Nyob rau hauv tsab xov xwm no, txoj kev suav hais tias yog ib txoj hauv kev los daws cov kab ke sib npaug (SLAE). Cov txheej txheem yog analytical, uas yog, nws tso cai rau koj los sau ib tug general kev daws teeb meem algorithm, thiab ces hloov qhov tseem ceeb los ntawm tej yam piv txwv muaj. Tsis zoo li cov qauv matrix lossis Cramer cov qauv, thaum daws cov kab sib npaug ntawm cov kab sib npaug siv Gauss txoj kev, koj tuaj yeem ua haujlwm nrog cov uas muaj ntau yam kev daws teeb meem. Los yog tsis muaj kiag li.
txhais li cas los daws los ntawm Gauss txoj kev?
Ua ntej, peb yuav tsum tau sau peb cov kab ke ntawm kev sib npaug ua ib qho matrix. Nws zoo li no. Qhov system tau txais:
Coefficients yog sau nyob rau hauv daim ntawv ntawm lub rooj, thiab nyob rau sab xis nyob rau hauv ib kem cais - dawb cov tswv cuab. Lub kem nrog cov tswv cuab pub dawb yog sib cais kom yooj yim los ntawm ib txoj kab ntsug. Lub matrix uas suav nrog kab no hu ua txuas ntxiv.
Tom ntej no, lub ntsiab matrix nrog coefficients yuav tsum raug txo kom qis rau sab sauv daim duab peb sab. Qhov no yog lub ntsiab lus tseem ceeb ntawm kev daws qhov system los ntawm txoj kev Gauss. Yooj yim muab, tom qab qee qhov kev hloov pauv, cov matrix yuav tsum zoo li qhov no, kom tsuas muaj xoom hauv nws sab laug sab:
Tom qab ntawd, yog tias koj sau cov matrix tshiab dua li qhov system ntawm kev sib npaug, koj yuav pom tias kab kawg twb muaj tus nqi ntawm ib qho ntawm cov hauv paus hniav, uas yog tom qab hloov mus rau qhov sib npaug saum toj no, lwm lub hauv paus pom., thiab lwm yam.
Qhov no yog ib qho kev piav qhia ntawm Gaussian tov nyob rau hauv feem ntau cov ntsiab lus. Thiab yuav ua li cas yog tias tam sim ntawd lub kaw lus tsis muaj kev daws teeb meem? Los yog muaj ib tus lej tsis kawg ntawm lawv? Txhawm rau teb cov lus nug no thiab ntau ntxiv, nws yog ib qho tsim nyog yuav tau txiav txim siab cais tag nrho cov ntsiab lus siv hauv kev daws teeb meem los ntawm txoj kev Gauss.
Matrices, lawv cov khoom
Tsis muaj lub ntsiab lus zais hauv matrix. Nws tsuas yog txoj hauv kev yooj yim los sau cov ntaub ntawv rau kev ua haujlwm tom qab. Txawm cov me nyuam kawm ntawv yuav tsum tsis txhob ntshai lawv.
Lub matrix yog ib txwm muaj plaub fab vim nws yooj yim dua. Txawm tias nyob rau hauv txoj kev Gauss, qhov twg txhua yam boils mus rau lub tsev ib daim duab peb sab matrix, ib daim duab plaub tshwm nyob rau hauv lub nkag, tsuas yog muaj zeros nyob rau hauv qhov chaw uas tsis muaj cov zauv. Zeros tuaj yeem raug tshem tawm, tab sis lawv txhais tau.
Matrix muaj qhov loj me. Nws "dav" yog tus naj npawb ntawm kab (m), nws "ntev" yog tus naj npawb ntawm kab (n). Tom qab ntawd qhov loj ntawm matrix A (cov tsiaj ntawv Latin feem ntau yog siv rau lawv lub npe) yuav raug suav tias yog A m ×n. Yog tias m=n, ces qhov matrix no yog square, thiabm=n - nws order. Raws li, txhua lub caij ntawm matrix A tuaj yeem txhais tau los ntawm tus lej ntawm nws kab thiab kab: axy; x - kab zauv, hloov [1, m], y - kab zauv, hloov [1, n].
