðàòíàÿ
ìàòðèöà ðàçìåðîì n.
94.
if(cond, x, y) – x, åñëè cond íå ðàâíî 0, èíà÷å – y (ðèñ. 2.7, 2.8).
95.
ifft(v) – îáðàòíîå ïðåîáðàçîâàíèå
Ôóðüå, ñîîòâåòñòâóþùåå fft. Áåðåòñÿ âåêòîð ðàçìåðîì 1+2n-1, ãäå n öåëîå ÷èñëî. Âîçâðàùàåò
äåéñòâèòåëüíûé âåêòîð ðàçìåðîì 2n.
96.
IFFT(v) – îáðàòíîå ïðåîáðàçîâàíèå,
ñîîòâåòñòâóþùåå FFT. Áåðåòñÿ âåêòîð ðàçìåðîì 1+2n-1, ãäå
n öåëîå ÷èñëî.
Âîçâðàùàåò äåéñòâèòåëüíûé âåêòîð ðàçìåðîì 2n.
97.
Im(z) – ìíèìàÿ ÷àñòü êîìïëåêñíîãî
÷èñëà z.
98.
In(m, x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ ïåðâîãî ðîäà ïîðÿäêà m.
99.
intercept(vx, vy) – êîýôôèöèåíò a ëèíåéíîé ðåãðåññèè y = a + b×x âåêòîðîâ vx è vy (ðèñ. 4.2).
100.
interp(vs, vx, vy, x) – èíòåðïîëèðóåìîå çíà÷åíèå y â òî÷êå x ïî èñõîäíûì âåêòîðàì vx è vy (âåêòîðû èìåþò îäèíàêîâîå
÷èñëî ýëåìåíòîâ) è ïî êîýôôèöèåíòàì ñïëàéíà vs (ðèñ. 4.7).
101.
interp(vs, Mxy, Mz, n) – èíòåðïîëèðóåìîå çíà÷åíèå
z, ñîîòâåòñòâóþùåå
òî÷êàì õ=n0 è ó=n1. Âåêòîð vs
âû÷èñëÿåòñÿ bspline, lspline, pspline, èëè cspline
íà îñíîâå
äàííûõ èç Mxy è Mz (ìàòðè÷íûå àðãóìåíòû ñ îäèíàêîâûì ÷èñëîì ñòðîê).
102.
IsArray(x) – 1, åñëè x – ìàòðèöà èëè âåêòîð, 0 – â äðóãèõ ñëó÷àÿõ (ðèñ. 1.30).
103.
IsScalar(x) – 1, åñëè x – âåùåñòâåííîå èëè
êîìïëåêñíîå ÷èñëî, 0 – â äðóãèõ ñëó÷àÿõ (ðèñ.
1.30).
104.
IsString(x) – 1, åñëè x – òåêñòîâàÿ ïåðåìåííàÿ, 0 – â äðóãèõ ñëó÷àÿõ (ðèñ. 1.30).
105.
iwave(v) – îáðàòíîå âîëíîâîå
ïðåîáðàçîâàíèå îòíîñèòåëüíî ïðåîáðàçîâàíèÿ wave. v – âåùåñòâåííûé âåêòîð, ðàçìåðîì 2n
(n>0).
106.
J0(x) – ôóíêöèÿ Áåññåëÿ ïåðâîãî
ðîäà íóëåâîãî ïîðÿäêà.
107.
J1(x) – ôóíêöèÿ Áåññåëÿ ïåðâîãî
ðîäà ïåðâîãî ïîðÿäêà.
108.
Jac(n, a, b, x) – ïîëèíîì ßêîáè ñòåïåíè n ñ ïàðàìåòðàìè a è b â òî÷êå x.
109.
Jn(m, x) – ôóíêöèÿ Áåññåëÿ ïåðâîãî
ðîäà ïîðÿäêà m; 0<m<100.
110.
js(n, x) – ñôåðè÷åñêàÿ ôóíêöèÿ
Áåññåëÿ ïåðâîãî ðîäà ïîðÿäêà n â òî÷êå x.
111.
K0(x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ âòîðîãî ðîäà íóëåâîãî ïîðÿäêà.
112.
K1(x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ âòîðîãî ðîäà ïåðâîãî ïîðÿäêà.
113.
Kn(m, x) – ìîäèôèöèðîâàííàÿ ôóíêöèÿ
Áåññåëÿ âòîðîãî ðîäà ïîðÿäêà m; 0<m<100.
114.
ksmooth(vx, vy, b) – n-ýëåìåíòíûé âåêòîð âîçâðàùåííûõ ñðåäíèõ vx, âû÷èñëåííûõ íà îñíîâå
ðàñïðåäåëåíèÿ Ãàóññà. vx è vy – n-ýëåìåíòíûå âåêòîðû
äåéñòâèòåëüíûõ ÷èñåë. Ïîëîñà ïðîïóñêàíèÿ b óïðàâëÿåò ñãëàæèâàþùèìè
îêíàìè.
115.
kurt(A) – ýêñöåññ ýëåìåíòîâ A.
116.
Lag(n, x) – ïîëèíîì Ëàãåððà ñòåïåíè n â òî÷êå x.
117.
last(v) – èíäåêñ ïîñëåäíåãî
ýëåìåíòà âåêòîðà v (ðèñ. ?.?).
118.
lcm(A) – íàèìåíüøåå îáùåå êðàòíîå:
íàèìåíüøåå ïîëîæèòåëüíîå öåëîå, äëÿ êîòîðîãî âñå çíà÷åíèÿ ìàññèâà ÿâëÿþòñÿ
ñîìíîæèòåëÿìè. Ýëåìåíòû ìàññèâà A äîëæíû áûòü öåëûìè íåîòðèöàòåëüíûìè ÷èñëàìè.
119.
Leg(n, x) – ïîëèíîì Ëåæàíäðà ñòåïåíè n â òî÷êå x.
120.
lenght(v) – ÷èñëî ýëåìåíòîâ â âåêòîðå v (ðèñ. ?.?).
121.
linfit(vx, vy, F) – êîýôôèöèåíòû ëèíåéíîé
àïïðîêñèìàöèè ôóíêöèé, õðàíÿùèõñÿ â ñèìâîëüíîì âåêòîðå F; èñõîäíûå òî÷êè õðàíÿòñÿ â âåêòîðàõ vx è vy (ðèñ. 4.5).
122.
linterp(vx, vy, x) – çíà÷åíèå â òî÷êå x ëèíåéíîãî èíòåðïîëÿöèîííîãî ìíîãî÷ëåíà âåêòîðîâ vx è vy (ðèñ. 4.7).
123.
ln(z) – íàòóðàëüíûé ëîãàðèôì äëÿ
íåíóëåâîãî âåùåñòâåííîãî ÷èñëà z. Äåéñòâèòåëüíàÿ ÷àñòü
(ìíèìàÿ ÷àñòü ìåæäó p è -p) äëÿ êîìïëåêñíîãî z (ðèñ. 4.?).
124.
LoadColormap(file) – ìíîæåñòâî, ñîäåðæàùåå
çíà÷åíèÿ öâåòîâîé äèàãðàììû file.
125.
loess(vx, vy, span) – âåêòîð, èñïîëüçóåìûé
ôóíêöèåé interp äëÿ íàõîæäåíèÿ ñîâîêóïíîñòè
ìíîãî÷ëåíîâ âòîðîé ñòåïåíè, êîòîðûå íàèëó÷øèì îáðàçîì àïïðîêñèìèðóþò ÷àñòü
äàííûõ èç âåêòîðîâ vx è vy. Àðãóìåíò span óêàçûâàåò ðàçìåð ÷àñòè
àïïðîêñèìèðóåìûõ äàííûõ.
126.
loess(Mxy, vz, span) – âåêòîð, èñïîëüçóåìûé
ôóíêöèåé interp äëÿ íàõîæäåíèÿ ñîâîêóïíîñòè
ìíîãî÷ëåíîâ âòîðîé ñòåïåíè, êîòîðûå íàèëó÷øèì îáðàçîì àïïðîêñèìèðóþò
çàâèñèìîñòü Z(x, y) ïî ìàññèâó Mxy (âåùåñòâåííàÿ ìàòðèöà ðàçìåðîì m´2, ñîäåðæàùàÿ êîîðäèíàòû (x,
y) äàííûõ
òî÷åê). Çíà÷åíèå Z â ìàññèâå vz.; span óêàçûâàåò ðàçìåð îáëàñòè,
íà êîòîðîé âûïîëíÿåòñÿ ëîêàëüíàÿ àïïðîêñèìàöèÿ.
127.
log(z, b) – ëîãàðèôì íåíóëåâîãî
âåùåñòâåííîãî ÷èñëà z ïî îñíîâàíèþ b. Äåéñòâèòåëüíàÿ ÷àñòü (ìíèìàÿ ÷àñòü ìåæäó p è -p) äëÿ êîìïëåêñíîãî z. Åñëè b íå çàäàíî – äåñÿòè÷íûé ëîãàðèôì z.
128.
lsolve(M, v) – âåêòîð ðåøåíèÿ ñèñòåìû
ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé âèäà M×x=v (ðèñ 1.7, 1.16, 4.9).
129.
lspline(vx, vy) – âåêòîð êîýôôèöèåíòîâ
ëèíåéíîãî ñïëàéíà, ïîñòðîåííîãî ïî âåêòîðàì vx è vy (ðèñ. 4.8).
130.
lspline(Mxy, Mz) – âåêòîð, èñïîëüçóåìûé
ôóíêöèåé interp äëÿ èíòåðïîëÿöèè äàííûõ èç Mxy è Mz. Èíòåðïîëèðóþùàÿ
ïîâåðõíîñòü èìååò íà ãðàíèöå ñåòêè, îïðåäåëÿåìîé Mxy, ðàâíûå íóëþ ïðîèçâîäíûå
âûøå ïåðâîãî ïîðÿäêà.
131.
lu(M) – òðåóãîëüíîå ðàçëîæåíèå
ìàòðèöû M: P×M=L×U, ãäå L è U – íèæíÿÿ è âåðõíÿÿ òðåóãîëüíûå ìàòðèöû ñîîòâåòñòâåííî. Âñå ÷åòûðå
ìàòðèöû êâàäðàòíûå, îäíîãî ïîðÿäêà.
132.
matrix(m, n, f) – ìàòðèöà, â êîòîðîé (i, j)-é ýëåìåíò ñîäåðæèò f(i, j), ãäå i=0, 1,... m - 1 è j=0, 1,... n - 1.
