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.PO 8 .MB 15 .PN 1 .HE -#- .OP `14 EXTRACALC-1 Release 1.2 ============================CONTENTS============================= 1. INTRODUCTION 2 2. SETTING UP ExtraCalc-1 SOFTWARE 3 a. Distribution Diskettes 4 b. System Configuration - Single Density 5 c. System Configuration - Double Density 8 3. PRINCIPLES OF ExtraCalc-1 OPERATION. DEFINITIONS 10 a. System Flowchart and Operation. 10 b. Definitions of Matrix Operands 12 c. Matrices. Definitions of Operations. 13 Determination of Operand sizes. 4. OPERATING INSTRUCTIONS 18 a. Manual Operation 18 b. Using SuperCalc's XQT file≤ 19 c« Usinτ Programmablσ Keys and XQT files 20 d. Automatic Operation 21 5. NUMERIC EXAMPLES 22 6. COMPOSITE OPERATIONS. APPLICATIONS 36 a. Least Squares Technique 36 b. Numeric Example: Dow Jones Regression model 37 7. ExtraCalc-1 ERROR MESSAGES 51 ================================================================= .PAè1. INTRODUCTION ExtraCalc-1TMùáá i≤á aεá add-oεá t∩á thσá SuperCalcTMùáá electroniπ ì spreadshee⌠á prograφ whicΦ add≤ matri° operation≤ t∩á SuperCalc'≤ ì rangσáá oµá operatioεá unde≥á CP/MTMùá operatinτáá system«áá Thesσ ì operation≤ arσ 1 - Transposition along main diagonal 2 - Transposition along secondary diagonal 3 - Reflection in a row 4 - Reflection in a column 5 - Inversion (including calculation of determinant) ╢ - Findinτ Eigenvalue≤ anΣ Eigenvector≤ oµ symmetriπ matrices 7 - Solution of system of linear equations 8 - Addition of general matrices 9 - Subtraction of general matrices 10 - Multiplication of general matrices T∩á initiatσ onσ oµ thσ abovσ operation≤ thσ use≥ ha≤ t∩ specif∙ ì entr∙áá rangσá anΣá operatioεá type«áá Afte≥áá tha⌠áá ExtraCalc-▒ ì automaticall∙á transfer≤á datß froφ electroniπ speadshee⌠ t∩á thσ ì datß processinτ softwarσ (.CO═ type)¼á perform≤ computations¼ anΣ ì return≤ bacδ t∩ thσ spreadshee⌠ anΣ load≤ result≤ oµá computatioε ì int∩ ß designateΣ area« Thσ abovσ basicÖ operation≤ allo≈ thσ use≥ ì t∩á perforφ compositσ matri° operation≤ usinτ separatσ steps«á A⌠ ì thσá enΣ oµ thi≤ guidσ wσ illustratσ usσ oµ thσ packagσá fo≥á Do≈ ì Jone≤ inde° forecastinτ usinτ Leas⌠ Square≤ Technique« Afte≥áá installatioεá ExtraCalc-▒á doe≤á no⌠á becomσá ßá par⌠á oµ ì Supercalπá bu⌠á remain≤á ß separatσá entit∙á consistinτá oµá fou≥ ì permanen⌠á program≤ anΣ numbe≥ oµ permanen⌠ anΣ transien⌠á files« ì Becausσá i⌠á i≤ no⌠ ß templateÖ bu⌠ ß systeφ oµá machinσá languagσ ì file≤á (programs)¼á i⌠ allow≤ t∩ havσ al∞ oµ SuperCalc'≤ interna∞ ì memor∙ fo≥ speadshee⌠ calculation≤ and/o≥ templates«á ExtraCalc-▒ ì work≤á witΦá AN┘á sizσ oµ workshee⌠á tha⌠á SupecCalπá caεá handlσ ì withou⌠ occupyinτ ß singlσ bi⌠ oµ valuablσ speadshee⌠ memory« ExtraCalc-▒á i≤á thσá firs⌠ prograφ iεá thσá ExtraCalc-nÖá series« ì Futurσá release≤á oµá ExtraCalc-2TM¼áá ExtraCalc-3TM¼áá etc«á arσ ì intendeΣá t∩á supplemen⌠á SuperCalπ iε area≤á othe≥á thaεá Matri° ì Algebra«á Thσá labe∞ oε you≥ ExtraCalc-▒ mus⌠ matcΦ you≥ compute≥ ì systeφ anΣ thσ releasσ numbe≥ oµ you≥ SuperCalc«á Iµ not¼á ge⌠ iε ì toucΦá witΦ you≥ deale≥ o≥ witΦ Smirno÷á Associates¼ (617⌐964-6607. TMùá SuperCalπ i≤ thσ registereΣ trademarδ oµ Sorcim¼á CP/═ i≤ thσ ì registereΣ trademarδ oµ Digita∞ ResearcΦ Inc.¼á anΣá ExtraCalc-1¼ ì ExtraCalc-2¼áá etc«áá arσáá registereΣáá trademark≤á oµáá Smirno÷ ì AssociatesR.è2. SETTING UP ExtraCalc-1 SOFTWARE Wσá recommenΣ tha⌠ yo⌡ makσ backup≤ oµ al∞ distributioε diskette≤ ì immediatelyÖá t∩á avoiΣ accidenta∞ los≤ o≥ damagσá oµá ExtraCalc-▒ ì files (consult user guide for your system)« Please¼á notσ tha⌠ therσ i≤ n∩ systeφ oε thσ systeφ track≤ oµ thσ ì ExtraCalc-▒áá distributioεá diskettes«áá D∩á no⌠á boo⌠á t∩á thesσ ì diskette≤ !!í D∩ no⌠ exi⌠ froφ you≥ COP┘ o≥ PI╨ utilit∙ whilσ thσ ì distributioε diskettσ i≤ iε drivσ A«á Doinτ eithe≥ wil∞ producσ ß ì screeε ful∞ oµ garbagσ anΣ possibl∙ overwritσ diskette. Thσ ExtraCalc-▒ distributioε diskette/diskette≤ ma∙ bσ iεá singlσ ì density«á Iµ yo⌡ arσ usinτ doublσ density¼á makσ ß doublσ densit∙ ì copy«á Afte≥á yo⌡á havσ madσ thσ copy¼á pu⌠ ß cop∙ oµá you≥á CP/═ ì systeφ oε it≤ systeφ tracks« .PAè a. Distribution Diskettes Diskettσá o≥á diskette≤ tha⌠ yo⌡ havσ receiveΣ witΦá thi≤á manua∞ ì contaiεá thσá followinτ file≤ (sizσ i≤ giveε fo≥á singlσá densit∙ ì versioε - roundeΣ t∩ highe≥ eveε #) OP.CO═ (8k)é i≤ thσ Matri° Operation≤ Manage≥ program«á I⌠á serve≤ ì a≤á interfacσ betweeε SuperCalπ anΣ thσ numbe≥ processinτ par⌠ oµ ì ExtraCalc-1« Thi≤ prograφ als∩ support≤ selectioε oµ ß particula≥ ì matri° operatioε b∙ user. OP04.CO═ (32k)é i≤ thσ firs⌠ oµ threσ number-crunchingÖ program≤ oµ ì ExtraCalc-1« I⌠ perform≤ operations 1 - Transposition along main diagonal 2 - Transposition along secondary diagonal 3 - Reflection in a row 4 - Reflection in a column 8 - Addition 9 - Subtraction 10 - Multiplication remark: ## correspond to ## in program module OP13.CO═á (36k)éá i≤á thσá seconΣ oµá thσá abovσá mentioneΣá threσ ì programs« I⌠ perform≤ operation≤ of ╡á - Inversioεá(includinτ calculatioεáoµ determinant) 7 - Solution of a system of linear equations OP2.CO═ (40k)é i≤ thσ las⌠ onσ oµ three« I⌠ perform≤ operation ╢á - Findinτáof Eigenvalue≤áanΣ Eigenvectors of symmetric matrices MAT1.PRN¼á MAT2.PRN¼ MAT3.PR╬ (0δ each)é arσ Matri° OperandÖ files« ì The∙ currentl∙ don't contaiε anything. $.▒á (0k)é i≤ temporar∙ defaul⌠ SuperCalπ (.CAL⌐ typσ file«á I⌠ i≤ ì als∩ initiall∙ set a≤ zer∩ file. SAV.XQ╘á (2k)é i≤ thσ SuperCalπ eXecutσ (.XQT⌐ filσ tha⌠á prepare≤ ì speadshee⌠á anΣ defaul⌠ $.▒ filσ iε preparatioε fo≥á transfe≥á t∩ ì ExtraCalc-1. RES.XQ╘á (2k)é i≤ SuperCalπ eXecutσ filσ tha⌠ control≤ loadinτá oµ ì result≤ oµ computatioε froφ ExtraCalc-1.