Nyob rau hauv txoj kev Gaussian, matrices tsis yog lub ntsiab lus tseem ceeb ntawm kev daws. Raws li txoj cai, txhua txoj haujlwm tuaj yeem ua ncaj qha nrog qhov sib npaug ntawm lawv tus kheej, txawm li cas los xij, cov ntawv sau yuav ntau dua cumbersome, thiab nws yuav yooj yim dua kom tsis meej pem hauv nws.
Qualifier
Matrix kuj muaj qhov txiav txim. Qhov no yog ib qho tseem ceeb heev. Tshawb nrhiav nws lub ntsiab lus tam sim no tsis tsim nyog, koj tuaj yeem qhia tau tias nws suav li cas, thiab tom qab ntawd qhia cov khoom ntawm matrix nws txiav txim siab li cas. Txoj kev yooj yim tshaj plaws los nrhiav tus txiav txim yog los ntawm kab pheeb ces kaum. Cov kab pheeb ces kaum yog kos rau hauv matrix; cov ntsiab lus nyob rau ntawm lawv txhua tus tau muab sib npaug, thiab tom qab ntawd cov khoom lag luam tau ntxiv: kab pheeb ces kaum nrog txoj kab nqes mus rau sab xis - nrog rau "plus" kos npe, nrog txoj kab nqes mus rau sab laug - nrog rau "muab" kos npe.
Nws yog ib qho tseem ceeb heev uas yuav tsum nco ntsoov tias tus txiav txim tsuas tuaj yeem suav rau ib square matrix xwb. Rau ib daim duab plaub, koj tuaj yeem ua tau raws li hauv qab no: xaiv qhov tsawg tshaj plaws ntawm cov kab thiab tus naj npawb ntawm kab (cia nws yog k), thiab tom qab ntawd randomly cim k kab thiab k kab hauv matrix. Cov ntsiab lus nyob rau ntawm qhov kev sib tshuam ntawm cov kab uas tau xaiv thiab kab yuav tsim ib lub square matrix tshiab. Yog hais tias tus txiav txim ntawm xws li ib tug matrix yog ib tug xov tooj uas tsis yog xoom, ces nws yuav raug hu ua lub hauv paus me ntawm lub qub matrix.
Ua ntejYuav ua li cas pib daws qhov system ntawm kev sib npaug los ntawm Gauss txoj kev, nws tsis raug mob los xam tus txiav txim. Yog tias nws hloov mus rau xoom, ces peb tuaj yeem hais tam sim ntawd tias lub matrix muaj ib qho kev daws teeb meem tsis kawg, lossis tsis muaj ib qho twg. Hauv qhov xwm txheej zoo li no, koj yuav tsum tau mus ntxiv thiab paub txog qib ntawm matrix.
Kev faib cov tshuab
Muaj ib yam li qib ntawm matrix. Qhov no yog qhov kev txiav txim siab tshaj plaws ntawm nws qhov kev txiav txim tsis yog xoom (nco txog lub hauv paus me, peb tuaj yeem hais tias qib ntawm matrix yog qhov kev txiav txim ntawm lub hauv paus me).
Txoj kev uas muaj nyob nrog qib, SIAB tuaj yeem muab faib ua:
- Joint. Rau kev sib koom ua ke, qib ntawm lub ntsiab matrix (tsuas yog cov coefficients nkaus xwb) coincides nrog qib ntawm qhov txuas ntxiv (nrog rau kab lus pub dawb). Cov kab ke no muaj kev daws teeb meem, tab sis tsis tas yuav yog ib qho, yog li ntawd, cov kab ke sib koom ua ke tau muab faib ua:
- - meej - muaj kev daws teeb meem tshwj xeeb. Hauv qee lub tshuab, qib ntawm matrix thiab tus lej ntawm cov tsis paub yog sib npaug (lossis tus lej ntawm kab, uas yog tib yam);
- - indefinite - nrog tus lej ntawm kev daws teeb meem tsis kawg. Qhov qeb duas ntawm matrices hauv cov tshuab no tsawg dua li cov tsis paub.
- Incompatible. Rau cov tshuab zoo li no, cov qib ntawm lub ntsiab thiab txuas ntxiv matrices tsis sib xws. Cov tshuab tsis sib xws tsis muaj kev daws teeb meem.