133.
max(A) – íàèáîëüøèé ýëåìåíò â
ìàòðèöå A. Åñëè A – êîìïëåêñ, âîçâðàùàåò max(Re(A))+i×max(Im(A)) (ðèñ. 2.8, 3.14).
134.
Maximize(f,
var1, var2,...) – çíà÷åíèÿ var1, var2,..., êîòîðûå çàñòàâëÿþò ôóíêöèþ f ïðèíÿòü ñâîå ìàêñèìàëüíîå çíà÷åíèå. Ïîëó÷àåì ñêàëÿð, åñëè òîëüêî îäíà
íåèçâåñòíàÿ, â äðóãîì ñëó÷àå – âåêòîð îòâåòà. Ìàêñèìèçàöèÿ ìîæåò ïðîâîäèòüñÿ ñ
îãðàíè÷åíèÿìè, êîòîðûå çàïèñûâàþòñÿ ìåæäó êëþ÷åâûì ñëîâîì Given è ôóíêöèåé Maximize (ðèñ. 2.2, 2.5, 2.6, 2.8, 2.9 è 6.33).
135.
mean(A) – àðèôìåòè÷åñêîå ñðåäíåå
çíà÷åíèå ýëåìåíòîâ ìàññèâà A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðîì m´n (ðèñ. 3.14).
136.
median(A) – ìåäèàíà ýëåìåíòîâ ìàññèâà
A. Ìåäèàíà – ýòî âåëè÷èíà, áîëüøå è ìåíüøå
êîòîðîé îäèíàêîâîå ÷èñëî ýëåìåíòîâ. A – âåùåñòâåííàÿ èëè
êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðîì m´n.
137.
medsmooth(vy, n) – m-ìåðíûé âåêòîð, ñãëàæèâàþùèé vy ìåòîäîì ñêîëüçÿùåé ìåäèàíû.
vy – m-ìåðíûé âåêòîð âåùåñòâåííûõ
÷èñåë. n – øèðèíà îêíà, ïî êîòîðîìó
ïðîèñõîäèò ñãëàæèâàíèå.
138.
mhyper(a, b, x) – îáúåäèíåííàÿ (ñëèÿíèå)
ãèïåðãåîìåòðè÷åñêàÿ ôóíêöèÿ 1F1(a, b, x) èëè M(a, b, x) ñ ïàðàìåòðàìè a è b â òî÷êå x.
139.
min(A) – íàèìåíüøèé ýëåìåíò â
ìàññèâå A. Åñëè A õðàíèò è êîìïëåêñíûå ÷èñëà, òî
âîçâðàùàåò min(Re(A))+i×min(Im(A)) (ðèñ. 3.14).
140.
MinErr(var1, var2,...) – âåêòîð çíà÷åíèé äëÿ var1, var2,..., êîòîðûå ïðèâîäÿò ê ìèíèìàëüíîé îøèáêå â ñèñòåìå
óðàâíåíèé è íåðàâåíñòâ, íà÷èíàþùèõñÿ îò êëþ÷åâîãî ñëîâà Given. ×èñëî íåèçâåñòíûõ ðàâíî
÷èñëó àðãóìåíòîâ. Âîçâðàùàåò ñêàëÿð, åñëè òîëüêî îäèí àðãóìåíò, â äðóãîì ñëó÷àå
– âåêòîð îòâåòà (ðèñ. 2.4, 2.7, 2.8, 3.3, 3.4, 3.5).
141.
Minimize(f, var1,
var2,...) –
çíà÷åíèÿ var1, var2,..., êîòîðûå çàñòàâëÿþò ôóíêöèþ
f ïðèíÿòü ñâîå ìèíèìàëüíîå çíà÷åíèå. Ïîëó÷àåì
ñêàëÿð, åñëè òîëüêî îäíà íåèçâåñòíàÿ, â äðóãîì ñëó÷àå – âåêòîð îòâåòà..
Ìèíèìèçàöèÿ ìîæåò ïðîâîäèòñÿ ñ îãðàíè÷åíèÿìè, êîòîðûå çàïèñûâàþòñÿ ìåæäó
êëþ÷åâûì ñëîâîì Given è
ôóíêöèåé Minimize
(ðèñ. 2.7, 2.8, 2.10, 3.3 è 3.4).
142.
mod(n, k) – îñòàòîê îò äåëåíèÿ n íà k (n, k – öåëûå ÷èñëà). Àðãóìåíòû äîëæíû áûòü äåéñòâèòåëüíûìè. Ðåçóëüòàò èìååò
òàêîé æå çíàê, êàê è n.
143.
mode(A) – çíà÷åíèå â ìàññèâå A, êîòîðîå âñòðå÷àåòñÿ íàèáîëåå ÷àñòî.
144.
multigrid(M, ncycle) – ìàòðèöà ðåøåíèÿ óðàâíåíèÿ
Ïóàññîíà, ãäå ðåøåíèå ðàâíî íóëþ íà ãðàíèöàõ.
145.
norm1(M) – L1 íîðìà ìàòðèöû M.
146.
norm2(M) – L2 íîðìà ìàòðèöû M.
147.
norme(M) – åâêëèäîâà íîðìà ìàòðèöû M.
148.
normi(M) – íåîïðåäåëåííàÿ íîðìà ìàòðèöû
M.
149.
num2str(z) – òåêñòîâûå ïåðåìåííûå, ÷üÿ õàðàêòåðèñòèêà ñîîòâåòñòâóåò äåñÿòè÷íîìó çíà÷åíèþ z (ðèñ. 1.30).
150.
pbeta(x,s1,s2) – çíà÷åíèå â òî÷êå x ôóíêöèè ñòàíäàðòíîãî b-ðàñïðåäåëåíèÿ ñ ïàðàìåòðàìè ôîðìû s1,
s2 (0<x<1, s1>0, s2>0).
151.
pbinom(k, n, p) – ôóíêöèÿ áèíîìèàëüíîãî
ðàñïðåäåëåíèÿ äëÿ k óñïåõîâ â ñåðèè n èñïûòàíèé (k – öåëîå, 0£k£n, p – âåùåñòâåííîå ÷èñëî, 0£p£1).
152.
pcauchy(x, l, s) – çíà÷åíèå â òî÷êå x ðàñïðåäåëåíèÿ Êîøè ñî øêàëîé ïàðàìåòðîâ l è s (s>0).
153.
pchisq(x, d) – çíà÷åíèå â òî÷êå x c-êâàäðàò-ðàñïðåäåëåíèÿ, â êîòîðîì d – ñòåïåíü ñâîáîäû (x – âåùåñòâåííîå, x³0).
154.
permut(n, k) – ÷èñëî ïóòåé ïîðÿäêà n òî÷íûõ îáúåêòîâ k, âçÿòûõ îäíîâðåìåííî. N è k – öåëûå ÷èñëà (0£k£n).
155.
pexp(x, r) – çíà÷åíèå â òî÷êå x ýêñïîíåíöèàëüíîãî ðàñïðåäåëåíèÿ.
156.
pF(x, d1, d2) – çíà÷åíèå â òî÷êå x ðàñïðåäåëåíèÿ Ôèøåða.
157.
pgamma(x, s) – çíà÷åíèå â òî÷êå x g-ðàñïðåäåëåíèÿ.
158.
pgeom(k, p) – Pr(X£k), ãäå X – ñëó÷àéíàÿ âåëè÷èíà,
èìåþùàÿ ãåîìåòðè÷åñêîå ðàñïðåäåëåíèå ñ ïàðàìåòðîì p.
159.
phypergeom(m, a, b,
n) – Pr(X£m), ãäå X – ñëó÷àéíàÿ âåëè÷èíà, èìåþùàÿ ãåîìåòðè÷åñêîå ðàñïðåäåëåíèå ñ
ïàðàìåòðàìè a, b è n.
160.
plnorm(x, m, s) – çíà÷åíèå â òî÷êå x ëîãíîðìàëüíîãî
ðàñïðåäåëåíèÿ, â êîòîðîì m – ëîãàðèôì ñðåäíåãî
çíà÷åíèÿ, s>0 – ëîãàðèôì ñòàíäàðòíîãî îòêëîíåíèÿ.
161.
plogis(x, l, s) – çíà÷åíèå â òî÷êå x ïîñëåäîâàòåëüíîãî ðàñïðåäåëåíèÿ, ãäå l – ïàðàìåòð ïîëîæåíèÿ; s>0 – ïàðàìåòð øêàëû.
162.
pnbinom(k, n, p) – çíà÷åíèå â òî÷êå x îòðèöàòåëüíîãî áèíîìèàëüíîãî ðàñïðåäåëåíèÿ, â êîòîðîì n>0 è 0<p£1.
163.
pnorm(x, m, s) – çíà÷åíèå â òî÷êå x íîðìàëüíîãî ðàñïðåäåëåíèÿ
ñî ñðåäíèì çíà÷åíèåì m è ñòàíäàðòíûì îòêëîíåíèåì s>0.
164.
polyroots(v) – (n+1) ýëåìåíòíûé âåêòîð êîðíåé ìíîãî÷ëåíà ñòåïåíè n, êîýôôèöèåíòû ìíîãî÷ëåíà íàõîäÿòñÿ â âåêòîðå v (ðèñ 1.11, 3.1, 3.3).
165.
ppois(k, l) – çíà÷åíèå â òî÷êå x ðàñïðåäåëåíèÿ Ïóàññîíà (k³0, l>0).
166.
predict(v, m, n) – ïðîãíîç. Âåêòîð,
ñîäåðæàùèé ðàâíîîòñòîÿùèå ïðåäñêàçàííûå çíà÷åíèÿ n ïåðåìåííûõ, âû÷èñëåííûõ ïî m çàäàííûì â âåêòîðå v äàííûì (ðèñ. 4.14).
167.
pspline(vx, vy) – âåêòîð êîýôôèöèåíòîâ
ïàðàáîëè÷åñêîãî ñïëàéíà, ïîñòðîåííîãî ïî âåêòîðàì vx è vy. Ïîëó÷åííûé âåêòîð ñòàíîâèòñÿ
ïåðâûì àðãóìåíòîì äëÿ ôóíêöèè interp (ðèñ. 4.8).
168.
pspline(Mxy, Mz) – âåêòîð âòîðûõ ïðîèçâîäíûõ
äëÿ äàííûõ Mxy è Mz. Ýòîò âåêòîð ñòàíîâèòñÿ ïåðâûì àðãóìåíòîì â ôóíêöèè interp. Ðåçóëüòèðóþùàÿ ïîâåðõíîñòü ÿâëÿåòñÿ ïàðàáîëè÷åñêîé â ãðàíèöàõ
îáëàñòè, îãðàíè÷åííîé õîðäîé Mxy.