è b. ExtraCalc-1 Configuration - Single Density T∩ proceed¼á yo⌡ wil∞ neeΣ fou≥ blanδ diskettes« Wσ wil∞ cal∞ thσ ì firs⌠á - Maste≥ diskettσ (D1⌐ anΣ seconΣ througΦ fourtΦ - D2¼á D│ ì anΣ D┤ diskettes. 1«á Placσá you≥á CP/═á diskettσ iε drivσ ┴ anΣá thσá firs⌠á blanδ ì formatteΣá diskettσá (D1⌐á iε drivσá B«á Pres≤á RESE╘á anΣ ì carriagσ returε <CR╛ (o≥ d∩ othe≥ appropriatσ step≤ t∩ invokσ ì CP/═ - consul⌠ you≥ systeφ use≥ manual) 2. When you see A> prompt, type PIP<CR>. 3«á Wheεá yo⌡áseσ ¬ promp⌠, placσ firs⌠ distributioε diskettσá iε ì drive A and type *B:=A:OP.COM<CR> *B:=A:RES.XQT<CR> *B:=A:SAV.XQT<CR> note║á firs⌠ characte≥ (*⌐ iε abovσ threσ line≤ i≤ PI╨ prompt« ì D╧ NO╘ TYP┼ I╘ IN. 4«á Placσ diskettσ containinτ SuperCalπ (v1.1▓ anΣ up⌐ iε drivσ ┴ ì and type *B:=A:SC.*<CR>ì note║ S├ i≤ thσ namσ oµ SuperCalπ oε you≥ diskette 5«á Iµá yo⌡á havσ SUBMIT.CO═ o≥ simila≥ batcΦ processinτá utilit∙ ì placσ diskettσ containinτ i⌠ int∩ drivσ ┴ aεd type *B:=A:SUBMIT.COM 6«á Next¼á cop∙á CP/═ ont∩ you≥ Maste≥ disδ b∙ usinτá thσá SYSGE╬ ì utilit∙á (o≥á similar⌐ oε you≥ CP/═ disk«á T∩ d∩á this,inser⌠ ì you≥á CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσ A╛ ì prompt¼ type A>SYSGEN<CR> note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. 7. Put away your Master disk now (D1)è8«á Placσá you≥á CP/═á diskettσá iε drivσá ┴á anΣá seconΣá blanδ ì formatteΣá diskettσá (D2⌐ iε drivσ B«á Pres≤ ^├ o≥á d∩á othe≥ ì appropriatσ step≤ t∩ REINITIALIZ┼ CP/═ - consul⌠ you≥á systeφ ì use≥ manual. 9. When you see A> prompt, type PIP<CR>. 10«áWheεá yo⌡ seσ ¬ promp⌠ placσ firs⌠ distributioε diskettσá iε ì drivσ ┴ anΣ type *B:=A:OP04.COM<CR> *B:=A:*.PRN<CR> *B:=A:$.1<CR> 11«áNext¼á cop∙á CP/═ ont∩ you≥ D▓ disδ b∙ usinτá thσá SYSGE╬ ì utilit∙á (o≥ similar⌐ oε you≥ CP/═ disk«á T∩ d∩ this¼á inser⌠ ì you≥á CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσ A╛ ì prompt¼ type A>SYSGEN<CR> note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. 12. Put away your D2 disk. 13« Placσ you≥ CP/═ diskettσ iε drivσ ┴ anΣ third blanδ formatteΣ ì diskettσá (D3⌐ iε drivσ B«á Pres≤ ^├ o≥ d∩ othe≥á appropriatσ ì step≤ t∩ REINITIALIZ┼ CP/═ - consul⌠ you≥ systeφ use≥ manual. 14. When you see A> prompt, type PIP<CR>. 15«áWheεá yo⌡ seσ ¬ promp⌠ placσ firs⌠ distributioε diskettσá iε ì drivσ ┴ anΣ type *B:=A:OP13.COM<CR> *B:=A:*.PRN<CR> *B:=A:$.1<CR> 16«áNext¼á cop∙á CP/═ ont∩ you≥ D│ disδ b∙ usinτ thσá SYSGE╬ ì utilit∙ (o≥ similar⌐ oε you≥ CP/═ disk«á T∩ d∩á this¼á inser⌠ ì you≥ CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσá A╛ ì prompt¼ type A>SYSGEN<CR>è note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. 17. Put away your D3 disk. 18« Placσ you≥ CP/═ diskettσ iε drivσ ┴ anΣ fourth blanδ formatteΣ ì diskettσá (D4⌐ iε drivσ B«á Pres≤ ^├ o≥ d∩ othe≥á appropriatσ ì step≤ t∩ REINITIALIZ┼ CP/═ - consul⌠ you≥ systeφ use≥ manual. 19. When you see A> prompt, type PIP<CR>. 20«áWheεá yo⌡ seσ ¬ promp⌠ placσ seconΣ distributioε diskettσ iε ì drivσ ┴ anΣ type *B:=A:OP2.COM<CR> *B:=A:*.PRN<CR> *B:=A:$.1<CR> 21«áNext¼á cop∙á CP/═ ont∩ you≥ D┤ disδ b∙ usinτá thσá SYSGE╬ ì utilit∙á (o≥ similar⌐ oε you≥ CP/═ disk«á T∩ d∩ this¼á inser⌠ ì you≥á CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσ A╛ ì prompt¼ type A>SYSGEN<CR> note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. 22«áPu⌠á awa∙á you≥ D┤ disk«á You≥á ExtraCalc-▒á SINGL┼á densit∙ ì configuratioε i≤ no≈ complete« .PAè c. System Configuration - Double Density T∩ proceed¼á yo⌡ wil∞ neeΣ tw∩ blanδ diskettes«á Wσ wil∞ cal∞ thσ ì firs⌠ - Maste≥ diskettσ (D1⌐ anΣ thσ seconΣ D2. 1«á Placσ you≥ CP/═ diskettσ iε drivσ ┴ anΣ firs⌠ blanδ formatteΣ ì diskettσá (D1⌐á iε drivσ B«á Pres≤ RESE╘ anΣ carriagσá returε ì <CR╛ (o≥ d∩ othe≥ appropriatσ step≤ t∩ invokσ CP/═á - consul⌠ ì you≥ systeφ use≥ manual) 2. When you see A> prompt, type PIP<CR>. 3«á Wheεáyo⌡á seσ ¬ promp⌠, placσ firs⌠ distributioε diskettσá iε ì drive A and type *B:=A:*.COM<CR> *B:=A:RES.XQT<CR> *B:=A:SAV.XQT<CR> note║ firs⌠ characte≥ (*⌐ iε abovσ threσ line≤ i≤ PI╨ prompt« ì D╧ NO╘ TYP┼ I╘ IN. 4«á Placσ seconΣ distributioε diskettσ (witΦ OP2.COM⌐ iε drivσ ┴ ì (iµ yo⌡ receiveΣ onl∙ onσ diskettσ iε doublσ densityÖ leavσ i⌠ ì iε drivσ A:⌐ anΣ type *B:=A:*.COM 5«á Placσ diskettσ containinτ SuperCalπ (v1.1▓ anΣ up⌐ iε drivσ ┴ ì and type *B:=A:SC.*<CR>ì note║ S├ i≤ thσ namσ oµ SuperCalπ oε you≥ diskette 6«á Iµ yo⌡ havσ SUBMIT.CO═ o≥ a simila≥ batcΦ processinτáutilit∙, ì placσ diskettσ containinτ i⌠ int∩ drivσ ┴ aε type *B:=A:SUBMIT.COM 7«á Next¼á cop∙á CP/═ ont∩ you≥ Maste≥ disδ b∙ usinτ thσá SYSGE╬ ì utilit∙ (o≥ similar⌐ oε you≥ CP/═ disk«á T∩ d∩á this¼á inser⌠ ì you≥ CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσá A╛ ì prompt¼ type A>SYSGEN<CR>è note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. 8. Put away your Master disk now (D1) 9« Placσ you≥ CP/═ diskettσ iε drivσ ┴ anΣ seconΣ blanδ formatteΣ ì diskettσá (D2⌐ iε drivσ B«á Pres≤ ^├ o≥ d∩ othe≥á appropriatσ ì step≤ t∩ REINITIALIZ┼ CP/═ - consul⌠ you≥ systeφ use≥ manual. 10. When you see A> prompt, type PIP<CR>. 11«áWheεáyo⌡ seσ ¬ promp⌠, placσ firs⌠ distributioε diskettσá iε ì drivσ ┴ anΣ type *B:=A:*.PRN<CR> *B:=A:$.1<CR> 12«áNext¼á cop∙á CP/═ ont∩ you≥ D▓ disδ b∙ usinτ thσá SYSGE╬ ì utilit∙á (o≥ similar⌐ oε you≥ CP/═ disk«á T∩ d∩ this¼á inser⌠ ì you≥á CP/═ diskettσ iε drivσ ┴ anΣ pres≤ ^C«á Wheε yo⌡ seσ A╛ ì prompt¼ type A>SYSGEN<CR> note 1: Do not type in CP/M prompt A> Specify A as the source and B as the destination note 2║ Yo⌡ ma∙ als∩ wan⌠ t∩ usσ thσ SETU╨ utilit∙ (o≥ similar⌐ ì t∩ specif∙ thσ prope≥ interfacσ fo≥ you≥ printer. No≈á pu⌠á awa∙á you≥ D▓ disk«á You≥á ExtraCalc-▒á DOUBL┼á densit∙ ì configuratioε i≤ no≈ complete« .PAè4. PRINCIPLES OF ExtraCalc-1 OPERATION. DEFINITIONS a. System Flowchart and Operation. Flowchar⌠ oµ ExtraCalc-▒ i≤ showε iε Figurσ ▒ below« Blacδ arrow≤ ì stanΣá fo≥á connection≤ betweeε prograφ file≤ whilσá grayÖá arrow≤ ì sho≈á connection≤ betweeε datß file≤ anΣ programs«á Thσ arro≈á i≤ ì blacδá anΣá gra∙ betweeε SuperCalπ anΣ $.