Txoj kev Gauss yog qhov zoo vim tias nws tso cai rau koj kom tau txais ib qho pov thawj tsis meej ntawm qhov tsis sib xws ntawm lub kaw lus (tsis suav cov kev txiav txim siab ntawm cov matrices loj) lossis cov kev daws teeb meem dav dav rau lub kaw lus nrog cov kev daws teeb meem tsis kawg.
Elementary transformations
Ua ntejyuav ua li cas mus ncaj qha mus rau qhov kev daws teeb meem ntawm qhov system, koj tuaj yeem ua rau nws tsis tshua muaj teeb meem thiab yooj yim dua rau kev suav. Qhov no yog ua tiav los ntawm kev hloov pauv hauv theem pib - xws li tias lawv qhov kev siv tsis hloov cov lus teb kawg hauv txhua txoj kev. Nws yuav tsum raug sau tseg tias qee qhov kev hloov pauv hauv theem pib tsuas yog siv tau rau matrices, qhov chaw uas yog qhov tseeb ntawm SLAE. Nov yog cov npe ntawm cov kev hloov pauv no:
- Hloov cov hlua. Nws yog qhov pom tseeb tias yog tias peb hloov qhov kev txiav txim ntawm qhov sib npaug hauv cov ntaub ntawv kaw lus, ces qhov no yuav tsis cuam tshuam rau kev daws teeb meem txhua txoj kev. Yog li ntawd, nws kuj tseem tuaj yeem sib pauv kab hauv matrix ntawm qhov system no, tsis txhob hnov qab, ntawm chav kawm, txog kab ntawm cov tswv cuab dawb.
- Muab tag nrho cov ntsiab lus ntawm txoj hlua los ntawm qee yam. Muaj txiaj ntsig heev! Nrog nws, koj tuaj yeem txo cov lej loj hauv matrix lossis tshem tawm xoom. Cov txheej txheem kev daws teeb meem, raws li niaj zaus, yuav tsis hloov, thiab nws yuav yooj yim dua los ua haujlwm ntxiv. Qhov tseem ceeb tshaj plaws yog qhov coefficient yuav tsum tsis txhob sib npaug rau xoom.
- Txhob cov kab nrog qhov sib npaug sib npaug. Qhov no yog ib feem ntawm cov kab lus dhau los. Yog tias ob lossis ntau kab hauv matrix muaj qhov sib npaug sib npaug, ces thaum sib faib / faib ib qho ntawm cov kab los ntawm qhov sib npaug sib npaug, ob (lossis, dua, ntau dua) cov kab zoo tib yam tau txais, thiab koj tuaj yeem tshem tawm cov ntxiv, tawm hauv xwb. ib.
- Txhob tus kab null. Yog hais tias nyob rau hauv lub chav kawm ntawm transformations ib txoj hlua tau txais qhov chaw nyob rau hauv uas tag nrho cov ntsiab lus, nrog rau cov neeg dawb, yog xoom, ces xws li ib txoj hlua yuav raug hu ua xoom thiab pov tawm ntawm lub matrix.
- Ntxiv rau cov ntsiab lus ntawm ib kab ntawm lwm qhov (raws licov kab sib txuas) muab faib los ntawm qee qhov coefficient. Qhov kev hloov pauv tsis meej tshaj plaws thiab tseem ceeb tshaj plaws ntawm txhua tus. Nws yog tsim nyog nyob ntawm nws nyob rau hauv ntau nthuav dav.
Ntxiv ib txoj hlua sib npaug los ntawm qhov xwm txheej
Kom nkag siab yooj yim, nws tsim nyog disassembling cov txheej txheem no ib kauj ruam. Ob kab yog muab los ntawm matrix:
a11 a 12 … a1n | b1
a21 a22 …a2n | b2
Cia hais tias koj yuav tsum tau ntxiv qhov thib ib qhov sib npaug los ntawm tus coefficient "-2" rau tus thib ob.
a'21 =a21 + -2×a11
a'22 =a22 + -2×a12
a'2n =a2n + -2×a1n
Tom qab ntawd kab thib ob hauv matrix hloov nrog tus tshiab, hos thawj tus tseem tsis hloov.