169.
pt(x, d) – çíà÷åíèå â òî÷êå x ðàñïðåäåëåíèÿ Ñòüþäåíòà, ãäå d – ñòåïåíü ñâîáîäû; x>0 è d>0.
170.
punif(x, a, b) – çíà÷åíèå â òî÷êå x ðàâíîìåðíîãî ðàñïðåäåëåíèÿ, ãäå b è a – ãðàíèöû èíòåðâàëà (a<b).
171.
pweibull(x, s) – çíà÷åíèå â òî÷êå x ðàñïðåäåëåíèÿ Âåéáóëëà (s>0).
172.
qbeta(p, s1, s2) – êâàíòèëè îáðàòíîãî
áåòà-ðàñïðåäåëåíèÿ ñ ïàðàìåòðàìè ôîðìû s1 è s2 (0£p£1 è s1, s2>0).
173.
qbinom(p,n,q) – ôóíêöèÿ îáðàòíîãî
áèíîìèíàëüíîãî ðàñïðåäåëåíèÿ, òî åñòü íàèìåíüøåå öåëîå k, òàêîå ÷òî pnbinom(k, n,
p)³p (0£q£1 è 0£p£1).
174.
qcauchy(p, l, s) – ôóíêöèÿ îáðàòíîãî
ðàñïðåäåëåíèÿ Êîøè ñî øêàëîé ïàðàìåòðîâ l è s (s>0 è 0<p<1).
175.
qchisq(p, d) – ôóíêöèÿ îáðàòíîãî c-êâàäðàò-ðàñïðåäåëåíèÿ, ïðè êîòîðîì d>0; õàêòåðèñòèêà ñòåïåíåé ñâîáîäû
(0£p<1).
176.
qexp(p, r) – ôóíêöèÿ îáðàòíîãî
ýêñïîíåíöèàëüíîãî ðàñïðåäåëåíèÿ, ïðè êîòîðîì r>0 îïðåäåëÿåò ÷àñòîòó (0£p<1).
177.
qF(p, d1, d2) – ôóíêöèÿ îáðàòíîãî
ðàñïðåäåëåíèÿ Ôèøåðà, â êîòîðîì d1 è d2 – ñòåïåíè ñâîáîäû (d1, d2>0, 0£p<1).
178.
qgamma(p, s) – ôóíêöèÿ îáðàòíîãî g-ðàñïðåäåëåíèÿ, ïðè êîòîðîì s>0 – ïàðàìåòðû ôîðìû (0£p<1).
179.
qgeom(p, q) – ôóíêöèÿ îáðàòíîãî
ãåîìåòðè÷åñêîãî ðàñïðåäåëåíèÿ, òî åñòü íàèìåíüøåå öåëîå k, òàêîå ÷òî qgeom(p, q)³p (0£q£1 è 0£p£1).
180.
qhypergeom(p, a, b,
n) – ôóíêöèÿ
îáðàòíîãî ãèïåðãåîìåòðè÷åñêîãî ðàñïðåäåëåíèÿ, òî åñòü íàèìåíüøåå öåëîå k, òàêîå ÷òî qhypergeom(p,
a, b, n)³p (0£a, 0£b, 0£n£(a+b) è 0£p£1).
181.
qlnorm(p, m, s) – ôóíêöèÿ îáðàòíîãî ëîãíîðìàëüíîãî ðàñïðåäåëåíèÿ, ïðè êîòîðîì m – ëîãàðèôì ñðåäíåãî ÷èñëà. s>0 – ëîãàðèôì ñòàíäàðòíîãî îòêëîíåíèÿ (0£p<1).
182.
qlogis(p, l, s) – ôóíêöèÿ îáðàòíîãî ïîñëåäîâàòåëüíîãî
ðàñïðåäåëåíèÿ, ãäå l – ïàðàìåòð ïîëîæåíèÿ; s>0 – ïàðàìåòð øêàëû (0<p<1).
183.
qnbinom(p, n, q) – ôóíêöèÿ îáðàòíîãî
îòðèöàòåëüíîãî áèíîìèàëüíîãî ðàñïðåäåëåíèÿ, òî åñòü íàèìåíüøåå öåëîå k, òàêîå ÷òî qnbinom(p, n,
q)³p (0<n, 0<p<1, 0<q<1).
184.
qnorm(p, m, s) – ôóíêöèÿ îáðàòíîãî íîðìàëüíîãî ðàñïðåäåëåíèÿ ñî ñðåäíèì çíà÷åíèåì m è ñòàíäàðòíûì îòêëîíåíèåì s (0<p<1 è s>0).
185.
qpois(p, l) – ôóíêöèÿ îáðàòíîãî ðàñïðåäåëåíèÿ Ïóàññîíà, òî åñòü íàèìåíüøåå öåëîå k, òàêîå ÷òî qpois(p, l)³p (l>0 è 0£p£1).
186.
qr(A) – ðàçëîæåíèå ìàòðèöû A, A=Q×R, ãäå Q – îðòîãîíàëüíàÿ ìàòðèöà è R – âåðõíÿÿ òðåóãîëüíàÿ ìàòðèöà.
187.
qt(p, d) – ôóíêöèÿ îáðàòíîãî
ðàñïðåäåëåíèÿ Ñòüþäåíòà, ãäå d îïðåäåëÿåò ñòåïåíè ñâîáîäû
(d>0 è 0<p<1).
188.
qunif(p, a, b) – ôóíêöèÿ îáðàòíîãî
ðàâíîìåðíîãî ðàñïðåäåëåíèÿ. b è a – êîíå÷íûå çíà÷åíèÿ èíòåðâàëà
(a<b è 0£p£1).
189.
qweibull(p, s) – ôóíêöèÿ îáðàòíîãî
ðàñïðåäåëåíèÿ Âåéáóëëà (s>0 è 0<p<1).
190.
rank(A) – ðàíã ìàòðèöû A. Ìàêñèìàëüíîå ÷èñëî ëèíåéíî íåçàâèñèìûõ ñòîëáöîâ â A.
191.
rbeta(m, s1, s2) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ b-ðàñïðåäåëåíèå. s1, s2 (áîëüøå íóëÿ) – ïàðàìåòðû
ôîðìû.
192.
rbinom(m, n, p) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ áèíîìèàëüíîå ðàñïðåäåëåíèå (0£p£1, n – öåëîå ÷èñëî,
óäîâëåòâîðÿþùåå óñëîâèþ n>0).
193.
rcauchy(m, l, s) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ðàñïðåäåëåíèå Êîøè. l è s>0 – ïàðàìåòðû øêàëû.
194.
rchisq(m, d) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ c-êâàäðàò-ðàñïðåäåëåíèå. d>0 îïðåäåëÿåò ñòåïåíè ñâîáîäû.
195.
Re(z) – äåéñòâèòåëüíàÿ ÷àñòü
êîìïëåêñíîãî ÷èñëà z (2).
196.
READ_BLUE(file) – ìàòðèöà, ñîäåðæàùàÿ òîëüêî
ñèíèé öâåò êîìïîíåíòà, ñîäåðæàùåãîñÿ â ôàéëå file,
èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ
ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ ìàòðèöû, âîçâðàùåííîé ôóíêöèåé READRGB(file).
197.
READBMP(file) – ìàòðèöà, ñîäåðæàùàÿ
÷åðíî-áåëîå ïðåäñòàâëåíèå èçîáðàæåíèÿ, ðàñïîëîæåííîãî â ôàéëå file. Êàæäûé ýëåìåíò ìàòðèöû ñîîòâåòñòâóåò îäíîìó ïèêñåëó. Êàæäûé ýëåìåíò –
öåëîå ÷èñëî îò 0 (÷åðíûé) äî 255 (áåëûé).
198.
READ_GREEN(file) – ìàòðèöà, ñîäåðæàùàÿ òîëüêî
çåëåíûé öâåò êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ
ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ ìàòðèöû, âîçâðàùåííîé ôóíêöèåé READRGB(file).
199.
READ_HLS(file) – ìàòðèöà, â êîòîðîé öâåòîâàÿ
èíôîðìàöèÿ, ñîäåðæàùàÿñÿ â ôàéëå file, ïðåäñòàâëåíà ñîîòâåòñòâóþùèìè çíà÷åíèÿìè
îòòåíêà öâåòà, ÿðêîñòüþ, è íàñûùåííîñòüþ. File èìååò BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò.
200.
READ_HLS_HUE(file) – ìàòðèöà, ñîäåðæàùàÿ òîëüêî
öâåòîâûå îòòåíêè êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HLS(file).
201.
READ_HLS_LIGHT(file)
– ìàòðèöà,
ñîäåðæàùàÿ òîëüêî ÿðêîñòü öâåòîâîãî êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HLS(file).
202.
READ_HLS_SAT(file) – ìàòðèöà, ñîäåðæàùàÿ òîëüêî
íàñûùåííîñòü öâåòîâîãî êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HLS(file).
203.
READ_HSV(file) – ìàòðèöà, â êîòîðîé öâåòîâàÿ
èíôîðìàöèÿ, ñîäåðæàùàÿñÿ â ôàéëå file, ïðåäñòàâëåíà
ñîîòâåòñòâóþùèìè çíà÷åíèÿìè îòòåíêà öâåòà, íàñûùåííîñòè è âåëè÷èíû. File èìååò BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò.
204.
READ_HSV_HUE(file) – ìàòðèöà, ñîîòâåòñòâóþùàÿ
îòòåíêó öâåòîâîãî êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HSV(file).
205.
READ_HSV_SAT(file) – ìàòðèöà, ñîîòâåòñòâóþùàÿ
òîëüêî íàñûùåííîñòè öâåòîâîãî êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HSV(file).
206.
READ_HSV_VALUE(file)
– ìàòðèöà,
ñîîòâåòñòâóþùàÿ âåëè÷èíå öâåòîâîãî êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé
READ_HSV(file).
207.
READ_IMAGE(file) – ìàòðèöà, ñîäåðæàùàÿ
÷åðíî-áåëîå ïðåäñòàâëåíèå, ñîäåðæàùååñÿ â ôàéëå file.
Êàæäûé ýëåìåíò
ìàòðèöû ñîîòâåòñòâóåò îäíîìó ïèêñåëó. Êàæäûé ýëåìåíò – öåëîå ÷èñëî îò 0 (÷åðíûé) äî 255 (áåëûé). file ìîæåò áûòü â BMP-, GIF-, JPG- èëè TGA-öâåòîâîì ôîðìàòå.
208.
READPRN(file) – ïðèñâàèâàíèå ìàòðèöå
çíà÷åíèé èç ôàéëà ñ èìåíåì READPRN(file).