▒ filσ becausσá oµá thσ ì structurσá oµá SuperCalπá files«á Pleasσ notσá tha⌠á gra∙á (data⌐ ì connection≤ arσ eithe≥ unidirectiona∞ (operanΣ MAT1¼ operanΣ MAT│ ì - inpu⌠ o≥ outpu⌠ file≤ only⌐ o≥ bidirectiona∞ (operanΣ MAT2). T∩ operatσ ExtraCalπ yo⌡ shoulΣ alway≤ havσ you≥ Maste≥á diskettσ ì iεá drivσ ┴ anΣ DnÖ diskettσ iε drivσ ┬ (nÖ ╜ ▓ fo≥ doublσ density¼ ì o≥ 2¼ 3¼ ┤ fo≥ singlσ densit∙ versions)« Thσ numbe≥ nÖ (fo≥ singlσ ì density⌐ i≤ defineΣ b∙ thσ typσ oµ operation≤ i⌠ performs: n = 2 for operations ## 1-4 and ## 8-10 n = 3 for operations # 5 and # 7 n = 4 for operations # 6 Figure 1èInteractioεá witΦá ExtraCalπá usuall∙á begin≤á iεá thσá SuperCalπ ì environment«á Iεá spreadshee⌠á yo⌡á shoulΣá specif∙á you≥á matri° ì operand≤á - MAT1¼á MAT▓ anΣ maybσ MAT│ (seσ nex⌠ section)«á Afte≥ ì tha⌠á yo⌡á havσá t∩á leavσá SuperCalπá anΣá ente≥á (manuall∙áá o≥ ì automatically⌐á thσá firs⌠ prograφ oµ ExtraCalc-1¼á whicΦ b∙á thσ ì way¼ alway≤ reside≤ oε drivσ ┴ (OP.COM). Afte≥ loadinτ itself¼á OP.CO═ wil∞ displa∙ thσ maiε men⌡ a≤ showε ì below: MATRIX OPERATIONS 1) TRANSPOSITION (MAIN DIAGONAL) 2) TRANSPOSITION (SECONDARY DIAGONAL) 3) REFLECTION IN A COLUMN 4) REFLECTION IN A ROW 5) INVERSION 6) EIGENVALUES AND EIGENVECTORS 7) SOLUTION OF SYSTEM OF LINEAR EQUATIONS 8) ADDITION 9) SUBTRACTION 10) MULTIPLICATION 11) EXIT TO SUPERCALC ENTER YOUR CHOICE: Dependinτá oε thσ choice≤ yo⌡ make¼á OP.CO═ wil∞ routσ yo⌡á t∩á ß ì numbe≥á oµá differen⌠á programs¼á tha⌠á (durinτá execution⌐á wil∞ ì providσáá yo⌡á witΦá differen⌠á run-time¼áá diagnostiπá o≥á erro≥ ì messages« Example≤ oµ typica∞ message≤ showε below CHAINING TO OPERATION # n SIZING THE MATRIX ... THE MATRIX IS p\q SIZING THE MATRIX ... THE MATRIX IS P1\Q1 ERROR: WRONG OUTPUT MATRIX DIMENSIONS Afte≥á succesfu∞ computation≤ (n∩ ERRO╥ messages⌐ ExtraCalπá wil∞ ì returεá t∩ you≥ spreadshee⌠ (seσ sectioε oε automatiπá operation⌐ ì anΣ reaΣ iε result≤ int∩ you≥ spreadshee⌠ automatically«á Iε casσ ì oµá unsuccessfu∞á computation≤ (errors)¼á ExtraCalπ wil∞ iεá mos⌠ ì case≤á returεá yo⌡á t∩ thσ OP.CO═ prograφá anΣá asδá fo≥á furthe≥ ì instructions. .PAè b. Definitions of Matrix Operands ExtraCalc-▒ ha≤ provision≤ fo≥ threσ matri° operanΣ file≤ - MAT1¼ ì MAT▓ anΣ MAT3« OperanΣ file≤ arσ createΣ oε SuperCalπ leve∞ usinτ ì /Outpu⌠ commanΣ anΣ thereforσ havσ .PR╬ type. Thσá use≥á ha≤á t∩ creatσ tw∩ operanΣ file≤ (MAT▒ anΣá MAT2⌐á t∩ ì perforφáá unitaryÖáá operation≤áá (transpositions¼ááá reflections¼ ì inversion) MAT2 = operation {MAT1} o≥ threσ operanΣ file≤ (MAT1¼á MAT2¼ MAT3⌐ fo≥ binaryÖ operation≤ ì (addition¼ subtraction¼ multiplication¼ solutioε oµ systems) MAT3 = operation {MAT1, MAT2} anΣá fo≥á unitaryÖ operatioε witΦ binaryÖ outpu⌠á (eigenvalue≤á anΣ ì eigenvectors) {MAT2, MAT3} = operation {MAT1} Worksheet≤á iε SuperCalπ tha⌠ arσ useΣ t∩ derivσ matri°á operand≤ ì shoulΣ consis⌠ oµ number≤ witΦ o≥ withou⌠ underlyinτ formulae« N∩ ì BLANK≤á o≥á TEX╘ i≤ allowed«á BLANK≤ iεá workshee⌠á wil∞á producσ ì ExtraCalc-▒á ERRO╥ messagσ whilσ TEXT≤ wil∞ bσ interpreteΣ a≤ 0s« ì Dimension≤á oµá matrice≤á (size≤ oµ matri°á operands⌐á shoulΣá bσ ì consisten⌠ witΦ eacΦ othe≥ a≤ wel∞ a≤ witΦ matri° operatioε t∩ bσ ì performeΣ (seσ nex⌠ section). .PAè c. Matrices. Definitions of Operations. Determination of Operand sizes. Matrices:éá Herσ wσ formulatσ onl∙ basiπ definition≤ anΣá concept≤ ì oµ matri° algebrß tha⌠ werσ useΣ iε ExtraCalc-▒ design«á Iεá casσ ì use≥á doe≤á no⌠ completel∙ understanΣ thi≤ materia∞ o≥á need≤á t∩ ì kno≈ abou⌠ morσ advanceΣ concept≤ tha⌠ arσ mentioneΣ herσ withou⌠ ì explanation¼á wσá recommenΣ readinτ thσ firs⌠ fe≈ chapter≤ oµ an∙ ì booδ oε linea≥ o≥ matri° algebrß « ┴ matri° i≤ ß rectangula≥ arra∙ oµ term≤ calleΣ elements¼ sucΦ as 1 2 3 7 `` a11 a12 `` 3 4 0 4 or `` a21 a22 `` 5 6 7 -1 ` a31 a32 ` ┴á rea∞ matrixÖ anΣ ß comple° matrixÖ arσ matrice≤á whosσá element≤ ì arσá rea∞á number≤ o≥ comple° number≤á respectively«á ExtraCalc-▒ ì work≤á witΦá rea∞ matrice≤ only«á T∩á perforφá calculation≤á witΦ ì comple°á number≤ onσ shoulΣ usσ compositeÖ matri° operation≤á (seσ ì sectioε below)« Thσ orderÖ o≥ dimensionÖ oµ ß matri° i≤ giveε b∙ statinτ thσ numbe≥ ì oµá row≤ (N⌐ anΣ theε thσ numbe≥ oµ column≤ (M⌐ iε thσ matri°á a≤ ì N\M«á Therefore¼á thσá abovσá matrice≤ arσ oµ 3\┤ anΣá 3\▓á orde≥ ì respectively. ┴ squareÖ matri° i≤ ß matri° fo≥ whicΦ thσ numbe≥ oµ row≤ i≤ equa∞ ì t∩ thσ numbe≥ oµ columns« Thσ diagona∞ froφ thσ uppe≥ lef⌠ corne≥ t∩ thσ lowe≥ righ⌠ corne≥ ì i≤á thσ principalÖ o≥ mainÖ diagonal«á Thσ diagona∞ froφ thσá lowe≥ ì lef⌠ corne≥ t∩ thσ uppe≥ righ⌠ corne≥ i≤ thσ secondaryÖ diagonal« Thσ determinantÖ oµ ß squarσ matri° i≤ thσ determinan⌠ obtaineΣ b∙ ì considerinτ thσ arra∙ oµ element≤ iε thσ matri° a≤ ß determinant« ┴ squarσ matri° i≤ singularÖ iµ it≤ determinan⌠ i≤ equa∞ t∩á zero« ì Otherwisσ i⌠ i≤ nonsingular. ┴á diagonalÖá matri°á i≤á ß squarσ matri°á witΦá al∞á it≤á nonzer∩ ì element≤ iε thσ principa∞ diagonal« Aεá identityÖ (o≥ unit⌐ matri° i≤ ß diagona∞ matri° whosσ element≤ ì iε thσ principa∞ diagona∞ arσ al∞ unity. èDefinition≤ oµ Operations« Determinatioε oµ OperanΣ sizes. 