a11 a12 … a1n | b1
a'21 a'22 …a'2n | b2
Nws yuav tsum tau muab sau tseg tias qhov sib npaug ntawm qhov sib npaug tuaj yeem xaiv tau raws li qhov tshwm sim ntawm kev ntxiv ob txoj hlua, ib qho ntawm cov hlua tshiab yog sib npaug rau xoom. Yog li ntawd, nws yog qhov ua tau kom tau txais qhov sib npaug hauv qhov system, qhov twg yuav muaj ib qho tsis paub. Thiab yog tias koj tau txais ob qhov sib npaug, ces qhov kev ua haujlwm tuaj yeem ua tiav thiab tau txais ib qho kev sib npaug uas twb muaj ob qhov tsis paub. Thiab yog tias txhua zaus peb tig mus rau xoom ib coefficient rau txhua kab uas qis dua qhov qub, ces peb tuaj yeem, zoo li cov kauj ruam, nqes mus rau hauv qab ntawm lub matrix thiab tau txais qhov sib npaug nrog ib qho tsis paub. Qhov no hu uadaws qhov system siv txoj kev Gauss.
Nyob zoo
Cia kom muaj qhov system. Nws muaj m sib npaug thiab n tsis paub keeb kwm. Koj sau tau li no:
Lub ntsiab matrix yog muab tso ua ke los ntawm cov coefficients ntawm lub system. Ib kem ntawm cov tswv cuab dawb tau ntxiv rau qhov nthuav dav matrix thiab sib cais los ntawm ib qho bar kom yooj yim.
Tom ntej:
- thawj kab ntawm matrix yog muab los ntawm coefficient k=(-a21/a11);
- thawj kab hloov kho thiab kab thib ob ntawm cov matrix ntxiv;
- hloov ntawm kab thib ob, qhov tshwm sim ntawm qhov sib ntxiv los ntawm kab lus dhau los tau muab tso rau hauv matrix;
- tam sim no thawj coefficient hauv kab thib ob tshiab yog a11 × (-a21/a11) + a21 =-a21 + a21=0.
Tam sim no tib txoj kev hloov pauv tau ua tiav, tsuas yog thawj kab thiab thib peb koom nrog. Raws li, nyob rau hauv txhua kauj ruam ntawm lub algorithm, lub caij a21 yog hloov los ntawm a31. Tom qab ntawd txhua yam rov ua dua rau a41, … am1. Qhov tshwm sim yog lub matrix qhov twg thawj lub caij hauv kab [2, m] yog sib npaug rau xoom. Tam sim no koj yuav tsum tsis nco qab txog kab thib ib thiab ua tib lub algorithm pib los ntawm kab thib ob:
- k coefficient=(-a32/a22);
- kab thib ob hloov kho yog ntxiv rau "tam sim no" kab;
- qhov tshwm sim ntawm qhov sib ntxiv yog hloov mus rau qhov thib peb, plaub thiab lwm yam kab, thaum thawj thiab thib ob tseem tsis hloov pauv;
- nyob rau hauv kab [3, m] ntawm matrix, thawj ob lub ntsiab twb sib npaug rau xoom.
Cov algorithm yuav tsum tau rov ua dua kom txog thaum lub coefficient k=(-am, m-1/amm tshwm). Qhov no txhais tau hais tias lub algorithm kawg nkaus nkaus xwb rau qhov sib npaug qis. Tam sim no lub matrix zoo li daim duab peb sab, los yog muaj ib tug stepped duab. Cov kab hauv qab no muaj cov kab zauv amn × x =bm. Lub coefficient thiab dawb lub sij hawm paub, thiab lub hauv paus yog qhia los ntawm lawv: x =bm/amn. Cov hauv paus ntsiab lus tau hloov mus rau hauv kab saum toj kawg nkaus kom pom xn-1=(bm-1 - am-1, n ×(bm/amn) ÷am-1, n-1. Thiab yog li ntawd los ntawm kev sib piv: nyob rau hauv txhua kab tom ntej no muaj lub hauv paus tshiab, thiab, tau mus txog "sab saum toj" ntawm lub kaw lus, ib tus tuaj yeem nrhiav tau cov kev daws teeb meem [x1, … x ]. Yuav yog tus xwb.