209.
READ_RED(file) – ìàòðèöà, ñîîòâåòñòâóþùàÿ
òîëüêî êðàñíîìó öâåòó êîìïîíåíòà, ðàñïîëîæåííîãî â ôàéëå file, èìåþùåì BMP-, GIF-, JPG- èëè TGA-öâåòîâîé ôîðìàò. Ïîëó÷åííàÿ ìàòðèöà ñîäåðæèò òðåòü îò ÷èñëà ñòîëáöîâ
ìàòðèöû, âîçâðàùåííîé ôóíêöèåé READRGB(file).
210.
READRGB(file) – ìàòðèöà, ñîñòîÿùàÿ èç
òðåõ ïîäìàòðèö, êîòîðûå ïðåäñòàâëÿþò êðàñíûé, çåëåíûé è ñèíèé êîìïîíåíòû
öâåòíîãî èçîáðàæåíèÿ, íàõîäÿùåãîñÿ â ôàéëå file.
211.
regress(vx, vy, n) – âîçâðàùàåò âåêòîð, òðåáóþùèé
interp, ÷òîáû íàéòè ïîëèíîì
ïîðÿäêà n, êîòîðûé íàèëó÷øèì îáðàçîì
ïðèáëèæàåò äàííûå èç vx è vy. vx –m-ýëåìåíòíûé âåêòîð,
ñîäåðæàùèé êîîðäèíàòû x. vy – m-ýëåìåíòíûé âåêòîð, ñîäåðæàùèé
êîîîðäèíàòû y, ñîîòâåòñòâóþùèå m òî÷êàì, îïðåäåëåííûì â vx.
212.
regress(Mxy,vz,n) – âåêòîð, çàïðàøèâàåìûé
ôóíêöèåé interp äëÿ âû÷èñëåíèÿ ìíîãî÷ëåíà n-é ñòåïåíè, êîòîðûé íàèëó÷øèì îáðàçîì ïðèáëèæàåò ìíîæåñòâà Mxy è vz. Mxy – ìàòðèöà m´2, ñîäåðæàùàÿ êîîðäèíàòû (x;y) m äàííûõ òî÷åê. vz – m-ýëåìåíòíûé âåêòîð, ñîäåðæàùèé
z êîîðäèíàò,
ñîîòâåòñòâóþùèõ m òî÷êàì, óêàçàííûì â Mxy.
213.
relax(A,B, C, D, E,
F, U, rjac) – êâàäðàòíàÿ
ìàòðèöà ðåøåíèÿ óðàâíåíèÿ Ïóàññîíà.
214.
reverse(v) – ïîðÿäîê ýëåìåíòîâ âåêòîðà
v.
215.
reverse(A) – ïîðÿäîê ñòðîê ìàòðèöû A.
216.
rexp(m, r) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ýêñïîíåíöèàëüíîå ðàñïðåäåëåíèå. r>0 ÿâëÿåòñÿ ÷àñòîòîé.
217.
rF(m, d1, d2) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ðàñïðåäåëåíèå Ôèøåðà. d1>0, d2>0 – ñòåïåíè ñâîáîäû.
218.
rgamma(m, s) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ g-ðàñïðåäåëåíèå. s>0 – ïàðàìåòð ôîðìû.
219.
rgeom(m, p) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ãåîìåòðè÷åñêîå ðàñïðåäåëåíèå (0<p£1).
220.
rhypergeom(m, a, b,
n) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ãèïåðãåîìåòðè÷åñêîå ðàñïðåäåëåíèå (a³0, b³0, 0£n£(a+b)).
221.
rkadapt(y, x1, x2,
acc, D, kmax, save) – ìàòðèöà, ñîäåðæàùàÿ çíà÷åíèÿ ðåøåíèÿ çàäà÷è Êîøè íà èíòåðâàëå îò x1 äî x2 äëÿ ñèñòåìû îáûêíîâåííûõ
äèôôåðåíöèàëüíûõ óðàâíåíèé, âû÷èñëåííûõ ìåòîäîì Ðóíãå ¾ Êóòòû ñ ïåðåìåííûì øàãîì; y – âåùåñòâåííûé âåêòîð íà÷àëüíûõ çíà÷åíèé. Kmax
– ìàêñèìàëüíîå
÷èñëî ïðîìåæóòî÷íûõ òî÷åê, ïî êîòîðûì äîëæíî áûòü àïïðîêñèìèðîâàíî ðåøåíèå. Save – íàèìåíüøåå äîïóñòèìîå ïðîñòðàíñòâî ìåæäó âåëè÷èíàìè, ïî êîòîðûì
àïïðîêñèìèðîâàíî ðåøåíèå. D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé.
222.
Rkadapt(v, x1, x2,
npts, D) –
ìàòðèöà ðåøåíèé ìåòîäîì Ðóíãå ¾ Êóòòû (ñ ïåðåìåííûì øàãîì)
ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé (ïî èíòåðâàëó îò x1 äî x2); y – âåùåñòâåííûé âåêòîð íà÷àëüíûõ çíà÷åíèé. Npts
– êîëè÷åñòâî
òî÷åê, íå ÿâëÿþùèõñÿ èçíà÷àëüíûìè ïî êîòîðûì àïïðîêñèìèðîâàíî ðåøåíèå. D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé.
223.
rkfixed(y, x1, x2,
npts, D) –
ìàòðèöà ðåøåíèé ìåòîäîì Ðóíãå ¾ Êóòòû ñèñòåìû îáûêíîâåííûõ
äèôôåðåíöèàëüíûõ óðàâíåíèé (ïî èíòåðâàëó îò x1
äî x2); y – âåùåñòâåííûé âåêòîð íà÷àëüíûõ çíà÷åíèé. npts
– êîëè÷åñòâî òî÷åê,
íå ÿâëÿþùèõñÿ èçíà÷àëüíûìè, ïî êîòîðûì àïïðîêñèìèðîâàíî ðåøåíèå. D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé (ðèñ. 5.2).
224.
rlnorm(m, m, s)– âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ
ëîãíîðìàëüíîå ðàñïðåäåëåíèå, â êîòîðîì m – ëîãàðèôì ñðåäíåãî
çíà÷åíèÿ. s>0 – ëîãàðèôì ñòàíäàðòíîãî îòêëîíåíèÿ.
225.
rlogis(m, l, s) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ïîñëåäîâàòåëüíîå ðàñïðåäåëåíèå, â êîòîðîì l – ëîêàëèçàöèîííûé ïàðàìåòð è
s>0 – ïàðàìåòð øêàëû.
226.
rnbinom(m, n, p) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ îòðèöàòåëüíîå áèíîìèàëüíîå ðàñïðåäåëåíèå (0<p£1). n – öåëîå ÷èñëî, êîòîðîå
óäîâëåòâîðÿåò óñëîâèþ n>0.
227.
rnd(x) – ïñåâäîñëó÷àéíîå ÷èñëî â
äèàïàçîíå îò íóëÿ äî x (ðèñ. 3.6, 3.7, 6.9).
228.
rnorm(m, m, s) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ
íîðìàëüíîå ðàñïðåäåëåíèå (s>0).
229.
root(f(var), var) – çíà÷åíèå ïåðåìåííîé var, ïðè êîòîðîì âûðàæåíèå f(var) ðàâíî íóëþ (ðèñ 1.10, 2.3, 2.7, 3.1).
230.
round(x) – îêðóãëåíèå âåùåñòâåííîãî x äî öåëîãî ÷èñëà.
231.
rows(A) – ÷èñëî ñòðîê â ìàññèâå A, ãäå A – ìàòðèöà èëè âåêòîð (ðèñ.
4.10).
232.
rpois(m, l) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ
ðàñïðåäåëåíèå Ïóàññîíà (l>0).
233.
rref(A) – ñòóïåí÷àòûé âèä
âåùåñòâåííîé ìàòðèöû A.
234.
rsort(A, i) – ñîðòèðîâêà ñòîëáöîâ
ìàòðèöû A ïî ðàñïîëîæåíèþ ýëåìåíòîâ â
ñòðîêå i (ïåðåñòàíîâêà ñòîëáöîâ ïî
âîçðàñòàíèþ çíà÷åíèé ýëåìåíòîâ â ñòðîêå i). Ïîëó÷èì ìàòðèöó òàêîãî æå
ðàçìåðà, êàê A (0£i£(m-1)).
235.
rt(m, d) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ðàñïðåäåëåíèå Ñòüþäåíòà. d>0 – ñòåïåíü ñâîáîäû.
236.
runif(m, a, b) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ðàâíîìåðíîå ðàñïðåäåëåíèå, â êîòîðîì b è a – ãðàíèöû èíòåðâàëà è a<b.
237.
rweibull(m,S) – âåêòîð m ñëó÷àéíûõ ÷èñåë, èìåþùèõ ðàñïðåäåëåíèå Âåéáóëëà, â êîòîðîì S>0 – ïàðàìåòð ôîðìû.
238.
SaveColormap(file, M) – öâåòîâàÿ äèàãðàììà file, ñîäåðæàùàÿ çíà÷åíèÿ ìàòðèöû M, âîçâðàùàåò êîëè÷åñòâî ñòðîê, íàïèñàííûõ â file.
239.
sbval(v, x1, x2, D,
load, score) –
óñòàíîâêà íà÷àëüíûõ óñëîâèé äëÿ êðàåâîé çàäà÷è, D – âåùåñòâåííûé n-ýëåìåíòíûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. Âåêòîð
v – íà÷àëüíûå
óñëîâèÿ ïî èíòåðâàëó (x1, x2). Load(x1, v) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, n ýëåìåíòîâ êîòîðîãî
àíàëîãè÷íû çíà÷åíèÿì n íåèçâåñòíûõ ôóíêöèé â òî÷êå
x1. Score(x2, y) – âåùåñòâåííûé n-ýëåìåíòíûé âåêòîð çíà÷åíèé ôóíêöèè c èçìåðåííûì ðàçëè÷èåì, âû÷èñëåííûì
â òî÷êå x2 (ðèñ. 5.5).
240.
search(S, SubS, m) – íà÷àëüíîå ïîëîæåíèå
òåêñòîâîé ïîäïåðåìåííîé SubS â S íà÷èíàÿ ñ ïîçèöèè m. Âîçâðàùàåò ìèíóñ åäèíèöó,
åñëè òåêñòîâàÿ ïîäïåðåìåííàÿ íå íàéäåíà (ðèñ. 1.30).
241.
sec(z) – ñåêàíñ z (â ðàäèàíàõ).
242.
sech(z) – ãèïåðáîëè÷åñêèé ñåêàíñ z.