1) TRANSPOSITION (MAIN DIAGONAL) unitary operation Thσ transposσ oµ ß matri° alonτ maiε diagona∞ i≤ thσ matri°á (AT⌐ ì resultinτá froφá interchanginτ thσ row≤ anΣ column≤ iε thσá giveε ì matri°á (A⌐á alonτá thσ diagona∞ drawε froφ to≡á lef⌠á corne≥á t∩ ì bottoφ righ⌠ corner« Iµ A = {ai,j} then AT = {aj,i), i=1,2,...N, j=1,2,...M Herσá ╬ i≤ numbe≥ oµ row≤ iε ┴ anΣ ═ i≤ numbe≥ oµ column≤á iεá A« ì Therefore¼á iµá MAT▒á i≤ N\═ theε matri° MAT▓ shoulΣá bσá oµá M\╬ ì dimension« MAT│ i≤ no⌠ requireΣ fo≥ thi≤ operation« 2) TRANSPOSITION (SECONDARY DIAGONAL) unitary operation Thσ transposσ oµ ß matri° alonτ secondar∙ diagona∞ i≤ thσá matri° ì (At⌐á resultinτá froφ interchanginτ thσ row≤ anΣ column≤á iεá thσ ì giveεá matri° (A⌐ alonτ thσ diagona∞ drawε froφ to≡ righ⌠á corne≥ ì t∩ bottoφ lef⌠ corner« Iµ A = {ai,j} then At = {aM-j+1,N-i+1), i=1,2,...N, j=1,2,...M Herσá ╬ i≤ numbe≥ oµ row≤ iε ┴ anΣ ═ i≤ numbe≥ oµ column≤á iεá A« ì Therefore¼á iµá MAT▒á i≤á N\═ theε matri° MAT▓ shoulΣ bσá oµá M\╬ ì dimension« MAT│ i≤ no⌠ requireΣ fo≥ thi≤ operation. 3) REFLECTION IN A COLUMN unitary operation Thσá reflectioεá oµá ßá matri° iε ß columεá i≤á thσá matri°á (AC⌐ ì resultinτ froφ interchanginτ thσ row≤ iε thσ giveε matri° (A)« If A = {ai,j} then AC = {aN-i+1,j), i=1,2,...N, j=1,2,...M Herσá ╬ i≤ numbe≥ oµ row≤ iε ┴ anΣ ═ i≤ numbe≥ oµ column≤á iεá A« ì Therefore¼á iµ MAT▒ i≤ N\═ matri° theε MAT▓ shoulΣ als∩ bσ oµ N\═ ì dimension« MAT│ i≤ no⌠ requireΣ fo≥ thi≤ operation. .PAè4) REFLECTION IN A COLUMN unitary operation Thσá reflectioε oµ ß matri° iε ß ro≈ i≤ thσ matri° (AR⌐ resultinτ ì froφ interchanginτ thσ column≤ iε thσ giveε matri° (A)« If A = {ai,j} then AR = {ai,M-j+1), i=1,2,...N, j=1,2,...M Herσá ╬ i≤ numbe≥ oµ row≤ iε ┴ anΣ ═ i≤ numbe≥ oµ column≤á iεá A« ì Therefore¼á iµ MAT▒ i≤ N\═ matri° theε MAT▓ shoulΣ als∩ bσ oµ N\═ ì dimension« MAT│ i≤ no⌠ requireΣ fo≥ thi≤ operation. 5) INVERSION and calculation of determinants unitary operation Fo≥á ß nonsingularÖ squarσ matri° (A)¼á thσ inversσ (A-1⌐á i≤á thσ ì quotien⌠á oµ thσ adjointÖ oµ thσ matri° anΣ thσ determinantÖ oµ thσ ì matrix« Iµ A-1ù i≤ thσ inversσ oµ A¼ theε produc⌠ AA-1ù ╜ A-1┴ ╜ I¼ ì wherσ ╔ i≤ thσ identit∙ matrix«á Thσ inversσ i≤ defineΣ onl∙á fo≥ ì nonsingula≥á squarσá matrices«á MAT▒ shoulΣ thereforσ bσá oµá N\╬ ì dimensioε anΣ havσ ß non-zer∩ determinant« MAT▓ i≤ als∩ N\N« Iε ß ì thσá coursσá oµá thi≤á operatioε thσ determinan⌠á oµá ┴á i≤á als∩ ì calculated¼á checkeΣá fo≥á non-zero¼á anΣ storeΣ fo≥ usσá iεá thσ ì spreadsheet. 6) Eigenvalues and Eigenvectors. unitary operation with binary result Fo≥ ß squarσ matri° ANxN¼á thσ eigenvaluσ i≤ ß scala≥ ∞ fo≥ whicΦ ì therσ i≤ ß nonzer∩ columε matri° ° ╜ {x1,x2,...,xN² anΣ fo≥ which A.x = l.x Thσá vecto≥ ° i≤ aε eigenvectorÖ o≥ Öá characteristicÖá vector«á Thσ ì matri°á ┴ caε havσ ╬ eigenvalue≤ tha⌠ arσ a⌠ thσ samσ timσá root≤ ì oµ characteristicÖ equation det |B[ lI - A |E] = 0 Characteristiπá root≤ arσ als∩ calleΣ latentÖá roots«á ExtraCalc-▒ ì caεá calculatσá botΦá thσ eigenvalue≤ anΣ thσ eigenvector≤á oµá ß ì symmetricÖá matri° A«á MAT▒ anΣ MAT│ shoulΣ bσ squarσ matrice≤á oµ ì N\╬á order«á MAT▓ shoulΣ havσ numbe≥ oµ element≤ greate≥ thaεá o≥ ì equa∞ t∩ N.è7) SOLUTION OF SYSTEM OF LINEAR EQUATIONS binary operation ┴á systeφá oµá simultaneousÖá linea≥á equation≤á i≤á ßá systeφá oµ ì equation≤ tha⌠ arσ linea≥ (oµ thσ firs⌠ degree⌐ iε thσ variables« ì Matri°á (A⌐á oµá coefficient≤ oµ ßá se⌠á oµá simultaneou≤á linea≥ ì equation≤á i≤á thσá rectangula≥á arra∙ lef⌠á afte≥á droppinτá thσ ì variable≤á froφá thσ equation≤ s∩ tha⌠ thσ coefficient≤á oµá likσ ì variable≤á arσ iε thσ samσ column≤ (zer∩ beinτ useΣ iµ ß terφá i≤ ì missing)« Iµ thσ systeφ oµ equation≤ is a11x1 + a12x2 + a13x3 + ... + a1MxM = d1 a21x1 + a22x2 + a23x3 + ... + a2MxM = d2 a31x1 + a32x2 + a33x3 + ... + a3MxM = d3 ........................................ aN1x1 + aN2x2 + aN3x3 + ... + aNMxM = dN then A = {ai,j} , i=1,2,3,...,N; j=1,2,3,...,M. Columε matri° (D⌐ oµ constan⌠ term≤ oµ thσ equation≤ above is D = {di}, i=1,2,3,...,N Thσ systeφ oµ linea≥ equatioε iε matri° forφ is¼ therefore¼ giveε ì by Ax = D wherσá ° ╜ {xi}¼á i=1,2,3,...,M«á AlthougΦ solutioε oµ thσ systeφ ì caεá als∩á bσ founΣ b∙ usinτ inversσ matri° A-1ù (a≤á °á ╜á A-1D)¼ ì ExtraCalc-▒ employ≤ ß differen⌠ procedurσ fo≥ solvinτ thσ system« ì Thσá methoΣ oµ solutioε i≤ b∙ elimination¼á usinτ larges⌠ pivota∞ ì divisor« EacΦ stagσ oµ eliminatioε consist≤ oµ interchanginτ row≤ ì wheε necessar∙ t∩ avoiΣ divisioε b∙ zer∩ o≥ smal∞ elements« T∩á avoiΣ error≤ MAT▒ (matri° A⌐ shoulΣ bσ oµá N\╬á order¼á whilσ ì MAT▓ (columε matri° D⌐ oµ N\1¼ anΣ MAT│ (vecto≥ x⌐ oµ N\▒ o≥ 1\N. .PAè8. ADDDITION binary operation Thσá suφá ┴á ½á ┬ oµ tw∩ matrice≤ ┴ anΣ ┬á i≤á thσá matri°á whosσ ì element≤á arσá formeΣ b∙ thσ rulσ tha⌠ thσ elemen⌠ iε ro≈á iÖá anΣ ì columεá jÖá i≤á thσ suφ oµ thσ element≤ aijù anΣ bijù iε ro≈á iÖá anΣ ì columε jÖ oµ ┴ anΣ B« Or¼ if A = {ai,j} and B = {bi,j} then A + B = {ai,j + bi,j} Thi≤ operatioε i≤ defineΣ onl∙ iµ ┴ anΣ ┬ havσ thσ samσ numbe≥ oµ ì row≤á anΣ thσ samσ numbe≥ oµ columns«á Thereforσ MAT1¼á MAT▓á anΣ ì MAT│ shoulΣ al∞ bσ oµ samσ dimensioε N\M. 9. SUBTRACTION binary operation Thσá differencσ ┴ - ┬ oµ tw∩ matrice≤ aµ matrice≤ ┴ anΣ ┬ i≤á thσ ì matri°á whosσ element≤ arσ formeΣ b∙ thσ rulσ tha⌠ thσ elemen⌠ iε ì ro≈ iÖ anΣ columε jÖ i≤ thσ differencσ oµ thσ element≤ aijù anΣá bijù ì iε ro≈ iÖ anΣ columε jÖ oµ ┴ anΣ B« Or¼ if A = {ai,j} and B = {bi,j} then A - B = {ai,j - bi,j} Thi≤ operatioε i≤ defineΣ onl∙ iµ ┴ anΣ ┬ havσ thσ samσ numbe≥ oµ ì row≤á anΣ thσ samσ numbe≥ oµ columns«á Thereforσ MAT1¼á MAT▓á anΣ ì MAT│ shoulΣ al∞ bσ oµ samσ dimensioε N\M. 10«áMULTIPLICATION binary operation Thσá produc⌠ A┬ oµ matrice≤ ┴ anΣ ┬ i≤ thσ matri° whosσá element≤ ì arσá determineΣ b∙ thσ rulσ tha⌠ thσ elemen⌠ cijù oµ matri° resul⌠ ì iεá ro≈á Θ anΣ columε Ω i≤ thσ suφ ove≥ kÖ oµ thσ produc⌠á oµá thσ ì elemen⌠ aikù iε ro≈ Θ anΣ columε kÖ oµ ┴ b∙ thσ elemen⌠ bkjù iεá ro≈ ì kÖ anΣ columε Ω oµ B: P C = ci,j = $#% aikbkj = A.B k=1 herσá i=1,2,...,N╗áá j=1,2,...,M╗á k=1,2,...,P«á Thσá produc⌠á i≤ ì defineΣá onl∙á iµá thσ numbe≥ ═ oµ column≤ iε ┴ i≤ equa∞á t∩á thσ ì numbe≥ oµ row≤ iε B«á Therefore¼ MAT▒ shoulΣ bσ oµ N\╨ dimension¼ ì MAT▓ oµ P\═ anΣ MAT│ oµ N\M«á Iε al∞ othe≥ situation≤ ExtraCalc-▒ ì wil∞ senΣ an erro≥ messagσ t∩ thσ terminal..PO 8è4. OPERATING INSTRUCTIONS a. Manual Operation Iεá thi≤á sectioε wσ describσ al∞ thσ entrie≤ t∩ perforφá onσá oµ ì abovσá matri°á operation≤ wheε yo⌡ d∩ no⌠ wan⌠ t∩á usσá SuperCalπ ì eXecutσá file≤ and/o≥ programmablσ keys«á Keyinτ oµ operatioεá i≤ ì ver∙á slo≈ anΣ tediou≤ iε thi≤ case«á I⌠ i≤ no⌠ expecteΣ tha⌠ thσ ì averagσá use≥á wil∞ emplo∙ thi≤ optioεá often«á Manua∞á entr∙á i≤ ì presenteΣáá herσáá t∩á providσá completσá understandinτá oµáá ho≈ ì ExtraCalc-▒ work≤ anΣ interact≤ witΦ SuperCalπ anΣ CP/M. Al∞ expanation≤ belo≈ arσ giveε t∩ thσ righ⌠ oµ ";"« Line≤ markeΣ ì witΦá *ùá iε explanationsÖ arσ no⌠ necessar∙ bu⌠á makσá runninτá oµ ì spreadshee⌠á and/o≥á program≤ smoother«á No≈ inser⌠á you≥á Maste≥ ì diskettσá iεá drivσá A║á anΣá DnÖ diskettσá iεá drivσá B:«á Invokσ ì SuperCalπ anΣ creatσ samplσ worksheet. note: 1> is the SuperCalc prompt ^ ; Position the cursor UP*. =A1<CR> ; Move cursor to top left corner*. /SB:$.1,B┴ ╗ Save curren⌠ workshee⌠ iε backu≡ file $.1. /CA1:BK254,A1,V ; Eliminate formulae and text in worksheet. =A1<CR> ; Move cursor to top left corner*. /OCn1:m1,DB:MAT1,B<CR> ; Create operand MAT1=n1:m1. /OCn2:m2,DB:MAT2,B<CR> ; Create operand MAT2=n2:m2. /OCn3:m3,DB:MAT3,B<CR> ; Create operand MAT3=n3:m3. /QY ; Leave SuperCalc. O╨<CR> ╗ Star⌠ firs⌠ prograφ oµ ExtraCalc« In ╗áresponsσ t∩ ExtraCalπ-1á promp⌠ ╗ use≥ wil∞ havσ t∩ ente≥ hi≤ choicσ ì ; of OPERATION (## 1:10) or EXIT code ╗áú 11«á ExtraCalc-▒ wil∞á terminatσ ì ╗áwitΦáSTO╨ámessage« A╛ i≤ CP/═ prompt SC<CR> ; Return to SuperCalc. /LB:$.1,A ; Load last version of spreadsheet. > ; Set cursor direction to the RIGHT. /GM ; Switch to manual recalculation. /XB:RES▒<CR> ╗áExecutσáfilσ RES1.XQ╘. It will load inì ; the result≤ oµ matri° operation. /XB:RES2<CR> ; Optional command used with operations # 5 ; and # 6. In case of # 5 it will load value ; of matrix determinant in the first row, just ; above the current worksheet« Wheε useΣ afte≥ ; operatioε ú ╢ i⌠ wil∞ loaΣ eigenvector≤ iε ; locatioε tha⌠ i≤ specified b∙ MAT3.è b. Using SuperCalc's XQT file≤ Significan⌠ numbe≥ oµ entrie≤ iε 4a« i≤ eliminateΣ iµ onσ employ≤ ì powerfu∞ SuperCalπ eXecutσ optioε (versioε 1.1▓ anΣ higher⌐ tha⌠ ì allow≤ yo⌡ t∩ ruε sequence≤ oµ SuperCalπ command≤á automatically« ì Wσá wil∞á translatσ thσ sequencσ oµ statement≤ describeΣá iεá 4a« ì usinτ tw∩ eXecutσ file≤ - RES.XQ╘ anΣ SAV.XQT. /XSAV<CR> /OCn1:m1,DB:MAT1,B<CR> /OCn2:m2,DB:MAT2,B<CR> /OCn3:m3,DB:MAT3,B<CR> /QY O╨ ; Choice of operation ## 1:10 or EXIT # 11. SC RES<CR> ; It is allowed to specify XQT filename when ; SuperCalc is invoked. /XB:RES2<CR> ; Optional for operations 5 and 6. Here file SAV.XQT consists of ^ =A1 /SB:$.1,B┴ /CA1:BK254,A1,V =A1 and file RES.XQT is /LB:$.1,A > /GM /XB:RES▒ì note║á N∩á space≤á arσ alloweΣ iε .XQ╘ file≤ afte≥á las⌠á (right⌐ ì characte≥ oε eacΦ oµ lines. .PAè c« Usinτ Programmablσ Keys and XQT files Usinτ XQ╘ file≤ alread∙ simplifieΣ ExtraCalc-▒ operatioε t∩ onl∙ ì seveε line≤ oµ entrie≤ pe≥ matri° operation« Herσ wσ arσ goinτ t∩ ì sho≈á thσ conveniencσ oµ usinτ programmablσ key≤ t∩ reducσ numbe≥ ì oµá character≤ iε eacΦ linσ oµ entries«á Wσ recommenΣ t∩ usσá thσ ì followinτ programminτ fo≥ you≥ keys 0: /XSAV<CR> 1: ,DB:MAT1,B<CR> 2: ,DB:MAT2,B<CR> 3: ,DB:MAT3,B<CR> 6: /XB:RES2<CR> 7: /QY 9: /OC Iε thi≤ casσ thσ abovσ sequencσ oµ entrie≤ fo≥ onσ operatioε wil∞ ì looδ a≤ follows ^0 ^9n1:m1^1 ; User should enter MATn ranges manually, of course. ^9n2:m2^2 ^9n3:m3^3 ^7 O╨<CR> ; Choice of operation ## 1:10 or EXIT # 11. SC RES<CR> notσá 1║á N∩ space≤ arσ alloweΣ iε .XQ╘ file≤ afte≥ las⌠á (right⌐ ì characte≥ oε eacΦ oµ lines. notσ 2║á N∩ space≤ arσ alloweΣ iε ke∙ definitioε line≤ afte≥ las⌠ ì (right⌐ characte≥ oε thσ linσ. .PAè d. Automatic Operation T∩á providσ ß trul∙ efficien⌠ anΣ automaticÖ runninτ oµ ExtraCalc-ì 1¼á onσ shoulΣ usσ CP/M'≤ SUBMIT.CO═ utility«á I⌠ wil∞ securσá aε ì automatiπá transitioε betweeε SuperCalπ anΣ OP.CO═ (firs⌠ prograφ ì oµ ExtraCalc-1⌐ anΣ bacδ t∩ SuperCalπ spreadsheet«á Wσá recommenΣ ì usσ oµ filσ O.SU┬ whicΦ consist≤ oµ tw∩ lines OP SC RES anΣ to reprograφ ke∙ 7║ as follows 7: /QY1SUBMIT O<CR> No≈ thσ sequencσ oµ operation≤ become≤ ß ver∙ shor⌠ anΣ efficien⌠ ì onσ indeed ^0 ; Worksheet initialization. ^9n1:m1^1 ; Operand (MAT1) specification. ^9n2:m2^2 ; Operand (MAT2) specification. ^9n3:m3^3 ; Operand (MAT3) specification. ^7 ; ExtraCalc-1 will be invoked automatically. User ; should select one of ExtraCalc-1 options (1:11). ; SuperCalc will be automatically invoked at the ╗ácompletitioεáoµ selecteΣámatri° operation. ; Results of computation will be automatically ╗áloaded iε thσ area specified by outpu⌠ámatri° ; operanΣ MAT2 and/or MAT3. Le⌠ u≤ reminΣ thσ use≥ tha⌠ file≤ RES.XQT¼á SAV.XQ╘ anΣ O.SU┬ arσ ì provideΣá oεá thσá distributioε diskette«á You≥á key≤á shoulΣá bσ ì programmeΣ a≤ summarizeΣ below 0: /XSAV<CR> 1: ,DB:MAT1,B<CR> 2: ,DB:MAT2,B<CR> 3: ,DB:MAT3,B<CR> 6: /XB:RES2<CR> 7: /QY1SUBMIT O<CR> 9: /OC notσá 3║á Maste≥ diskettσ shoulΣ bσ unprotecteΣ iµ SUBMIT.CO═á i≤ ì used..PO 8è5. NUMERIC EXAMPLES Iε thσ firs⌠ examplσ (transpositition⌐ aε initia∞ speadshee⌠ wil∞ ì bσá thσ onσ presenteΣ oε Figurσ 2«á Forma⌠ oµ eacΦá examplσá wil∞ ì consis⌠ of a. Sequence of entries as outlined in section 4d. (Auto. Operation). b. Answer to ExtraCalc-1 prompt (there is only one) c. Resulting Speadsheet d. Comments (sometimes) and additional entry for operations 5, 6. .PO 0 | A || B || C || D || E || F || G || H | 1| 1 2 3 4 5 6 7 8 2| 5 5 5 5 5 5 5 5 3| -1 1 -1 1 -1 1 -1 1 4| 2 4 2 1 4 5 5 6 5| 6| 7| 0 