Read e-book online ALGOL 60 Implementation: The Translation and Use of ALGOL 60 PDF

By Brian Randell

ALGOL 60 implementation: the interpretation and use of ALGOL 60 courses on a computer
Volume five of A.P.I.C. experiences in facts processing
Issue five of ALGOL 60 Implementation, Brian Randell

Show description

Read Online or Download ALGOL 60 Implementation: The Translation and Use of ALGOL 60 Programs on a Computer PDF

Best mathematics books

Download PDF by P. T. Johnstone: Topos Theory

Post 12 months be aware: First released in 1977
------------------------

Focusing on topos theory's integration of geometric and logical rules into the principles of arithmetic and theoretical computing device technological know-how, this quantity explores inner classification concept, topologies and sheaves, geometric morphisms, different subjects.

Download PDF by Marco Schnurr: Steigungen hoherer Ordnung zur verifizierten globalen

In dieser Arbeit wird die automatische Berechnung von Steigungstupeln zweiter und h? herer Ordnung behandelt. Die entwickelten Techniken werden in einem Algorithmus zur verifizierten globalen Optimierung verwendet. Anhand von Testbeispielen auf einem Rechner wird der neue Algorithmus mit einem Algorithmus aus der Literatur verglichen.

Additional info for ALGOL 60 Implementation: The Translation and Use of ALGOL 60 Programs on a Computer