243.
sign(x) – âîçðàùàåò 0, åñëè x=0, 1 – åñëè x>0, ìèíóñ åäèíèöó – â äðóãèõ ñëó÷àÿõ (x – âåùåñòâåííîå ÷èñëî).
244.
signum(z) – âîçâðàùàåò 1 – åñëè z=0, è z/|z| – â äðóãèõ ñëó÷àÿõ (z – êîìïëåêñíîå ÷èñëî) (ðèñ.
7.13).
245.
sin(z) – ñèíóñ z (ðèñ 1.15, 1,16, 1,29).
246.
sinh(z) – ãèïåðáîëè÷åñêèé ñèíóñ z.
247.
skew(A) – êîýôôèöèåíò àñèììåòðèè
ìàññèâà A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðà m´n, m×n³3.
248.
slope(vx, vy) – êîýôôèöèåíò a ëèíåéíîé ðåãðåññèè y = a×x + b âåêòîðîâ vx è vy (vx è vy èìåþò îäèíàêîâûé ðàçìåð –
ðèñ. 4.2).
249.
sort(v) – ñîðòèðîâêà ýëåìåíòîâ
âåêòîðà v ïî âîçðàñòàíèþ èõ çíà÷åíèÿ.
250.
stack(A, B) – ìàòðèöà, ñôîðìèðîâàííàÿ
ïóòåì ðàñïîëîæåíèÿ A íàä B. Ìàòðèöû (èëè âåêòîðû) A, B äîëæíû èìåòü îäèíàêîâîå ÷èñëî ñòîëáöîâ (ðèñ. 6.9).
251.
stderr(vx, vy) – ñòàíäàðòíàÿ îøèáêà,
ñâÿçàííàÿ ñ ëèíåéíîé ðåãðåññèåé, ïîêàçûâàþùåé, íàñêîëüêî äàííûå òî÷êè
ðàçáðîñàíû îòíîñèòåëüíî ëèíèè ðåãðåññèè.
252.
stdev(A) – ñòàíäàðòíîå îòêëîíåíèå
ýëåìåíòîâ ìàññèâà A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðà m´n. m´n-1 èñïîëüçóåòñÿ â çíàìåíàòåëå (äåëèòåëå): Stdev(A)
= Ö var(A).
253.
Stdev(A) – ñòàíäàðòíîå îòêëîíåíèå
ýëåìåíòîâ A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà (âåêòîð) ðàçìåðà m´n. m´n-1 èñïîëüçóåòñÿ â çíàìåíàòåëå (äåëèòåëå): Stdev(A)
= Ö Var(A).
254.
stiffb(y, x1, x2,
acc, D, J, kmax, save) – ìàòðèöà ðåøåíèé æåñòêîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ñ
èñïîëüçîâàíèåì ìåòîäà Bulirsch-Stoer;
y – âåêòîð
íà÷àëüíûõ çíà÷åíèé íà èíòåðâàëå (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. J(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, âîçâðàùàþùèé ìàòðèöó ðàçìåðîì n´(n+1), â ïåðâîì ñòîëáöå êîòîðîé ñîäåðæàòñÿ ÷àñòíûå ïðîèçâîäíûå D ïî x, à â îñòàëüíûõ ñòîëáöàõ –
÷àñòíûå ïðîèçâîäíûå D ïî y. Kmax – ìàêñèìàëüíîå ÷èñëî
ïðîìåæóòî÷íûõ òî÷åê, ïî êîòîðûì äîëæíî áûòü àïïðîêñèìèðîâàíî ðåøåíèå. Save – íàèìåíüøåå äîïóñòèìîå ïðîñòðàíñòâî ìåæäó âåëè÷èíàìè, ïî êîòîðûì
àïïðîêñèìèðîâàíî ðåøåíèå. D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé.
255.
Stiffb(y, x1, x2,
npts, D, J) –
ìàòðèöà ðåøåíèé æåñòêîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ñ èñïîëüçîâàíèåì ìåòîäà
Bulirsch-Stoer; y – âåêòîð íà÷àëüíûõ çíà÷åíèé
ïî èíòåðâàëó (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. J(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, âîçâðàùàþùèé ìàòðèöó ðàçìåðîì n´(n+1), â ïåðâîì ñòîëáöå êîòîðîé ñîäåðæàòñÿ ÷àñòíûå ïðîèçâîäíûå D ïî x, à â îñòàëüíûõ ñòîëáöàõ –
÷àñòíûå ïðîèçâîäíûå D ïî y. Npts – êîëè÷åñòâî òî÷åê, íå
ÿâëÿþùèõñÿ èçíà÷àëüíûìè, ïî êîòîðûì àïïðîêñèìèðîâàíî ðåøåíèå.
256.
stiffr(ó, x1, x2,
acc, D, J, kmax, save) – ìàòðèöà ðåøåíèé æåñòêîãî
äèôôåðåíöèàëüíîãî
óðàâíåíèÿ ñ èñïîëüçîâàíèåì ìåòîäà Ðîçåíáðîêà;
y – âåêòîð
íà÷àëüíûõ çíà÷åíèé ïî èíòåðâàëó (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. J(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, âîçâðàùàþùèé ìàòðèöó ðàçìåðîì n´(n+1), â ïåðâîì ñòîëáöå êîòîðîé ñîäåðæàòñÿ ÷àñòíûå ïðîèçâîäíûå D ïî x, à â îñòàëüíûõ ñòîëáöàõ –
÷àñòíûå ïðîèçâîäíûå D ïî y. Kmax – ìàêñèìàëüíîå ÷èñëî ïðîìåæóòî÷íûõ òî÷åê, ïî êîòîðûì äîëæíî áûòü
àïïðîêñèìèðîâàíî ðåøåíèå. Save – íàèìåíüøåå äîïóñòèìîå
ïðîñòðàíñòâî ìåæäó âåëè÷èíàìè, ïî êîòîðûì àïïðîêñèìèðîâàíî ðåøåíèå. D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé.
257.
Stiffr(y, x1, x2,
npts, D, J) –
ìàòðèöà ðåøåíèé æåñòêîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ñ èñïîëüçîâàíèåì ìåòîäà Ðîçåíáðîêà; y – âåêòîð íà÷àëüíûõ çíà÷åíèé ïî èíòåðâàëó (x1, x2); D(x, y) – âåùåñòâåííûé âåêòîð çíà÷åíèé ôóíêöèè, ñîäåðæàùèé ïðîèçâîäíûå (ïî y) íåèçâåñòíûõ ôóíêöèé. J(x, y) – âåùåñòâåííûé âåêòîð
çíà÷åíèé ôóíêöèè, âîçâðàùàþùèé ìàòðèöó ðàçìåðîì n´(n+1), â ïåðâîì ñòîëáöå êîòîðîé ñîäåðæàòñÿ ÷àñòíûå ïðîèçâîäíûå D ïî x, à â îñòàëüíûõ ñòîëáöàõ –
÷àñòíûå ïðîèçâîäíûå D ïî y. Npts – êîëè÷åñòâî òî÷åê, íå ÿâëÿþùèõñÿ èçíà÷àëüíûìè, ïî êîòîðûì
àïïðîêñèìèðîâàíî ðåøåíèå.
258.
str2num(S) – ïîñòîÿííàÿ, îáðàçîâàííàÿ
ïóòåì îáðàùåíèÿ çíàêîâ èç S â ÷èñëî. Çíàêè â S äîëæíû ñîñòàâëÿòü ðåàëüíîå (èëè êîìïëåêñíîå) ÷èñëî ñ ïëàâàþùåé çàïÿòîé
èëè ÷èñëî ñ e-ôîðìàòîì. Ïðîáåëû â
òåêñòîâîé ïåðåìåííîé èãíîðèðóþòñÿ (ðèñ. 1.30).
259.
str2vec(S) – âåêòîð ñ ASCll-êîäàìè, ñîîòâåòñòâóþùèìè çíà÷åíèÿì òåêñòîâîé ïåðåìåííîé S (ðèñ. 1.30).
260.
strlen(S) – ÷èñëî çíàêîâ â òåêñòîâîé
ïåðåìåííîé S (ðèñ. 1.30).
261.
submatrix(A, ir, jr,
ic, jc) –
ïîäìàòðèöà A, ñîñòîÿùàÿ èç ýëåìåíòîâ,
îáùèõ äëÿ ñòðîê îò ir äî jr è ñòîëáöîâ îò ic äî jc. Äëÿ òîãî ÷òîáû ñîõðàíèòü ïîðÿäîê ñòðîê è (èëè) ñòîëáöîâ, íóæíî áûòü
óâåðåííûì, ÷òî ir£jr è ic£jc, â ïðîòèâíîì ñëó÷àå ïîðÿäîê ñòðîê è (èëè) ñòîëáöîâ áóäåò èçìåíåí (ðèñ.
4.10).
262.
substr(S, m, n) – òåêñòîâàÿ ïîäïåðåìåííóþ S, íà÷èíàþùàÿñÿ ñî çíàêîâ â ïîçèöèè m
è èìåþùàÿ ñàìîå
áîëüøåå n çíàêîâ (m, n³0) (ðèñ. 1.30).
263.
supsmoot(vx, vy) – n-ìåðíûé âåêòîð, ñãëàæèâàþùèé çàâèñèìîñòü y îò x. Çíà÷åíèÿ y è x â âåêòîðàõ vy è vx (vx, vy – n-ýëåìåíòíûå âåêòîðû) (ðèñ.
4.15).
264.
svd(A) – ñèíãóëÿðíîå ðàçëîæåíèå
ìàòðèöû A ðàçìåðîì n´m: A=U×S×VT, ãäå U è V – îðòîãîíàëüíûå ìàòðèöû ðàçìåðîì m´m è n´×n ñîîòâåòñòâåííî. S – äèàãîíàëüíàÿ ìàòðèöà, íà
äèàãîíàëè – ñèíãóëÿðíûå ÷èñëà ìàòðèöû A.
265.
svds(A) – âåêòîð, ñîäåðæàùèé
ñèíãóëÿðíûå ÷èñëà ìàòðèöû A, èìåþùåé ðàçìåð m´n, ãäå m³n.
266.
tan(z) – òàíãåíñ z (â ðàäèàíàõ).
267.
tanh(z) – ãèïåðáîëè÷åñêèé òàíãåíñ z.
268.
Tcheb(n, x) – ïîëèíîì ×åáûøåâà ïåðâîãî
ðîäà ñòåïåíè n â òî÷êå x.
269.
tr(M) – ñóììà äèàãîíàëüíûõ
ýëåìåíòîâ êâàäðàòíîé ìàòðèöû M (ñëåä ìàòðèöû).