0 0 0 1 0 8| 0 0 0 0 2 0 9| 0 0 0 0 3 0 10| 0 0 0 0 4 0 11| 0 0 0 0 0 12| 0 0 0 0 0 13| 0 0 0 0 0 14| 0 0 0 0 0 15| 16| 17| 1 4 -6 -2 1 2 -3 4 18| 4 3 5 7 2 -3 4 7 19| -6 5 1 -1 5 3 0 6 20| -2 7 -1 3 7 -1 -5 -2 .PO 8 Figure 2 Resultinτá spreadshee⌠á afte≥á eacΦ operatioε wil∞á servσá a≤á aε ì initia∞ speadshee⌠ oµ operatioε tha⌠ follow≤ it. note║á Kee≡á workshee⌠ number≤ iε DEFAUL╘ forma⌠ only«á Otherwisσ ì thσ ExtraCalc-▒ wil∞ misinterpre⌠ o≥ misusσ thσ data. .PAè1) TRANSPOSITION (MAIN DIAGONAL) ================================================================= a. `` b. `` ^0 `` 1<CR> ^9A1:H4^1 `` ^9A7:D14^2 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1| 1 2 3 4 5 6 7 8 2| 5 5 5 5 5 5 5 5 3| -1 1 -1 1 -1 1 -1 1 4| 2 4 2 1 4 5 5 6 5| 6| 7| 1 5 -1 2 1 0 8| 2 5 1 4 2 0 9| 3 5 -1 2 3 0 10| 4 5 1 1 4 0 11| 5 5 -1 4 0 12| 6 5 1 5 0 13| 7 5 -1 5 0 14| 8 5 1 6 0 15| 16| 17| 1 4 -6 -2 1 2 -3 4 18| 4 3 5 7 2 -3 4 7 19| -6 5 1 -1 5 3 0 6 20| -2 7 -1 3 7 -1 -5 -2 .PO 8 Figure 3 .PAè2) TRANSPOSITION (SECONDARY DIAGONAL) ================================================================= a. `` b. `` ^0 `` 2<CR> ^9A7:D14^1 `` ^9A1:H4^2 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1| 6 5 5 4 1 2 4 2 2| 1 -1 1 -1 1 -1 1 -1 3| 5 5 5 5 5 5 5 5 4| 8 7 6 5 4 3 2 1 5| 6| 7| 1 5 -1 2 1 0 8| 2 5 1 4 2 0 9| 3 5 -1 2 3 0 10| 4 5 1 1 4 0 11| 5 5 -1 4 0 12| 6 5 1 5 0 13| 7 5 -1 5 0 14| 8 5 1 6 0 15| 16| 17| 1 4 -6 -2 1 2 -3 4 18| 4 3 5 7 2 -3 4 7 19| -6 5 1 -1 5 3 0 6 20| -2 7 -1 3 7 -1 -5 -2 .PO 8 Figure 4 .PAè3) REFLECTION IN A COLUMN ================================================================= a. `` b. `` ^0 `` 3<CR> ^9A7:D14^1 `` ^9A7:D14^2 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1| 6 5 5 4 1 2 4 2 2| 1 -1 1 -1 1 -1 1 -1 3| 5 5 5 5 5 5 5 5 4| 8 7 6 5 4 3 2 1 5| 6| 7| 8 5 1 6 1 0 8| 7 5 -1 5 2 0 9| 6 5 1 5 3 0 10| 5 5 -1 4 4 0 11| 4 5 1 1 0 12| 3 5 -1 2 0 13| 2 5 1 4 0 14| 1 5 -1 2 0 15| 16| 17| 1 4 -6 -2 1 2 -3 4 18| 4 3 5 7 2 -3 4 7 19| -6 5 1 -1 5 3 0 6 20| -2 7 -1 3 7 -1 -5 -2 .PO 8 Figure 5 .PAè4) REFLECTION IN A ROW ================================================================= a. `` b. `` ^0 `` 4<CR> ^9A7:D14^1 `` ^9A7:D14^2 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1| 6 5 5 4 1 2 4 2 2| 1 -1 1 -1 1 -1 1 -1 3| 5 5 5 5 5 5 5 5 4| 8 7 6 5 4 3 2 1 5| 6| 7| 6 1 5 8 1 0 8| 5 -1 5 7 2 0 9| 5 1 5 6 3 0 10| 4 -1 5 5 4 0 11| 1 1 5 4 0 12| 2 -1 5 3 0 13| 4 1 5 2 0 14| 2 -1 5 1 0 15| 16| 17| 1 4 -6 -2 1 2 -3 4 18| 4 3 5 7 2 -3 4 7 19| -6 5 1 -1 5 3 0 6 20| -2 7 -1 3 7 -1 -5 -2 .PO 8 Figure 6 .PAè5) INVERSION ================================================================= a. `` b. `` ^0 `` 5<CR> ^9A17:D20^1 `` ^9E17:H20^2 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1| 6 5 5 4 1 2 4 2 2| 1 -1 1 -1 1 -1 1 -1 3| 5 5 5 5 5 5 5 5 4| 8 7 6 5 4 3 2 1 5| 6| 7| 6 1 5 8 1 0 8| 5 -1 5 7 2 0 9| 5 1 5 6 3 0 10| 4 -1 5 5 4 0 11| 1 1 5 4 0 12| 2 -1 5 3 0 13| 4 1 5 2 0 14| 2 -1 5 1 0 15| 16| 17| 1 4 -6 -2 .2206572 .1924882 .0892019 -.272300 18| 4 3 5 7 .1924882 .157277 .1948356 -.173709 19| -6 5 1 -1 .0892019 .1948357 .2488263 -.312207 20| -2 7 -1 3 -.272300 -.173709 -.312207 .4530516 .PO 8 Figure 7 .PAè ================================================================= d. To display value of determinant one should type in ^6 ================================================================= Resulting spreadsheet. .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 1 2 4 2 3| 1 -1 1 -1 1 -1 1 -1 4| 5 5 5 5 5 5 5 5 5| 8 7 6 5 4 3 2 1 6| 7| 8| 6 1 5 8 1 0 9| 5 -1 5 7 2 0 10| 5 1 5 6 3 0 11| 4 -1 5 5 4 0 12| 1 1 5 4 0 13| 2 -1 5 3 0 14| 4 1 5 2 0 15| 2 -1 5 1 0 16| 17| 18| 1 4 -6 -2 .2206572 .1924882 .0892019 -.272300 19| 4 3 5 7 .1924882 .157277 .1948356 -.173709 20| -6 5 1 -1 .0892019 .1948357 .2488263 -.312207 21| -2 7 -1 3 -.272300 -.173709 -.312207 .4530516 .PO 0 Figure 8 .PAè6) EIGENVALUES AND EIGENVECTORS ================================================================= a. `` b. `` ^0 `` 6<CR> ^9E18:H21^1 `` ^9E12:H15^2 `` ^9E2:H5^3 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 1 2 4 2 3| 1 -1 1 -1 1 -1 1 -1 4| 5 5 5 5 5 5 5 5 5| 8 7 6 5 4 3 2 1 6| 7| 8| 6 1 5 8 1 0 9| 5 -1 5 7 2 0 10| 5 1 5 6 3 0 11| 4 -1 5 5 4 0 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| 1 4 -6 -2 .2206572 .1924882 .0892019 -.272300 19| 4 3 5 7 .1924882 .157277 .1948356 -.173709 20| -6 5 1 -1 .0892019 .1948357 .2488263 -.312207 21| -2 7 -1 3 -.272300 -.173709 -.312207 .4530516 .PO 8 Figure 9 .PAè ================================================================= d. To display Eigenvectors one should type in ^6 ================================================================= Resulting spreadsheet. .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 .4196722 -.742605 -.021986 -.521469 3| 1 -1 1 -1 .3744284 -.191104 .7270643 .5428264 4| 5 5 5 5 .4711288 .6290308 .3184456 -.530048 5| 8 7 6 5 -.679497 -.127817 .6078543 -.390461 6| 7| 8| 6 1 5 8 1 0 9| 5 -1 5 7 2 0 10| 5 1 5 6 3 0 11| 4 -1 5 5 4 0 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| 1 4 -6 -2 .2206572 .1924882 .0892019 -.272300 19| 4 3 5 7 .1924882 .157277 .1948356 -.173709 20| -6 5 1 -1 .0892019 .1948357 .2488263 -.312207 21| -2 7 -1 3 -.272300 -.173709 -.312207 .4530516 .PO 8 Figure 10 .PAè 7) SOLUTION OF SYSTEM OF LINEAR EQUATIONS ================================================================= a. `` b. `` ^0 `` 7<CR> ^9A18:D21^1 `` ^9F8:F11^2 `` ^9H8:H11^3 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 .4196722 -.742605 -.021986 -.521469 3| 1 -1 1 -1 .3744284 -.191104 .7270643 .5428264 4| 5 5 5 5 .4711288 .6290308 .3184456 -.530048 5| 8 7 6 5 -.679497 -.127817 .6078543 -.390461 6| 7| 8| 6 1 5 8 1 -.215962 9| 5 -1 5 7 2 .3967136 10| 5 1 5 6 3 -.023474 11| 4 -1 5 5 4 .2558685 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| 1 4 -6 -2 .2206572 .1924882 .0892019 -.272300 19| 4 3 5 7 .1924882 .157277 .1948356 -.173709 20| -6 5 1 -1 .0892019 .1948357 .2488263 -.312207 21| -2 7 -1 3 -.272300 -.173709 -.312207 .4530516 .PO 8 Figure 11 ExtraCalc-▒á wil∞á displa∙ (durinτ execution⌐ ß valuσ oµá minima∞ ì pivo⌠á tha⌠ wa≤ useΣ iε computations«á Iµ i⌠ think≤ tha⌠ pivo⌠ i≤ ì to∩ smal∞, thσ warning SYSTEM IS ALMOST SINGULAR wil∞á bσá displayed«á I⌠ doe≤ no⌠ alway≤ meaεá tha⌠á solutioεá i≤ ì incorrec⌠ o≥ no⌠ precisσ bu⌠ serve≤ t∩ aler⌠ thσ user. .PAè 8) ADDITION ================================================================= a. `` b. `` ^0 `` 8<CR> ^9A2:H5^1 `` ^9A18:H21^2 `` ^9A18:H21^3 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 .4196722 -.742605 -.021986 -.521469 3| 1 -1 1 -1 .3744284 -.191104 .7270643 .5428264 4| 5 5 5 5 .4711288 .6290308 .3184456 -.530048 5| 8 7 6 5 -.679497 -.127817 .6078543 -.390461 6| 7| 8| 6 1 5 8 1 -.215962 9| 5 -1 5 7 2 .3967136 10| 5 1 5 6 3 -.023474 11| 4 -1 5 5 4 .2558685 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| 7 9 -1 2 .6403294 -.550117 .0672158 -.793769 19| 5 2 6 6 .5669166 -.033827 .9218999 .3691175 20| -1 10 6 4 .5603307 .8238665 .5672719 -.842254 21| 6 14 5 8 -.951798 -.301526 .2956478 .0625908 .PO 8 Figure 12 .PAè 9) SUBTRACTION ================================================================= a. `` b. `` ^0 `` 9<CR> ^9A2:H5^1 `` ^9A18:H21^2 `` ^9A18:H21^3 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 .4196722 -.742605 -.021986 -.521469 3| 1 -1 1 -1 .3744284 -.191104 .7270643 .5428264 4| 5 5 5 5 .4711288 .6290308 .3184456 -.530048 5| 8 7 6 5 -.679497 -.127817 .6078543 -.390461 6| 7| 8| 6 1 5 8 1 -.215962 9| 5 -1 5 7 2 .3967136 10| 5 1 5 6 3 -.023474 11| 4 -1 5 5 4 .2558685 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| -1 -4 6 2 -.220657 -.192488 -.089202 .2723004 19| -4 -3 -5 -7 -.192488 -.157277 -.194836 .1737089 20| 6 -5 -1 1 -.089202 -.194836 -.248826 .3122066 21| 2 -7 1 -3 .2723004 .1737089 .3122065 -.453052 .PO 8 Figure 13 .PAè 10) MULTIPLICATION ================================================================= a. `` b. `` ^0 `` 10<CR> ^9A18:D21^1 `` ^9E18:H21^2 `` ^9E2:H5^3 `` ^7 ` ================================================================= c.` == .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 .9999994 -2.09e-7 -.000001 .0000004 3| 1 -1 1 -1 .0000001 1 .0000002 -2.38e-7 4| 5 5 5 5 0 .0000004 .9999997 -2.98e-7 5| 8 7 6 5 -1.19e-7 .0000003 -2.38e-7 .9999999 6| 7| 8| 6 1 5 8 1 -.215962 9| 5 -1 5 7 2 .3967136 10| 5 1 5 6 3 -.023474 11| 4 -1 5 5 4 .2558685 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| -1 -4 6 2 -.220657 -.192488 -.089202 .2723004 19| -4 -3 -5 -7 -.192488 -.157277 -.194836 .1737089 20| 6 -5 -1 1 -.089202 -.194836 -.248826 .3122066 21| 2 -7 1 -3 .2723004 .1737089 .3122065 -.453052 .PO 8 Figure 14 .PAèAfte≥á reformattinτá /FE2:H5,$<CR> .PO 0 | A || B || C || D || E || F || G || H | 1|DET= -852.000 2| 6 5 5 4 1.00 .00 .00 .00 3| 1 -1 1 -1 .00 1.00 .00 .00 4| 5 5 5 5 .00 .00 1.00 .00 5| 8 7 6 5 .00 .00 .00 1.00 6| 7| 8| 6 1 5 8 1 -.215962 9| 5 -1 5 7 2 .3967136 10| 5 1 5 6 3 -.023474 11| 4 -1 5 5 4 .2558685 12| 1 1 5 4 .9334186 13| 2 -1 5 3 .1477654 14| 4 1 5 2 .0915645 15| 2 -1 5 1 -.092936 16| 17| 18| -1 -4 6 2 -.220657 -.192488 -.089202 .2723004 19| -4 -3 -5 -7 -.192488 -.157277 -.194836 .1737089 20| 6 -5 -1 1 -.089202 -.194836 -.248826 .3122066 21| 2 -7 1 -3 .2723004 .1737089 .3122065 -.453052 .PO 8 Figure 14a I⌠ i≤ clea≥ tha⌠ thσ resul⌠ (E2:H5⌐ i≤ aε identit∙ matrix« .PO 8 .PAè6. COMPOSITE OPERATIONS. APPLICATIONS. Usinτá successivσá matri°á operation≤ a≤ outlineΣ abovσá onσá ma∙ ì solvσá ßá tremendou≤ numbe≥ oµ morσ comple°á problem≤á oµá matri° ì algebra¼á operation≤ research¼ anΣ iε general¼ man∙ problem≤ tha⌠ ì allo≈á matri°á description«á Iε thi≤ sectioε wσá wil∞á illustratσ ì time-serie≤á analysi≤á usinτ ß s∩ calleΣ Leas⌠ Square≤á Techiquσ ì whicΦá serve≤ a≤ ß foundatioε oµ linea≥ anΣ nonlinea≥á regressioε ì analysis. a. Least Squares Technique Suppose that we have made a series of observations t1,y1,t2,y2,...,tN,yN tk can be interpreted as time instants, while yk is observed value (price, temperature, etc.) I⌠á i≤ ofteε assumeΣ tha⌠ thσ observeΣ valuσ (y⌐ i≤ ß functioε oµ ì time (t), or y = y(t) (1) Defininτá thi≤á dependenc∙ iε ß morσ specifiπ wa∙á wσá ofteεá ma∙ ì writσ it down as M y = $#% ap.fp(t) (2) p=1 where ap are constants, and fp(t⌐á arσ choseε systeφ oµ function≤ sucΦ a≤á polynomials¼ ì trigonometriπ functions¼ exponential≤ anΣ s∩ on. Example≤ oµ fpù arσ t2ù ½ ⌠ -3¼á sin3t-con5t¼ e-2tù - -t1« Iε thσ casσ ì oµ polynomia∞ regressioε fp(t⌐ ╜ tp« Oµ coursσ wσ ma∙ rewritσ (2⌐ ì fo≥ aε arbitrar∙ k-tΦ observation M yk = $#% ap.fp(tk), k=1,2,...,N (3) p=1 Or in the matrix form ┘ ╜ ╞ «áa¼á FNxMù ╜ {fp(tk}¼áaMx1ù ╜ {ap} (4) Thσ probleφ oµ Leas⌠ Square≤ Techiquσ i≤ t∩ finΣ vecto≥ ß ╜ {ap}¼ ìèp=1,2,...,M╗ tha⌠ minimize≤ ß form N M |B[ Y - F.a |E]2 = $#% |B[ yk - $#% ap.fp(tk) |E]2 (5) k=1 p=1 B∙ takinτ partia∞ derivative≤ witΦ respec⌠ t∩ apù anΣ puttinτ theφ ì equa∞ t∩ zer∩ wσ have FT.F.a = FT.Y (6) Thereforσ apù caε bσ founΣ iε fivσ step≤ usinτ ExtraCalc-1 1) Calculate matrix F using SuperCalc built in functions 2) Calculate transpose of F using ExtraCalc-1 3) Calculate FT.F = A 4) Calculate FT.Y = D 5) Solve system of equations A.a = D with respect to a. Applications to forecasting and trend analysis T∩á usσá thσá obtaineΣá regresioεá model¼á onσá simpl∙á ma∙á pluτ ì differen⌠áá number≤áá specifyinτá tkùá outsidσá oµáá interva∞áá oµ ì observation«á Iµ thσ systeφ oµ fpù function≤ wa≤ choseεá correctl∙ ì (baseΣ oε somσ theoretica∞ analysi≤ oµ process¼á o≥ b∙ shee≥ lucδ ì thσ predictioε caε bσ ver∙ precise« Oµ coursσ onσ shoulΣ no⌠ takσ ì moment≤á oµá timσ tkù to∩ fa≥ iε thσ futurσ bu⌠ rathe≥ iε 5Ñá - 7Ñ ì rangσ oµ observatioε interva∞ length« Fo≥á morσá informatioεá oεá ho≈á t∩á usσá regressioεá model≤áá iε ì forecasting¼ interpolatioε anΣ trenΣ analysi≤ onσ shoulΣ refe≥ t∩ ì aε appropriatσ booδ oε Probability and Statistics. b. Numeric Example: Dow Jones Regression Model. A≤ aε illustratioε oµ abovσ techniquσ wσ wil∞ conside≥ ßá probleφ ì oµ D╩ forecasting«á Ou≥ initia∞ spreadshee⌠ i≤ showε oε Figurσ 1╡ ì anΣá consist≤á oµá tw∩ column≤ oµ ~10░á observations«á Thσá firs⌠ ì columεá i≤á ß datσ oµ observation¼á whilσ thσ seconΣá i≤á closinτ ì valuσ oµ D╩ inde° (Pk⌐ oε thi≤ (k-th⌐ week.