NSSE4  REMARKS T  GAUSSRE   GAUSSRED"  MATMULT ,  MATMULTD6  MATINV @  MATINVD J  GAUSSEL   GAUSSELD  FFT ^  5 GAUSS ELIMINATI PROGRAM N EQS IN N UNKNOWNS 8 ENTER THE AUGMENTED MATRIX OF COEFFICIENTS BY ROW 2 PUT\AUGMENTED MATRIX,UPPER DIAGAL MATRIX, 7( SOLUTI VECR, DETERMINANT OF THE N BY N MATRIX 2 "INPUT NO OF EQS : ",N < AN,N1) F E11E60 P D1 Z R1 N d "INPUT ROW NO",R,":", n C1 N1 x 1 AR,C)  C\\ R "THE AUGMENTED MATRIX IS:" - I1 N\ J1 N1\AI,J),\ J\\ I + WARD ELIMINATI, SEARCH PIVOT ROW C1\R1 PAR,C) P1R  IR N  AI,C))P)  PAI,C)\P1I  I  ERCHANGE ROWS  P)E1   P1R ,  DD  KC N1 *" BAP1,K)\AP1,K)AR,K)\AR,K)B\ K , NMALIZE PIVOT 6 DDP $@ KC N1\AR,K)AR,K)P\ K J RN  T COLUMN REDUCTI ^ I1R1 N\BAI1,C) h B)E1 | -r KC N1\AI1,K)AI1,K)AR,K)B\ K | I1  RR1\CC1\ $ "DET IS 0, NO UNIQUE SOLUTION"   BACK SUBSTITUTI % "THE UPPER DIAGONAL MATRIX IS:" - I1 N\ J1 N1\AI,J),\ J\\ I  K1 N1\ J1 K 4 ANK,N1)ANK,N1)ANK,N1J)AN1J,N1)  J\ K  "THE SOLUTION VECTOR IS:"  I1 N\AI,N1),\ I\  "THE DETERMINANT IS",D  @A/APAAP@$B A AFffgAdPA335@'I PA9A@A@Af wu\6j;0z t~2tHvvp0|~x~p0~vtr0>z2{zT|yY^@@v|fx@wVP2;zRr r ^z\f6vp4v_pr~trP@:~tr 6&6R6P0v<|0r0~|RP0R,d7~p27pP4w2pP2 ?k/!e㳗Gb!/{Ow@ 3'Aπ!aA`A ɭ͡ЀQ!P۽@တˇ- GAUSS ELIMINATI DOCUMENTATI   WRITTEN BY $( DAVID N. SAUNDERS )2 M.I.T. BRANCH P.O. BOX 227 '< CAMBRIDGE, MASS 02139 7F GAUSS ELIMINATI IS A PROCEDURE THE SOLUTI OF 5P A SYSTEM OF AR EQUATIS, WHERE THE NUMBER OF  ENTER THE ELEMENTS OF THE FIRST ROW OF THE FIRST MATRIX 7 THE SECD ROW, ETC, UNTIL ALL THE ELEMENTS ARE 1 ENTERED. DO THE SAME THE SECD MATRIX. 0" EACH ELEMENT IS FOLLOWED BY A CARRIAGE . ,, THE PRODUCT MATRIX IS COMPUTED ED. 6 PLE 'S DO @ "J 1 3 1 2 4 (T 2 1 0 X 1 3 ? "Y 0 1 ^ 9h THE FIRST MATRIX HAS ENSIS 2,3. THE SECD MATRIX 6r HAS ENSIS 3,2. THE PRODUCT MATRIX HAS ENSIS | 2,2.  THE PRODUCT MATRIX IS   5 14  5 11  1 FURTHER INMATI MATRIX MULTIPLICATI, 7 CSULT ANY STARD TEXT ELEMENTARY AR ALGEBRA.  