270.
trunc(x) – öåëàÿ ÷àñòü âåùåñòâåííîãî
÷èñëà x.
271.
Ucheb(n, x) – ïîëèíîì ×åáûøååâà âòîðîãî
ðîäà ñòåïåíè n, â òî÷êå x.
272.
var(A) – âàðèàöèÿ (äèñïåðñèÿ)
ýëåìåíòîâ ìàññèâà A, ãäå A – âåùåñòâåííàÿ èëè êîìïëåêñíàÿ ìàòðèöà ðàçìåðîì m´n èëè ìíîæåñòâî.
273.
vec2str(v) – òåêñòîâàÿ ïåðåìåííàÿ,
îáðàçîâàííàÿ êîíâåðòèðîâàíèåì âåêòîðà
v â ASCll-êîäàõ ê çíàêàì. Ýëåìåíòû v
äîëæíû áûòü
öåëûìè ÷èñëàìè â èíòåðâàëå îò 0 äî 255 (ðèñ. 1.30).
274.
wave(v) – äèñêðåòíîå âîëíîâîå
ïðåîáðàçîâàíèå äåéñòâèòåëüíûõ ÷èñåë ñ èñïîëüçîâàíèåì 4-êîýôôèöèåíòíîãî
âîëíîâîãî ôèëüòðà Äîáèøè (Daubechies). Âåêòîð
v äîëæåí
ñîäåðæàòü 2n äåéñòâèòåëüíûõ çíà÷åíèé, ãäå
n – öåëîå ÷èñëî áîëüøå 0.
275.
WRITEBMP(file) – øêàëà ÿðêîñòè âûõîäíîãî ôàéëà ìàòðèöû BMP.
276.
WRITE_HLS(file) – ìàòðèöà, èìåþùàÿ öâåòîâîé ôîðìàò BMP â ôàéëå file, ïîëó÷åííàÿ èç
ìàòðèöû, îáðàçîâàííîé ïóòåì ñëèÿíèÿ òðåõ ìàòðèö, äàþùèõ ñîîòâåòñòâóþùèå
çíà÷åíèÿ îòòåíêà öâåòà, ÿðêîñòè è íàñûùåííîñòè.
277.
WRITE_HSV(file) – ìàòðèöà, èìåþùàÿ öâåòîâîé
ôîðìàò BMP â ôàéëå file, ïîëó÷åííàÿ èç
ìàòðèöû, îáðàçîâàííîé ïóòåì ñëèÿíèÿ òðåõ ìàòðèö, äàþùèõ ñîîòâåòñòâóþùèå
çíà÷åíèÿ îòòåíêà öâåòà, íàñûùåííîñòè è âåëè÷èíû.
278.
WRITEPRN(file) – ìàòðèöà â ôàéëå file, èìåþùåì ñòðóêòóðó ASCII. Êàæäàÿ ñòðîêà ìàòðèöû
ñòàíîâèòñÿ íîâîé ñòðîêîé â äàííîì ôàéëå.
279.
WRITERGB(file) – ìàòðèöà, èìåþùàÿ öâåòîâîé
ôîðìàò BMP â ôàéëå file, ïîëó÷åííàÿ èç ìàòðèöû,
îáðàçîâàííîé ïóòåì ñëèÿíèÿ òðåõ ìàòðèö, äàþùèõ êðàñíîå, çåëåíîå è ñèíåå
çíà÷åíèÿ.
280.
Y0(x) – ôóíêöèÿ Áåññåëÿ âòîðîãî
ðîäà íóëåâîãî ïîðÿäêà; x – äåéñòâèòåëüíîå è
ïîëîæèòåëüíîå.
281.
Y1(x) – ôóíêöèÿ Áåññåëÿ âòîðîãî
ðîäà ïåðâîãî ïîðÿäêà; x – äåéñòâèòåëüíîå è
ïîëîæèòåëüíîå.
282.
Yn(m, x) – ôóíêöèÿ Áåññåëÿ âòîðîãî
ðîäà ïîðÿäêà
m; x – äåéñòâèòåëüíîå ïîëîæèòåëüíîå ÷èñëî; m – îò 0 äî 100.
283.
ys(n, x) – ñôåðè÷åñêàÿ ôóíêöèÿ
Áåññåëÿ âòîðîãî ðîäà ïîðÿäêà n.
284.
d(m, n) – d-ôóíêöèÿ Êðîíåêåðà (1, åñëè m=n, è 0 – â äðóãèõ ñëó÷àÿõ; x è y – öåëûå ÷èñëà).
285.
e(i, j, k) – àáñîëþòíî àñèììåòðè÷íûé òåíçîð ðàçìåðíîñòè òðè. i, j è k äîëæíû áûòü öåëûìè ÷èñëàìè
îò 0 äî 2 (èëè ìåæäó ORIGIN è ORIGIN+2, åñëè ORIGIN¹0). Ðåçóëüòàò ðàâåí 0, åñëè ëþáûå äâà ðàâíû, 1 – åñëè òðè àðãóìåíòà ÿâëÿþòñÿ ÷åòíîé ïåðåñòàíîâêîé (0, 1, 2), è -1, åñëè òðè àðãóìåíòà
ÿâëÿþòñÿ ïåðåñòàíîâêîé (0, 1, 2), êðàòíîé 2 è íå êðàòíîé 4.
286.
Ã(z) – g-ôóíêöèÿ Ýéëåðà (ðèñ. 7.6).
287.
Ã(x, y) – âûòÿíóòàÿ g-ôóíêöèÿ Ýéëåðà.
288.
F(x) – ôóíêöèÿ Õåâèñàéäà (1, åñëè x³0, è 0 – â äðóãèõ ñëó÷àÿõ) (ðèñ.
6.19).
·
ðåøåíèå
ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé:
gaussjInvert(A) – inverse of the square matrix A, found using Gauss-Jordan elimination.
gaussjSolve(A, b) – solution vector x for the linear system Ax = b, found using Gauss-Jordan elimination.
luSolve(A, b) – solution vector x for the linear system Ax = b,
found using LU decomposition.
tridag(a, b, c, r) – solution x of the tridiagonal system Ax = r; a
is the subdiagonal, b the diagonal and c the
superdiagonal of the matrix A, with the first entry of a and the last of c being 0.
improve(A, b, x) – improved solution vector for the system Ax = b, given
a solution x.
svdcmp(M) – Singular
Value Decomposition of the real m by n matrix M into a product UWV-transpose of matrices, where U is m by n, W is an n by n diagonal matrix and V is n by n. Returns a matrix of size max(m,n) by 2n+1, containing in columns 0
through n-1 the matrix U, in column n the
diagonal elements of the matrix W and in columns n+1 through 2n the matrix V. If m and n are
not equal, zeros are filled in for the unused entries.
svSolve(A, b) – Singular Value Decomposition of the real m by n matrix M into a product UWV-transpose of matrices, where U is m by n, W is an n by n diagonal matrix and V is n by n. Returns a matrix of size max(m, n) by 2n+1, containing in columns 0
through n-1 the matrix U, in column n the diagonal elements of the matrix W and
in columns n+1 through 2n the
matrix V. If m and n are
not equal, zeros are filled in for the unused entries.
·
èíòåðïîëÿöèÿ
è ýêñòðàïîëÿöèÿ:
polint(X, Y, p) – interpolated value at the point p of a function given by a
table of x-y pairs, using polynomial interpolation. The x-y data are in the vectors X and Y. Returns
a vector of length 2 containing the interpolated value and an
estimate of the error.
ratint(X, Y, p) – interpolated value at the point p of a function defined by a
table of x-y pairs, using rational function
interpolation. The x-y data are in the vectors X and Y. Returns a vector of length 2 containing the interpolated value and an
estimate of the error.
polcof(X, Y) – coefficients of the interpolating polynomial for the points with (x,y) coordinates in the vectors X and Y. The
coefficients start with the constant term.
polin2(X, Y, Z, a, b) – interpolated value at the point (a,b) of a function of two
variables defined by a table of x-y-z values, using polynomial
interpolation. The x-y-z data are in the vectors X, Y, and Z. Returns a vector of length 2 containing the interpolated value and an
estimate of the error.
·
èíòåãðèðîâàíèå ôóíêöèé:
Midpnt(f, a, b) – integral of f from a to b computed
using the extended midpoint rule.
Midinf(f, a, b) – integral of f from a to b, where
one of the limits is infinite.
Midsql(f, a, b) – integral of f from a to b where f has a "1 over the square root of x" singularity
at a.
Midsqu(f, a, b) – integral of f from a to b where f has a singularity at b.
Midexp(f, a) – integral of f from a to b where f has a singularity at b.
qgaus(f, a, b) – integral of f from a to b computed
using 10-point Gaussian quadrature.
Gauleg(f, a, b, n) – integral of f from a to b using n-point Gauss-Legendre quadrature.
Gaulag(f, alpha, n) – integral of x^alpha* e^(-x)* f from 0 to infinity computed using
n-point Gauss-Laguerre.
Gauher(f, n) –
integral of x^alpha* e^(-x)* f from 0 to infinity computed using n-point
Gauss-Laguerre
Gaujac(f, alpha, beta, n) – integral of (1 - x)^alpha* (1 + x)^beta* f from 0 to infinity computed using n-point
Gauss-Jacobi quadrature.
quad3d(f, X1, X2, Y, Z) – integral of f over the three dimensional region defined by
the x limits x1 and x2 and the y and z boundaries in Y (a two-vector giving the lower and upper y limits
as a function of x)
and Z (a two-vector giving the lower and upper z limits as a function of x and y).
·
ïðåîáðàçîâàíèå ôóíêöèé:
ddpoly(c, x, nd) – vector containing the value at x of the polynomial with
coefficients c, and the first nd derivatives evaluated at x.
poldiv(N, D) – quotient and remainder resulting from dividing the polynomial with
coefficients N by the polynomial with coefficients D. The coefficients of the quotient are in the first column of the result and the coefficients of the remainder in the second column.
chebft(a, b, n, f) – coefficients for an approximation to f over
the interval [a,b] as a sum of the first n Chebyshev polynomials.
chebev(a, b, c, x) – Chebyshev approximation to the value of a function at the point x. The Chebyshev coefficients c of a
Chebyshev approximation to f over the interval (a,b) were computed using the function chebft.