Thaum tsis muaj kev daws teeb meem
Yog hais tias nyob rau hauv ib qho ntawm cov kab matrix tag nrho cov ntsiab lus, tshwj tsis yog lub sij hawm pub dawb, sib npaug rau xoom, ces qhov sib npaug sib npaug rau kab no zoo li 0=b. Nws tsis muaj kev daws teeb meem. Thiab txij li qhov kev sib npaug no suav nrog hauv lub kaw lus, tom qab ntawd cov txheej txheem kev daws teeb meem ntawm tag nrho cov kab ke yog khoob, uas yog, nws yog degenerate.
Thaum muaj ntau qhov kev daws teeb meem tsis kawg
Nws yuav tig tawm tias hauv qhov txo qis daim duab peb sab tsis muaj kab nrog ib lub ntsiab lus - qhov sib npaug ntawm qhov sib npaug, thiab ib qho - tus tswv cuab dawb. Tsuas muaj cov hlua uas, thaum rov sau dua, yuav zoo li qhov sib npaug nrog ob lossis ntau qhov sib txawv. Qhov no txhais tau hais tias lub kaw lus muaj cov kev daws teeb meem tsis kawg. Hauv qhov no, cov lus teb tuaj yeem muab rau hauv daim ntawv ntawm kev daws teeb meem. Yuav ua li cas?
Txhua tusvariables nyob rau hauv lub matrix muab faib ua yooj yim thiab dawb. Basic - cov no yog cov uas sawv ntsug "ntawm ntug" ntawm cov kab hauv cov kauj ruam matrix. Tus so yog dawb. Hauv kev daws teeb meem, cov kev hloov pauv yooj yim tau sau ua cov lus pub dawb.
Rau qhov yooj yim, cov matrix yog thawj zaug rov sau rov qab rau hauv qhov system ntawm kev sib npaug. Tom qab ntawd nyob rau hauv qhov kawg ntawm lawv, qhov tseeb tsuas yog ib qho kev hloov pauv tseem ceeb, nws tseem nyob ntawm ib sab, thiab txhua yam hloov mus rau lwm qhov. Qhov no yog ua tiav rau txhua qhov sib npaug nrog ib qho kev hloov pauv yooj yim. Tom qab ntawd, nyob rau hauv qhov sib npaug ntawm qhov sib npaug, qhov twg ua tau, es tsis txhob ntawm qhov hloov pauv yooj yim, cov lus tau txais rau nws yog hloov pauv. Yog hais tias qhov tshwm sim yog ib qho kev nthuav qhia uas tsuas muaj ib qho kev hloov pauv yooj yim, nws tau nthuav tawm los ntawm qhov ntawd dua, thiab lwm yam, kom txog rau thaum txhua qhov hloov pauv tau sau ua ib qho kev qhia nrog dawb variables. Nov yog qhov kev daws teeb meem ntawm SLAE.
Koj tuaj yeem pom cov kev daws teeb meem yooj yim ntawm lub kaw lus - muab qhov hloov pauv dawb txhua qhov txiaj ntsig, thiab tom qab ntawd suav cov txiaj ntsig ntawm qhov hloov pauv yooj yim rau qhov tshwj xeeb no. Muaj ntau yam kev daws teeb meem tsis kawg.
Kev daws nrog cov piv txwv tshwj xeeb
Ntawm no yog qhov kev sib npaug.
Rau qhov yooj yim, nws yog qhov zoo dua los ua nws cov matrix tam sim ntawd
Nws paub tias thaum daws los ntawm Gauss txoj kev, qhov sib npaug sib npaug rau thawj kab yuav tsis hloov pauv thaum kawg ntawm kev hloov pauv. Yog li ntawd, nws yuav muaj txiaj ntsig ntau dua yog tias sab laug sab laug ntawm matrix yog qhov tsawg tshaj plaws - tom qab ntawd thawj cov ntsiab lustus so ntawm kab tom qab kev ua haujlwm yuav tig mus rau xoom. Qhov no txhais tau hais tias nyob rau hauv lub compiled matrix nws yuav muaj txiaj ntsig los muab cov kab thib ob nyob rau hauv qhov chaw thib ib.