TWO ARE  6 AR ALGEBRA ITS APPLICATIS BY GILBERT STRANG , MIT), ACADEMIC PRESS,N.Y.,N.Y., 1976  0 ELEMENTARY AR ALGEBRA BY PAUL C. SHIELDS : WAYNE STATE UNIV), WTH PUBLSHERS, N.Y.,N.Y., 1968  1,K)AR,K)\AR,K)B\ K  NMALIZE PIVOT  DDP $ KC 2N\AR,K)AR,K)P\ K  COLUMN REDUCTI (" I11 N\ I1R T\BAI1,C) , B)E1 T 06 & MATRIX INVERSI N BY N MATRIX  THE MATRIX BY ROWS $ "INPUT SIZE OF THE MATRIX :",N ( AN,2N) 2 E11E60 7 D1 < R1 N F "INPUT ROW NO",R,":", P C1 N Z 1 AR,C) d C\\ R n "THE MATRIX IS:" +x I1 N\ J1 N\AI,J),\ J\\ I %z I1 N\ JN1 2N\AI,J)0 | INJ AI,J)1 ~ J\ I  SEARCH PIVOT ROW C1\R1 PAR,C) P1R  IR N  AI,C))P)  PAI,C)\P1I  I  ERCHANGE ROWS  P)E1 r  P1R  DD  KC 2N * BAP1,K)\AP1,K)AR,K)\AR,K)B\ K  NMALIZE PIVOT  DDP $ KC 2N\AR,K)AR,K)P\ K  COLUMN REDUCTI (" I11 N\ I1R T\BAI1,C) , B)E1 T 06 KC 2N\ KC AI1,K)0\ KC J @ AI1,K)AI1,K)AR,K)B J K T I1 ^ RN  h RR1\CC1\ $r "DET IS 0, NO UNIQUE SOLUTION" | "THE INVERSE MATRIX IS:" / I1 N\ JN1 2N\AI,J),\ J\\ I  "THE DETERMINANT IS : ",D  W1XXWW XX)XWXW A/KA@AAI  A A9X0APA!Xfffp@ APA333 Y.,N.Y., 1968  P1,K)AR,K)\AR,K)B\ K  NMALIZE PIVOT  DDP $ KC 2N\AR,K)AR,K)P\ K  COLUMN REDUCTI (" I11 N\ I1R T\BAI1,C) , B)E1 T 06 , MATRIX INVERSI DOCUMENTATI   WRITTEN BY $( DAVID N. SAUNDERS )2 M.I.T. BRANCH P.O. BOX 227 '< CAMBRIDGE, MASS 02139 8F THIS PROGRAM DETERMINES THE INVERSE OF ANY SQUARE 2P MATRIX. NSQUARE MATRICES HAVE NO INVERSE. Z 'S FIND THE INVERSE OF d n 1 0 2 x 0 1 1  1 0 1 6 THE PROGRAM FIRST ASKS THE SIZE OF THE MATRIX 4 I.E., THE ENSI). IN THIS CASE WE ANSWER 3. 8 THE PROGRAM ASKS THE COEFFICIENTS ROW BY ROW. ; THE ELEMENTS OF THE FIRST ROW ARE ENTERED, OF THE < SECD ROW, ETC. EACH NUMBER IS FOLLOWED BY A CARRIAGE  . SO WE GO < 1CR),0CR),2CR),0CR),1CR),1CR),1CR),0CR),1CR) ; AFTER THE LAST ELEMENT IS ENTERED THE INVERSE MATRIX 5 IS COMPUTED. A MATRIX OF ENSI N, THERE ARE ! NN ELEMENTS BE ENTERED. 6 THE DETERMINANT OF THE MATRIX IS ALSO COMPUTED. : CALCULATE THE INVERSE,THE PROGRAM SETS UP A SECD 4 MATRIX WITH N ROWS 2N COLUMNS. THE LEFTH 9 N COLUMNS AIN THE MATRIX BE INVERTED. THE RIGHT 9 N COLUMNS AIN AN N BY N IDENTITY MATRIX THIS HAS :" 1'S THE DIAGAL WITH ALL OFFDIAGAL ELEMENTS 0). 