Chebcoef(c, a, b) – coefficients of a polynomial approximation for a function f, computed from the coefficients c of
the Chebyshev approximation for f over the interval (a,b). Use chebft to compute c.
pade(C) – coefficients
of the numerator and denominator of a Pade approximation to a function
that has a Taylor approximation with coefficients C. The
third column of the returned array contains the norm of the residual vector
followed by zeros.
ratlsq(f, a, b, m, k) – coefficients for a rational approximation to f over
the interval [a,b] with numerator degree m and
denominator degree n.
·
ðàáîòà
ñî ñïåöèàëüíûìè ôóíêöèÿìè:
gammln(x) – ln(Gamma(x))
bico(n, k) – binomial coefficient (n,k).
beta(x, w) – beta function B(x,w).
factln(n) – logarithm
of n factorial, ln(n!).
gammp(a, x) – incomplete gamma function P(a,x).
gammq(a, x) – complementary incomplete gamma function.
erffc(x) – complementary
error function.
ei(x) – exponential
integral.
betai(a, b, x) – incomplete beta function I(a,b,x).
bessjy(x, n) – Bessel functions Jn and Yn and
their derivatives for a positive real argument x. Returns
a 2 by 2 matrix containing Jn(x) and Yn(x) in the first row and Jn'(x) and Yn'(x) in the second row.
bessik(x, n) – Jn(x) and Yn(x) in the first row and Jn'(x) and Yn'(x) in the second row.
airy(x) – Airy functions Ai(x) and Bi(x) and their derivatives for a
real number x. Returns a 2 by 2 matrix containing Ai(x) and Bi(x) in
the first row and Ai'(x) and Bi'(x) in the second row.
sphbes(n, x) – spherical Bessel functions jn(x) and yn(x) and
their derivatives for a positive argument x. Returns a 2 by 2 matrix containing jn(x) and yn(x) in the first row and j'n(x) and y'n(x) in the second row.
plgndr(nu, mu, x) – value at x of the associated Legendre polynomial P(nu,mu,x).
FresnelC(x) – Fresnel cosine integral.
FresnelS(x) – Fresnel sine integral.
Ci(x) – cosine
integral.
Si(x) – sine
integral.
rf(x, y, z) – Carlson's elliptic integral of the first kind.
rd(x, y, z) – Carlson's elliptic integral of the second kind.
rj(x, y, z, p) – Carlson's elliptic integral of the third kind.
rc(x, y) – Carlson's
degenerate elliptic integral.
ellf(phi, k) – Legendre elliptic integral of the first kind.
elle(phi, k) – Legendre elliptic integral of the second kind.
ellpi(phi, k) –
Legendre elliptic integral of the third kind.
sncndn(u, m) – Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m), in a vector of length 3.
hypgeo(a, b, c,
z) – hypergeomtric function F(a,b;c;z) for
complex argument z.
·
ðàáîòà
ñî ñëó÷àéíûìè ÷èñëàìè:
InitializeExpdev(k) – input value. Initializes the function expdev with
a seed value for later calls.
expdev(n) – vector of n random numbers drawn from an exponential
distribution with mean 1.
InitializeGasdev(k)
– input value. Initializes the function gasdev with a seed value for later calls.
gasdev(n) – vector of n random numbers drawn from a normal distribution
with mean 0 and variance 1.
InitializeGamdev(k)
– input value. Initializes the function gamdev with a seed value for later calls.
gamdev(a, n) – vector of n random numbers drawn from a gamma distribution
of order a.
InitializePoidev(k)
– input value. Initializes the function poidev with a seed value for later calls.
poidev(y, n) – vector of n random numbers drawn from a Poisson
distribution with mean y.
InitializeBnldev(k)
– vector of n random numbers drawn from a
Poisson distribution with mean y.
bnldev(p, N, n)
– vector of n random numbers drawn from a
binomial distribution for N trials with success probability p.
irbit2(n) – vector n random bits (0 or 1) generated
by a recurrence based on the primitive polynomial.
InitializeSobseq(k)
– input value; used to initialize the function sobseq for later calls.
sobseq(n, m) – array of m points in n-space
generated using a Sobol' sequence; each column
represents a point, with all coordinates between 0 and 1.
vegas(rg, f,
init, n, im) – integral of f over
the region defined in the array rgn. Returns an array of length 3 containing the best estimate
for the integral, its standard deviation, and the value of the chi-square statistic for the set of estimates.
·
ïîèñê
êîðíåé íåëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé è ñèñòåì:
rtbis(f, a, b,
acc) – root of f between a and b, to accuracy acc, found
by bisection.
zriddr(f, a, b,
acc) – root of f between a and b, to accuracy acc, found
using Ridder's method.
zbrent(f, a, b,
acc) – root of f between a and b, to accuracy acc, found
by Brent's method.
laguer(C, z) – approximation for a root of the polynomial with coefficients C, using Laguerre's method starting from the
guess z.
zroots(C, polish)
– vector containing all the complex roots of the
polynomial with complex coefficients C. Set polish to 1 to refine the roots.
newt(x, F) – common root of the set of n functions of n variables given in F, using Newton's method. The length-n vector x is an initial guess.
broydn(x, F) – common root of the set of n functions of n variables given in F, using Broyden's method. The
length-n vector x is an initial guess.
·
ìèíèìèçàöèÿ
è ìàêñèìèçàöèÿ ôóíêöèé îäíîé è ìíîãèõ ïåðåìåííûõ (âêëþ÷àÿ çàäà÷ó ëèíåéíîãî
ïðîãðàììèðîâàíèÿ è çàäà÷ó êîììèâîÿæåðà):
brent(a, b, c,
f, acc) – location of a minimum of f lying between a and c, where b is between a and c and f(b) is
less than f(a) and f(c); found by Brent's method with
precision acc.
amoeba(p, acc,
f) – location of a minimum of the scalar function of n variables f, with initial guess in the vector p; uses the downhill simplex method and returns a matrix whose rows are n + 1 points in n-space at which f is
within acc of a minimum.
powell(s, xi,
ftol, f) – location of a minimum of the scalar function of n variables f, with initial guess in the vector s and starting directions in the array xi; uses
Powell's method with convergence tolerance ftol.
frprmn(p, acc,
f, gradf) – location of a minimum of the scalar function of n variables f, with initial guess in the vector p and gradient computed by the vector function gradf; uses
Fletcher-Reeves-Polak-Ribiere minimization with convergence tolerance acc.
dfpmin(p, acc,
f, gradf) – location of a minimum of the scalar function of n variables f, with initial guess in the vector p and gradient computed by the vector function gradf; uses
Davidon-Fletcher-Powell minimization with convergence tolerance acc.
simplx(S, m1,
m2) – vector containing the values of the variables that
minimize the objective function subject to a set a inequalities. The matrix S contains the objective function and the
inequality constraints in restricted normal form; m1 and m2 are the numbers of "less than or equal to" and "greater
than or equal to" constraints.
anneal(X, Y) – solution to the traveling salesman problem. The vectors X and Y give the coordinates of the cities, and the
returned vector gives the order in which they should be visited for a nearly
minimal path.
·
ðàáîòà
ñ ñîáñòâåííûìè ñèñòåìàìè:
jacobi(M) – eigenvalues and eigenvectors of the real symmetric matrix M; the eigenvalues are in the first column of the answer and the
eigenvectors in the remaining columns.
tqli(d, e) – eigenvalues and eigenvectors of a real symmetric tridiagonal matrix; d is the diagonal and e is the subdiagonal of the
matrix (the first entry of e is not used). The
eigenvalues are in the first column of the answer and the eigenvectors in the
remaining columns.
·
áûñòðûå
ïðåîáðàçîâàíèÿ Ôóðüå:
cosft2(v) – staggered cosine transform of the data vector v.
icosft2(v) – inverse staggered cosine transform of v.
fourn(v, n) –
n-dimensional discrete Fourier transform of the data
stored in v.
ifourn(v, n) –
n-dimensional inverse discrete Fourier transform of the
points stored in v.
·
ñïåêòðàëüíûé àíàëèç:
memcof(v, m) – vector of m linear prediction coefficients for the data v.
fixrts(d) – modified vector of linear prediction coefficients; moves any zeros of
the characteristic polynomial that are outside the unit circle to inside.
predic(v, d, n)
– vector of n values predicted from the
data v using the linear prediction coefficients d.
evlmem(f, d, m)
– vector giving the power spectrum of the data d at the frequencies in f, computed using m coefficients generated by the function memcof.
period(X, Y,
of, hf) – Lomb periodogram of a set of x-y data points. The x-y data are in the vectors X and Y; hf is the highest frequency and of is the oversampling factor. The result is a matrix with frequencies in
the first column and the values of the normalized periodogram at these
frequencies in the second column.
fasper(X, Y,
of, hf) – Lomb periodogram of a set of x-y data points, using a fast algorithm. The x-y data
are in the vectors X and Y; hf is the highest frequency and of is the oversampling factor. The result is a matrix with
frequencies in the first column and the values of the normalized periodogram at
these frequencies in the second column.
wt1Daub4(v) –
1-dimensional discrete wavelet transform of v computed using the Daubechies 4-coefficient wavelet filter.
iwt1Daub4(v) – inverse 1-dimensional discrete wavelet transform of v computed using the Daubechies 4-coefficient wavelet filter.
wtnDaub4(v, n)
– n-dimensional discrete wavelet transform of the data
stored in v computed using the Daubechies 4-coefficient wavelet filter.
iwtnDaub4(v, n)
– inverse n-dimensional discrete
wavelet transform of the points stored in v computed using the Daubechies 4-coefficient wavelet filter.
wt1Pwt(v, w) –
1-dimensional discrete wavelet transform of v computed using the Daubechies 4-, 12- or 20-coefficient wavelet filter; w gives the order of the
filter.
iwt1Pwt(v, w) –
inverse 1-dimensional discrete wavelet
transform of v computed using the Daubechies 4-, 12- or 20-coefficient wavelet filter; w gives the order of the filter.
wtnPwt(v, w, m)
– n-dimensional discrete wavelet
transform of the data stored in v computed using the
Daubechies 4-, 12- or 20-coefficient wavelet filter;
the elements of w give the size of the
data in each dimension; m gives the order of the
filter.
iwtnPwt(v, w,
m) – inverse n-dimensional discrete wavelet
transform of the data stored in v computed using the
Daubechies 4-, 12- or 20-coefficient wavelet filter;
the elements of w give the size of the data in each dimension; m gives the order of the filter.
·
ñòàòèñòè÷åñêàÿ
îáðàáîòêà äàííûõ:
moment(v) – Moment information for the data vector v.