Tom ntej no, koj yuav tsum hloov kab thib ob thiab thib peb kom thawj cov ntsiab lus dhau los ua xoom. Txhawm rau ua qhov no, ntxiv lawv rau thawj tus, muab faib los ntawm tus coefficient:
second line: k=(-a21/a11)=(-3/1)=-3
a'21 =a21 + k×a11=3 + (-3) × 1=0
a'22 =a22 + k×a12 =-1 + (- 3) × 2=-7
a'23 =a23 + k×a13 =1 + (-3) × 4=-11
b'2 =b2 + k×b1=12 + (-3) × 12=-24
thib peb kab: k=(-a31/a11)=(- 5/1)=-5
a'31 =a31 + k×a11=5 + (-5)×1=0
a'32 =a32 + k×a12 =1 + (-5)×2=-9
a'33 =a33 + k×a13 =2 + (-5) × 4=-18
b'3=b3 + k×b1=3 + (-5) × 12=-57
Tam sim no, txhawm rau kom tsis txhob tsis meej pem, koj yuav tsum sau cov matrix nrog cov txiaj ntsig nruab nrab ntawm kev hloov pauv.
Pom tseeb, xws li matrix tuaj yeem ua kom nyeem tau ntau dua nrog kev pab ntawm qee qhov haujlwm. Piv txwv li, koj tuaj yeem tshem tag nrho "minuses" los ntawm kab thib ob los ntawm kev muab txhua lub ntsiab lus los ntawm "-1".
Nws tseem tsim nyog sau cia tias hauv kab thib peb txhua lub ntsiab lus yog qhov sib npaug ntawm peb. Ces koj ua tautxiav txoj hlua los ntawm tus lej no, muab txhua lub ntsiab lus los ntawm "-1/3" (muab - tib lub sijhawm kom tshem tawm qhov tsis zoo).
Zoo zoo dua. Tam sim no peb yuav tsum tau tawm ntawm ib leeg thawj kab thiab ua haujlwm nrog thib ob thiab thib peb. Txoj hauj lwm yog ntxiv kab thib ob mus rau kab thib peb, muab faib ua qhov zoo li qhov uas lub caij a32 ua xoom.
k=(-a32/a22)=(-3/7)=-3/7 (yog tias thaum muaj qee qhov kev hloov pauv nyob rau hauv cov lus teb tau muab tsis tau ib tug integer, nws yog pom zoo kom tso nws "raws li yog", nyob rau hauv daim ntawv ntawm ib tug zoo ib feem, thiab tsuas yog tom qab ntawd, thaum cov lus teb tau txais, txiav txim siab seb puas yuav hloov mus rau lwm daim ntawv. notation)
a'32=a32 + k×a22=3 + (-3 /7) × 7=3 + (-3)=0
a'33=a33 + k×a23=6 + (-3 /7) × 11=-9/7
b'3 =b3 + k×b2=19 + (-3 /7) × 24=-61/7
Matrix yog sau dua nrog cov nqi tshiab.
| 1 | 2 | 4 | 12 |
| 0 | 7 | 11 | 24 |
| 0 | 0 | -9/7 | -61/7 |
Raws li koj tuaj yeem pom, cov txiaj ntsig matrix twb muaj ib daim ntawv stepped. Yog li ntawd, kev hloov pauv ntxiv ntawm qhov system los ntawm txoj kev Gauss tsis tas yuav tsum tau ua. Yuav ua li cas ntawm no yog tshem tawm tag nrho cov coefficient "-1/7" los ntawm kab thib peb.
Tam sim no sawv dawszoo. Lub ntsiab lus me me - sau cov matrix dua nyob rau hauv daim ntawv ntawm ib qho kev sib npaug thiab xam cov hauv paus hniav
x + 2y + 4z=12 (1)
7y + 11z=24 (2)
9z=61 (3)
Cov algorithm uas cov hauv paus hniav tam sim no yuav pom yog hu ua qhov thim rov qab hauv txoj kev Gauss. Equation (3) muaj tus nqi z:
z=61/9
Tom ntej, rov qab mus rau qhov sib npaug thib ob:
y=(24 - 11 × (61/9))/7=-65/9
Thiab thawj kab zauv tso cai rau koj nrhiav x:
x=(12 - 4z - 2y) / 1=12 - 4 × (61/9) - 2 × (-65/9)=-6/9=-2/3
Peb muaj cai hu rau qhov kev sib koom ua ke, thiab txawm tias qhov tseeb, uas yog, muaj kev daws teeb meem tshwj xeeb. Cov lus teb yog sau rau hauv daim ntawv hauv qab no:
x1=-2/3, y=-65/9, z=61/9.