8, THIS N BY 2N MATRIX IS OPERATED BY GAUSSJDAN 96 REDUCTI. THE LEFTH N COLUMNS BECOME AN IDENTITY 6@ MATRIX. THE RIGHTH N COLUMNS AIN THE INVERSE J OF THE IGINAL MATRIX. )T THE INVERSE OF THE PLE MATRIX IS ^ 1 0 2 h 1 1 1 r 1 0 1 |  THE DETERMINANT IS 1 / FURTHER INMATI MATRIX INVERSI . PLES, CSULT ANY TEXT ELEMENTARY AR  ALGEBRA. TWO ARE 8 AR ALGEBRA ITS APPLICATIS BY GILBERT STRANG . MIT), ACADEMIC PRESS, N.Y.,N.Y. 1976  2 ELEMENTARY AR ALGEBRA BY PAUL C. SHIELDS = WAYNE STATE UNIV.), WTH PUBLISHERS, N.Y.,N.Y. 1968 LISHERS, N.Y.,N.Y. 1968 WAYNE STATE UNIV.), WTH P), ACADEMIC PRESS, N.Y.,N.Y. 1976  2 ELEMENTARY AR ALGEBRA BY PAUL C. SHIELDS = WAYNE STATE UNIV.), WTH PUBLISHERS, N.Y.,N.Y. 1968 STATE UNIV.), WTH PUBLISHERS, N.Y.,N.Y. 1968  PUBL 9 EXCEPT THE DIETE FOURIER TRANSM PROGRAM "FFT", 2 THIS IS A COLLECTI OF AR ALGEBRA PROGRAMS - WRITTEN BY DAVID N. SAUNDERS OF M.I.T. .( EACH ACCOMPANIED BY A DOCUMENTATI FILE ?2 WHOSE NAME IS THE NAME OF THE PROGRAM FOLLOWED BY A "D". 5, NO. SAMPLES 16 d F116), F216) n N 1 16 4x T .375N1) \ F1N) T)T) \ F2N) 0  N  K 4 \  6 M 1 16 \ %4I, M, %9F4, F1M), F2M) \ 6 THIS PROGRAM IS DERIVED FROM THE TRAN PROGRAM  PRESENTED IN APPIX B OF * DIGITAL SIGNAL ANALYSIS ( BY '2 SAMUEL D. STEARNS %< HAYDEN, 1975) F TEST DRIVER FFT RINE )P COMPUTES DFT OF FT) T)T) 'Z TIME .375, NO. SAMPLES 16 d F116), F216) n N 1 16 4x T .375N1) \ F1N) T)T) \ F2N) 0  N  K 4 \  6 M 1 16 \ %4I, M, %9F4, F1M), F2M) \ M  - M IS EQUIENT THE TRAN MOD FUNCTI  MX,Y)  X0 X XY)Y \ X0  8 FFT IS FAST FOURIER TRANSM UG TIME DECOMPOSITI  WITH BIT REVERSAL 0 IS IN F1 REAL) F2 IMAGINARY) ARRAYS / COMPUTATI IS IN PLACE PUT REPLACES ) # NUMBER OF POS MUST BE N2K 3 F1 F2 MUST BE ENSIED IN THE MAIN PROGRAM $ N 2 K \ M1 0 \ N0 N 1  M 1 N0 " L N ', L L 2) \ M1L N0 , )6 M1 MM1,L) L \ M1 M T 6@ T1 F1M1) \ F1M1) F1M11) \ F1M11) T1 6J T2 F2M1) \ F2M1) F2M11) \ F2M11) T2 T M ^ L 1 h L N r I3 2 L | L0 L  M 1 L  A 3.1415927 1M) L0  W1 A) \ W2 A) ! I M N I3 \ J I L ! T1 W1 F1J) W2 F2J) ! T2 W1 F2J) W2 F1J) , F1J) F1I) T1 \ F2J) F2I) T2 , F1I) F1I) T1 \ F2I) F2I) T2  I \ M \ L I3 \ h  \[\\c\\[S\[\\