è.PN 39 .OP Wσá wil∞ usσ firs⌠ 5░ observation≤ t∩ obtaiε thσ mode∞ anΣá late≥ ì usσ othe≥ 5▓ observation≤ t∩ comparσ actua∞ anΣ forecasteΣ value≤ ì of Dow index. 1⌐ Le⌠ u≤ firs⌠ blanδ thσ las⌠ 5▓ row≤ oµ observation≤ anΣ definσ ì fp(tk⌐ as fp(tk⌐ ╜ a1.Pk-1ù ½ a2.(Pk-2-Pk-1⌐ ½ a3.(Pk-3-Pk-2⌐ ½ a4.(Pk-4-Pk-3) I⌠á i≤ goinτ t∩ meaε tha⌠ Presen⌠ Valuσ oµ D╩ inde° i≤ ß functioε ì oµ last four week indices« Sequencσ oµ SupecCalπ statementsÖ fo≥ thσ abovσ wil∞ be /BA52:B102<CR> > =C6 B5<CR> B5-B4<CR> B4-B3<CR> B3-B2<CR> /RC6:F6,C7:C51<CR> ! Resultinτá matri° (F⌐ i≤ 46\┤ anΣ occupie≤ workshee⌠ C6:F5▒á (seσ ì Figurσ 16). .PAè.PN 41 .OP 2) Calculate transpose of F using ExtraCalc-1. Sequence of SuperCalc and ExtraCalc-1 statements for this will be =C2 0 /RC2,D2:AV2<CR> /RC2:AV2,C3:C5<CR> ^0 ^9C6:F51^1 ^9C2:AV5^2 ^7 Answer to ExtraCalc-1 prompt is 1. 3) Calculate FT.F = A Sequence of ExtraCalc-1 statements for this will be ^0 ^9C2:AV5^1 ^9C6:F51^2 ^9C6:F9^3 ^7 Answe≥ t∩ ExtraCalc-▒ promp⌠ i≤ 10«á Afte≥ thi≤ operatioε wσá ma∙ ì blanδá unnecessar∙á no≈ part≤ oµ worksheet«á Wheεá creatinτá MATnÖ ì (matri° operand⌐ KEE╨ WORKSHEE╘ I╬ DEFAULTÖ FORMA╘ ONL┘ !!! /BC10:F51<CR> 4) Calculate FT.Y = D Sequence of ExtraCalc-1 statements for this will be ^0 ^9B6:B51^2 ^9C2:C5^3 ^7 Answe≥á t∩ ExtraCalc-▒ promp⌠ i≤ 10«á Afte≥ thi≤ operatioε wσ ma∙ ì blanδá unnecessar∙á no≈ part≤ oµá worksheet«á KEE╨á WORKSHEE╘á I╬ ì DEFAULTÖ FORMA╘ ONL┘ !!! /BD2:AV5<CR> Resulting worksheet is shown below (Figure 17). .PAè.PN 43 .OP 5) Solve system of equations A.a = D with respect to a. Sequence of ExtraCalc-1 statements for this will be ^0 ^9C6:F9^1 ^9C2:C5^2 ^9C2:C5^3 ^7 Answe≥á t∩ ExtraCalc-▒ promp⌠ i≤ 7«á Afte≥ thi≤ operatioε wσá ma∙ ì now blanδ unnecessar∙ part≤ oµ workshee⌠. /BC6:F9<CR> Resulting worksheet is shown below (Figure 18). .PAè.PN 45 .OP Le⌠á u≤á no≈ creatσ ß workshee⌠ t∩ comparσ actua∞ anΣá forecasteΣ ì value≤ oµ Do≈ Jone≤ index«á First¼á wσ replacσ thσ analyzeΣá datß ì witΦá ne≈á se⌠ oµ datß usinτ thσ followinτ sequencσ oµá SuperCalπ ì statements /BA2:B51<CR> /LB:DOW.CAL,PA48:B102,A2,V Le⌠á u≤á theεá calculatσá forecasteΣ value≤á iεá columεá dÖá usinτ ì obtaineΣ value≤ oµ ap¼ p=1,2,3,┤ (entr∙ C2:C5)« =D6<CR> C2*B5+C3*(B5-B4)+C4*(B4-B3)+C5*(B3-B2)<CR> /RD6,D7:D56,ANYYNYYNYYNYY ! Resulting spreadsheet is shown on Figure 19. .PAè.PN 47 .OP Fo≥ comparisoε oµ actua∞ anΣ forecasteΣ value≤ oµ inde° yo⌡á havσ ì t∩ ente≥ followinτ (SuperCalc) > =C6 IF(((B6-B5)*(D6-B5)>0,C5+1,C5) /RC6,C7:C56<CR> =C6 IF(((B6-B5)*(D6-B5)>0,C1+1,C1) =C1 0 =B57<CR> " %= C56*100/(52-5) ! Tota∞á Ñá oµá correc⌠á prediction≤ i≤á ~52Ñá anΣá is¼á obviously¼ ì unsatisfactor∙á (Fiτá 20)¼á althougΦá onσá ma∙á noticσá tha⌠á thσ ì predictioεá wa≤á ver∙á gooΣá fo≥ firs⌠ fe≈á week≤á (seσá analysi≤ ì below). .PAè.PN 49 .OP Let'≤á looδá a⌠á ho≈ thσ predictioεá accurac∙á change≤á witΦá thσ ì distancσá froφ B╡ (las⌠ datß useΣ iε regressioε model)«á Let'≤ d∩ ì followinτ transformations =D5 "Dow.Fore =E5 " % = =F5 "# of Forecast =F6 1 =F7 F6+1 /RF7,F8:F56<CR> =E6 100*C6/F6 /RE6,E7:E56<CR> A≤á i⌠ caε bσ seeε (Figurσ 21)¼á forecasteΣ value≤ oµ D╩ movσá iε ì thσá samσá directioεá a≤á nex⌠ week actua∞ value≤á iεá 100Ñá case≤ ì THROUG╚ 4tΦ week«á Iε othe≥ words¼ wσ shoulΣ recalculatσ apù ever∙ ì 4-╡ week≤ at least t∩ ge⌠ ß gooΣ prediction. .PAè.PN 51 .OP 7. ExtraCalc-1 ERROR MESSAGES Thσ ExtraCalc-▒ detect≤ tw∩ kind≤ oµ errors║á Warning≤ anΣá Fata∞ ì Errors«á Wheεá ß Warninτ i≤ issued¼á executioε contro∞ return≤ t∩ ì ExtraCalc-▒ manage≥ prograφ OP.COM«á Wheε ß Fata∞ Erro≥ i≤ found¼ ì ExtraCalc-▒ cease≤ executioε anΣ contro∞ i≤ returneΣ t∩ operatinτ ì systeφ (batcΦ filσ under SUBMIT.COM, iµ yo⌡ usσ one). Examples of Warnings: OP.COM no error messages in OP.COM program OP04.COM ERROR: WRONG OUTPUT MATRIX DIMENSIONS ERROR: DIMENSIONS OF INPUT MATRICES DO NOT MATCH OP13.COM ERROR: NON-SQUARE INPUT MATRIX ERROR: NON-SQUARE OUTPUT MATRIX ERROR: NON-SQUARE SYSTEM MATRIX ERROR: NON-MATCHING SYSTEM VECTOR ERROR: NON-MATCHING OUTPUT VECTOR MATRIX DETERMINANT = 0.0 INVERSION CANNOT BE COMPLETED NO SOLUTION OBTAINED: SINGULAR SYSTEM) MINIMAL PIVOT = xxx !!! SYSTEM IS ALMOST SINGULAR AVERAGE PIVOT = xxx OP2.COM ERROR: NON-SQUARE INPUT MATRIX ERROR: WRONG OUTPUT MATRIX DIMENSIONS (MAT3) ERROR: OUTPUT FILE SIZE = N1 < N = REQUIRED èFatal Error Messages are surrounded by asterisks as follows **XX** at address XXXX** Iµá yo⌡á encountereΣ Fata∞ Erro≥ messagσ firs⌠ checδá ExtraCalc-▒ ì anΣá SuperCalπ file≤ location«á Imprope≥ occurencσ o≥ absencσá oµ ì thσ onσ oε specifieΣ drivσ i≤ onσ oµ thσ primar∙ reasoε fo≥ Fata∞ ì Error« Nex⌠ checδ logiπ oµ you≥ computatioε witΦ ExtraCalc-1. Iµá yo⌡ arσ no⌠ ablσ t∩ detec⌠ ß causσ fo≥ Fata∞á Error¼á please¼ ì recorΣáá al∞á circumstance≤á oµá it≤á occurencσá anΣá mai∞áá thi≤ ì informatioεá t∩ Smirno÷ Associates«á Normall∙ i⌠ i≤ no⌠á expecteΣ ì tha⌠ yo⌡ registe≥ Fata∞ Erro≥ messagσ iµ you≥ systeφ i≤ correctl∙ ì configured. ThirΣ kinΣ oµ erro≥ message≤ ma∙ comσ froφ you≥ operatinτ system« ì Please refer to your user's manual for guidance.