Returns the mean, average absolute deviation, standard deviation, variance, skewness and kurtosis of v.
ttest(v1, v2) –
value of Student's t to
test whether the two samples v1 and v2
are drawn from distributions with the same mean. The two distributions are
assumed to have equal variances. The result is a vector of length 2 containing the value of Student's t and
the probability that this value or larger would occur when the distributions
have the same mean.
tutest(v1, v2)
– value of Student's t to
test whether the two samples v1 and v2 are
drawn from distributions with the same mean. Equal variances are not assumed.
The result is a vector of length 2 containing the value of
Student's t and the probability that this value or larger
would occur when the distributions have the same mean.
tptest(v1, v2)
– value of Student's t to
test whether the paired data v1 and v2 are
drawn from distributions with the same mean. Equal variances are not assumed,
and the paired data may be correlated. The result is a vector of length 2 containing the value of Student's t and
the probability that this value or larger would occur when the distributions
have the same mean.
ftest(v1, v2) –
value of the F statistic to test whether
the two samples v1 and v2 are drawn from distributions
with the same variance. The result is a vector of length 2 containing the value of F and the probability that
this value or larger would occur when the distributions have the same variance.
chsone(bins,
ebins, k) – Chi-square for the comparison of an observed
distribution with a known distribution. The binned observed
data are in bins, the expected counts for the know distribution
are in ebins; k is the number of constraints. The result is a
vector of length 3 containing the number of degrees of freedom,
the value of the chi-square statistic and the probability that this
value or a higher value will occur when the observed data is drawn from a
population with the known distribution.
chstwo(bins1,
bins2, k) – Chi-square for two sets of binned observed
data. The binned observed data are in bins1 and bins2 and k is the number of constraints. The result is a
vector of length 3 containing the number of degrees of freedom,
the value of the chi-square statistic and the probability that this
value or a higher value will occur when the two sets of observed data are drawn
from the same distribution.
ksone(d, f) – Kolmogorov-Smirnov statistic comparing the distribution of the data in d with the cumulative distribution function f. The
result is a vector of length 2 containing the statistic D and its significance level.
kstwo(v1, v2) –
Kolmogorov-Smirnov statistic for two sets of data v1 and v2. The result is a vector of length 2 containing the value of the Kolmogorov-Smirnov statistic D and the probability that this value or a larger value would occur if the
two sets of observed data were drawn from the same distribution.
cntab1(T) – significance of association between two variables with observed data
stored in the two-dimensional contingency table T. The
result is a length 5 vector containing the value of the chi-square statistic, the number of degrees of freedom, the probability that
the value of the chi-square statistic occurs when the variables are
independent, Cramer's V, and the contingency
coefficient C.
pearsn(X, Y) – Information about the correlation of the paired data in the vectors X and Y. The result is a vector of length 3 containing the value of Pearson's r, the
significance level at which we reject the null hypothesis that the two samples
are uncorrelated and Fisher's z-transformation.
spear(X, Y) – rank-order correlation of the data vectors X and Y. The result is a vector of length 5 containing
the sum squared difference of ranks D, the number of standard
deviations by which D deviates from the expected value under the null
hypothesis, the probability that this value or a larger value occurs for
uncorrelated samples, Spearman's rank-order correlation coefficient rs and the probability that this value or one larger in absolute value
would occur if the samples were uncorrelated.
kendl1(X, Y) – Kendall's tau for the data sets X and Y. The result is a vector containing Kendall's tau, its
number of standard deviations away from zero and the probability that a value
this large or larger in absolute value would occur if the samples were
uncorrelated.
kendl2(T) – Kendall's tau for the data in the contingency table T. The result is a vector containing Kendall's tau, its
number of standard deviations away from zero and the probability that a value
this large or larger in absolute value would occur if the samples were
uncorrelated.
ks2d1s(x, y,
quad) – two-dimensional Kolmogorov-Smirnov statistic
comparing the x-y data points given by the vectors x and y with the target model given by the function quad. The result is a vector of length 2 containing
the K-S statistic and its significance level.
quadvl(x, y) – vector of length 4 containing the fraction of
the uniform distribution on [0,1] x [0,1] that
lies in each quadrant around (x,y).
ks2d2s(X, Y, Z,
W) – two-dimensional Kolmogorov-Smirnov statistic for the
two-dimensional data sets defined by the vectors X, Y and Z, W. The result is a length 2 vector containing the
largest percentage difference found between the two sets of data points and the
probability that this value is consistent with the two sets of data having the
same distribution.
·
ìîäåëèðîâàíèå äàííûõ:
fit(X, Y) – best straight-line fit to the x-y data in the vectors X and Y, assuming unknown errors in the first
coordinate. The result is a vector of length 5 containing
the values of a and b such that a + bx is the line of best fit for the given data, the probable uncertainties
of the values of a and b and the value of the chi-square statistic for the set of points.
fitmwt(X, Y, S)
– best straight-line fit to the x-y data
in the vectors X and Y, assuming no errors in the
first coordinate and standard deviations for the second coordinate given by S. The result is a vector of length 6 containing
the values of a and b such that a + bx is the line of best fit for the given data, the probable uncertainties
of the values of a and b, the value of the chi-square statistic for the set of points and the probability of a chi-square statistic this large or larger.
fitexy(X, Y,
SX, SY) – best straight-line fit to the x-y data in the vectors X and Y, with
standard deviations SX and SY. The result is a vector of
length 6 containing the values of a and b such that a + bx is
the line of best fit for the given data, the probable uncertainties of the
values of a and b, the value of the chi-square statistic for the set of points and the probability of a chi-square statistic this large or larger.
lfit(x, y, s,
a, ia, F) – coefficients for the linear combination of the
functions in F that best fit the x-y data
given in the vectors x and y, with standard deviations
given by s. The vector a gives
values for the coefficients and the vector ia indicates
which functions are to be fitted and which take their coefficients from a.
svdfit(x, y, s,
m, F) – coefficients for the linear combination of the m functions in F that best fit the x-y data
given in the vectors x and y, with standard deviations
for the y's given by s.
svdvar(x, y, s,
m, F) – covariance matrix for the fitting parameters
found by svdfit. The data vectors are x and y, with standard deviations for the y's in s. The fitting functions are in F.
Mrqmin(x, y, s,
a, ia, F) – coefficients for the best fit by the nonlinear
function defined by F to the x-y data
given in the vectors x and y, with standard deviations
given by s. The vector a
gives values for the parameters and the vector ia indicates
which parameters are to be fitted and which held fixed at the values in a. F is a vector function of x and the parameters, with the
fitting function in the first element and its derivative with respect to each
parameter in the remaining elements.
fgauss(x, a) – value of a sum of Gaussian curves specified by the parameters in the
vector a at the point x, followed
by the derivatives with respect to each parameter evaluated at x.
medfit(X, Y) – line with minimum absolute deviation from the x-y data
points in the vectors X and Y. The
result is a vector of length 3 containing the values a and b such that the minimizing line has the equation y = a + bx and the mean absolute deviation of the points from the computed line.
·
ðåøåíèå
îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé (çàäà÷à Êîøè):
rkdumb(y, x1,
x2, s, D) – solution to a system of n first-order
differential equations, computed over the interval x1 to x2 in s equal steps using 4th order
Runge-Kutta. The initial conditions are in the vector y and
the array D gives the derivatives of the system components.
The result is a matrix with the x values in the first column
and the corresponding values of the solution components in the following
columns.
BsstepR(y, x1,
x2, acc, ht, hm, D, J, km, s) – solution to a system of n first-order differential equations, computed over the interval x1 to x2 using the Bulirsch-Stoer method with rational
extrapolation. The initial conditions are in the vector y and
the array D gives the derivatives of the system components.
The result is a matrix with the x values in the first column
and the corresponding values of the solution components in the following
columns.
BsstepP(y, x1,
x2, acc, ht, hm, D, J, km, s) – solution to a system of n first-order differential equations, computed over the interval x1 to x2 using the Bulirsch-Stoer method with polynomial
extrapolation. The initial conditions are in the vector y and
the array D gives the derivatives of the system components.
The result is a matrix with the x values in the first column
and the corresponding values of the solution components in the following
columns.
Stiff(y, x1,
x2, acc, ht, hm, D, J, km, s) – solution to a stiff system
of n first-order differential equations, computed
over the interval x1 to x2, computed by Rosenbrock
methods. The initial conditions are in the vector y, the
array D gives the derivatives of the system components,
and the array J gives the Jacobian matrix. The result is a
matrix with the x values in the first column and the
corresponding values of the solution components in the following columns.
·
ðåøåíèå
êðàåâîé çàäà÷è:
shoot(v, x1,
x2, D, load, score) – Initial conditions providing
a solution to the two-point boundary value problem for a system of first-order
differential equations. The starting and end points are x1 and x2 and the system derivatives are in the array D.
·
ðåøåíèå
äèôôåðåíöèàëüíûõ óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ:
fred2(n, a, b, g, lambdaK) – solution for the Fredholm integral equation of the second kind; the integral
is over the interval [a,b] with kernel lambdaK and
added function g, and the answer is returned as a two-column
array giving abscissas for n-point Gaussian quadrature
and the value of the solution at these points.
fredin(x, n, a, b, g, lambdaK) – value at x of the solution of a Fredholm equation,
computed by interpolating the n-point answer returned by the
function fred2. The integral is over the interval [a,b] with kernel lambdaK and added function g.
voltra(n, m, a, h, g, K) – solution for a set of m linear Volterra equations of
the second kind, with matrix kernel K and added vector function g. The starting point for the integrals is a, and n-1 steps of size h are taken.
sor(a, b, c, d, e, f, u, rjac) – solution of a second-order difference equation over a rectangular mesh,
computed by relaxation. The matrices a b c d e and f define the differencing scheme, u contains the initial
conditions, and rjac is the Jacobian radius.
[1] Åñëè ãîâîðèòü î ñèìâîëüíîé
ìàòåìàòèêå (à êî ìíîãèì âñòðîåííûì ôóíêöèè Mathcad ïðèìåíèìû àíàëèòè÷åñêèå
ïðåîáðàçîâàíèÿ – ñì. ýòþä 7), òî òåðìèí «âîçâðàùàþò» òóò ñîâñåì íåóìåñòåí.
[2] Ôóíêöèè ðàçáèòû ïî ãðóïïàì
ñîãëàñíî Ìàñòåðó ôóíêöèé (ñì. ðèñ. 1.28). Íåêîòîðûå ôóíêöèè (lsolve, íàïðèìåð) óïîìèíàþòñÿ â
íåñêîëüêèõ ãðóïïàõ.
[3] Îíè ñòàíîâÿòñÿ âñòðîåííûìè
ñ Mathcad ïîñëå ïîäãðóçêè ñîîòâåòñòâóþùåãî ýëåêòðîííîãî ó÷åáíèêà.