Piv txwv ntawm qhov tsis muaj qhov kawg
Qhov sib txawv ntawm kev daws qee qhov system los ntawm Gauss txoj kev tau txheeb xyuas, tam sim no nws yog qhov tsim nyog los txiav txim siab qhov teeb meem yog tias qhov system tsis muaj qhov kawg, uas yog, ntau yam kev daws teeb meem tuaj yeem nrhiav tau rau nws.
x1 + x2 + x3 + x 4+ x5=7 (1)
3x1 + 2x2 + x3 + x 4 - 3x5=-2 (2)
x2 + 2x3 + 2x4 + 6x 5 =23 (3)
5x1 + 4x2 + 3x3 + 3x 4 - x5=12 (4)
Daim ntawv ntawm lub system twb ceeb heev, vim hais tias tus naj npawb ntawm cov tsis paub yog n=5, thiab qeb ntawm lub system matrix twb raws nraim li tus naj npawb no, vim tus naj npawb ntawm kab yog m=4, uas yog, qhov loj tshaj plaws kev txiav txim ntawm square determinant yog 4. Yog li,Muaj ntau qhov kev daws teeb meem tsis kawg, thiab peb yuav tsum nrhiav nws daim ntawv dav dav. Txoj kev Gauss rau kab sib npaug tso cai rau koj ua qhov no.
Ua ntej, raws li ib txwm, cov augmented matrix yog muab tso ua ke.
Second kab: coefficient k=(-a21/a11)=-3. Hauv kab thib peb, thawj lub ntsiab lus yog ua ntej kev hloov pauv, yog li koj tsis tas yuav kov dab tsi, koj yuav tsum tau tawm ntawm nws raws li nws yog. 4 kab: k=(-a41/a11)=-5
Muab cov ntsiab lus ntawm thawj kab los ntawm txhua tus ntawm lawv cov coefficients nyob rau hauv lem thiab ntxiv rau cov kab uas yuav tsum tau, peb tau txais ib tug matrix ntawm daim ntawv hauv qab no:
Raws li koj tuaj yeem pom, kab thib ob, thib peb thiab plaub muaj cov ntsiab lus sib npaug rau ib leeg. Qhov thib ob thiab thib plaub feem ntau yog tib yam, yog li ntawd ib qho ntawm lawv tuaj yeem raug tshem tawm tam sim ntawd, thiab tus so tau muab faib los ntawm cov coefficient "-1" thiab tau txais kab 3. Thiab dua, tawm ib ntawm ob kab tib yam.
Qhov tshwm sim yog xws li matrix. Lub kaw lus tseem tsis tau sau cia, nws yog qhov tsim nyog ntawm no los txiav txim siab qhov hloov pauv yooj yim - sawv ntawm cov coefficients a11=1 thiab a22=1, thiab dawb - tag nrho cov so.
Tsuas muaj ib qho kev sib txawv hauv qhov sib npaug thib ob - x2. Li no, nws tuaj yeem hais tawm los ntawm qhov ntawd, sau los ntawm kev hloov pauv x3, x4, x 5, uas yog dawb xwb.
Hloov cov lus qhia ua rau thawj qhov sib npaug.
Nws tig tawm qhov sib npaug uasqhov yooj yim sib txawv yog x1. Wb ua ib yam li nrog x2.
Txhua qhov hloov pauv yooj yim, uas muaj ob, tau hais tawm ntawm peb qhov pub dawb, tam sim no koj tuaj yeem sau cov lus teb rau hauv daim ntawv dav dav.
Koj tuaj yeem hais qhia ib qho ntawm cov kev daws teeb meem tshwj xeeb ntawm lub kaw lus. Rau cov xwm txheej zoo li no, raws li txoj cai, xoom raug xaiv raws li qhov tseem ceeb rau cov hloov pauv dawb. Ces cov lus teb yuav yog:
-16, 23, 0, 0, 0.
Ib qho piv txwv ntawm qhov tsis sib xws
Kev daws qhov tsis sib xws ntawm cov kab sib npaug los ntawm txoj kev Gauss yog qhov ceev tshaj plaws. Nws xaus sai li sai tau ntawm ib theem ib qho kev sib npaug tau txais uas tsis muaj kev daws teeb meem. Ntawd yog, theem nrog kev suav ntawm cov hauv paus hniav, uas yog ntev heev thiab dreary, disappears. Cov hauv qab no raug txiav txim siab:
x + y - z=0 (1)
2x - y - z=-2 (2)
4x + y - 3z=5 (3)
Raws li ib txwm, cov matrix tau muab tso ua ke:
| 1 | 1 | -1 | 0 |
| 2 | -1 | -1 | -2 |
| 4 | 1 | -3 | 5 |
Thiab txo mus rau ib daim ntawv kauj ruam:
k1 =-2k2 =-4
| 1 | 1 | -1 | 0 |
| 0 | -3 | 1 | -2 |
| 0 | 0 | 0 | 7 |
Tom qab thawj qhov kev hloov pauv, kab thib peb muaj qhov sib npaug ntawm daim ntawv
0=7, no solution. Yog li ntawd, lub systemyog inconsistent, thiab cov lus teb yog lub khoob.
Qhov zoo thiab qhov tsis zoo ntawm txoj kev
Yog tias koj xaiv txoj kev twg los daws SLAE ntawm daim ntawv nrog tus cwj mem, ces txoj kev uas tau txiav txim siab hauv kab lus no zoo li ntxim nyiam tshaj plaws. Hauv kev hloov pauv hauv theem pib, nws nyuaj dua kom tsis meej pem dua li qhov tshwm sim yog tias koj yuav tsum tau manually nrhiav tus txiav txim siab los yog qee qhov kev sib txawv ntawm cov matrix. Txawm li cas los xij, yog tias koj siv cov kev pab cuam rau kev ua haujlwm nrog cov ntaub ntawv ntawm hom no, piv txwv li, cov ntawv nthuav qhia, tom qab ntawd nws hloov tawm tias cov kev pab cuam no twb muaj cov algorithms rau kev xam lub ntsiab ntawm cov matrices - tus txiav txim, cov menyuam yaus, rov qab thiab hloov matrices, thiab lwm yam.. Thiab yog tias koj paub tseeb tias lub tshuab yuav xam cov nqi no nws tus kheej thiab yuav tsis ua yuam kev, nws yog qhov tsim nyog los siv txoj kev matrix lossis Cramer cov qauv, vim tias lawv daim ntawv thov pib thiab xaus nrog kev suav cov determinants thiab inverse matrices.
Daim ntawv thov
Vim tias Gaussian tov yog ib qho algorithm, thiab lub matrix yog, qhov tseeb, ib tug ob-dimensional array, nws yuav siv tau nyob rau hauv lub programming. Tab sis txij li tsab xov xwm tso nws tus kheej ua ib qho kev qhia "rau dummies", nws yuav tsum tau hais tias qhov chaw yooj yim tshaj plaws los tso rau hauv yog cov ntawv nthuav qhia, piv txwv li, Excel. Ib zaug ntxiv, txhua SLAE nkag mus rau hauv ib lub rooj nyob rau hauv daim ntawv ntawm matrix yuav raug txiav txim siab los ntawm Excel raws li ob sab array. Thiab rau kev ua haujlwm nrog lawv, muaj ntau ntau cov lus txib zoo: ntxiv (koj tuaj yeem ntxiv cov matrices ntawm tib qhov loj!), Kev sib tw los ntawm tus lej, matrix multiplication (tseem nrog.qee qhov kev txwv), nrhiav qhov hloov pauv thiab hloov pauv matrices thiab, qhov tseem ceeb tshaj plaws, suav cov determinant. Yog tias lub sijhawm siv sijhawm no tau hloov pauv los ntawm ib qho lus txib, nws nrawm dua los txiav txim siab qib ntawm matrix thiab, yog li ntawd, tsim kom muaj kev sib raug zoo lossis tsis sib xws.