Sample text

St δf (x; x0 ) = x1 + (x0 )1 · x22 1 (x0 )21 + 1 · x2 + (x0 )2 0 = δf (x; x0 ) . ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ❊s s❡✐ f : D ⊆ Rn → R st❡t✐❣ ✉♥❞ x0 , x1 ∈ D ❜❡❧✐❡❜✐❣✱ ❛❜❡r ❢❡st✱ ♠✐t x0 = x1 ✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δ2 f : D → Rn×n ♠✐t ❉❡✜♥✐t✐♦♥ ✶✳✸✳✸✵ f (x) = f (x0 ) + δf (x1 ; x0 ) (x − x0 ) + (x − x1 )T δ2 f (x; x1 , x0 ) (x − x0 ) , x ∈ D, ✭✶✳✶✹✮ ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 ✳ ❉❛❜❡✐ ✐st δf (x1 ; x0 ) ∈ R1×n ❞❡r ❲❡rt ❡✐♥❡r ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ δf (x; x0 ) ❡rst❡r ❖r❞✲ ♥✉♥❣ ✈♦♥ f ❛♥ ❞❡r ❙t❡❧❧❡ x = x1 ✳ ❊✐♥❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ δ2 f ([x] ; x1 , x0 ) ✈♦♥ δf (x; x1 , x0 ) ❛✉❢ ❞❡♠ ■♥t❡r✈❛❧❧ [x]✱ ❞✳ ❤✳ ❡✐♥ δ2 f ([x] ; x1 , x0 ) ∈ IRn×n ♠✐t δ2 f ([x] ; x1 , x0 ) ⊇ {δ2 f (x; x1 , x0 ) | x ∈ [x] } , ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 ✳ ❋ür ❋✉♥❦t✐♦♥❡♥ f = (fi ) : D ⊆ Rn → Rm ✇✐r❞ ❡✐♥❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ❞❡✜♥✐❡rt✱ ✐♥❞❡♠ ❞✐❡ ●❧❡✐❝❤✉♥❣ ✭✶✳✶✹✮ ❦♦♠♣♦♥❡♥t❡♥✇❡✐s❡ ❛♥❣❡✇❡♥❞❡t ✇✐r❞✳ ✷✽ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ■st δ2 f ([x] ; x1 , x0 ) ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x]✱ s♦ ❣✐❧t ✇✐❡ ✐♥ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✷ ♠✐t ❍✐❧❢❡ ❞❡r ❊✐♥s❝❤❧✐❡ÿ✉♥❣s❡✐❣❡♥s❝❤❛❢t ✶✳✷✳✸ s♦✇✐❡ ❞❡r ❙✉❜❞✐str✐❜✉t✐✈✐tät ❞❡r ■♥t❡r✈❛❧❧r❡❝❤♥✉♥❣ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✸✶ f (x) ∈ f (x0 ) + δf (x1 ; x0 ) + ([x] − x1 )T δ2 f ([x] ; x1 , x0 ) ([x] − x0 ) ⊆ f (x0 ) + δf (x1 ; x0 ) ([x] − x0 ) + ([x] − x1 )T δ2 f ([x] ; x1 , x0 ) ([x] − x0 ) ❢ür ❛❧❧❡ x ∈ [x]✳ ✶✳✸✳✹ ❆♥✇❡♥❞✉♥❣❡♥ ◆❡❜❡♥ ❞❡r ❇❡st✐♠♠✉♥❣ ❜③✇✳ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ❡✐♥❡r ❋✉♥❦t✐♦♥ ✉♥❞ ♥❡❜❡♥ ❆♥✇❡♥❞✉♥❣❡♥ ✐♥ ❞❡r ❣❧♦❜❛❧❡♥ ❖♣t✐♠✐❡r✉♥❣ ✭s✐❡❤❡ ❑❛♣✐t❡❧ ✺✮ ❦ö♥♥❡♥ ❙t❡✐❣✉♥❣s✲ ❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❛✉❝❤ ③✉♠ ❊①✐st❡♥③♥❛❝❤✇❡✐s ✈♦♥ ◆✉❧❧st❡❧❧❡♥ ❡✐♥❡r ❋✉♥❦t✐♦♥ ✈❡r✇❡♥❞❡t ✇❡r❞❡♥✳ ❲✐r ❣❡❜❡♥ ✐♠ ❋♦❧❣❡♥❞❡♥ ❡✐♥✐❣❡ ❊①✐st❡♥③sät③❡ ❛♥✱ ✐♥ ❞❡♥❡♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ❜③✇✳ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ✈❡r✇❡♥❞❡t ✇❡r❞❡♥✳ ❇❛s✐❡r❡♥❞ ❛✉❢ ❞❡♠ ❋✐①♣✉♥❦ts❛t③ ✈♦♥ ❇r♦✉✇❡r ❣✐❧t ❞❡r ❢♦❧❣❡♥❞❡ ❙❛t③ ✈♦♥ ▼♦♦r❡✳ ❙❛t③ ✶✳✸✳✸✷ ✭❙❛t③ ✈♦♥ ▼♦♦r❡✱ ❬✷✻❪✮ ❊s s❡✐ f : D ⊆ Rn → Rn st❡t✐❣ ❛✉❢ ❞❡r ♦✛❡♥❡♥✱ ❦♦♥✈❡①❡♥ ▼❡♥❣❡ D ✉♥❞ [x] ⊆ D✳ ❋ür ❢❡st❡s x0 ∈ [x] s❡✐ δf ([x] ; x0 ) ∈ IRn×n ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ●✐❜t ❡s ❡✐♥❡ ♥✐❝❤ts✐♥❣✉❧är❡ ▼❛tr✐① A ∈ Rn×n ✱ s♦ ❞❛ss ❞❡r ❑r❛✇❝③②❦✲❖♣❡r❛t♦r K([x] , x0 , A) := x0 − A f (x0 ) + I − A δf ([x] ; x0 ) ([x] − x0 ) ❞✐❡ ■♥❦❧✉s✐♦♥ K([x] , x0 , A) ⊆ [x] , ✭✶✳✶✺✮ ❡r❢ü❧❧t✱ ❞❛♥♥ ❡♥t❤ä❧t [x] ❡✐♥❡ ◆✉❧❧st❡❧❧❡ ✈♦♥ f ✳ ❊✐♥ ✇❡✐t❡r❡r ❙❛t③✱ ❞❡r ❞✐❡ ❊①✐st❡♥③ ❡✐♥❡r ◆✉❧❧st❡❧❧❡ ✈♦♥ f : D ⊆ Rn → Rn ❣❛r❛♥t✐❡rt✱ ✐st ❞❡r ❙❛t③ ✈♦♥ ▼✐r❛♥❞❛✳ ❙❛t③ ✶✳✸✳✸✸ ✭▼✐r❛♥❞❛✱ ❬✷✺❪✮ ❊s s❡✐ f : D ⊆ Rn → Rn st❡t✐❣✱ x0 ∈ D ✉♥❞ [x] = [x0 − s, x0 + t] ⊆ D ♠✐t s, t ∈ Rn ✱ si ≥ 0✱ ti ≥ 0✳ ❉✐❡ s✐❝❤ ❣❡❣❡♥ü❜❡r❧✐❡❣❡♥❞❡♥✱ ♣❛r❛❧❧❡❧❡♥ ❙❡✐t❡♥ ✈♦♥ [x] s❡✐❡♥ ♠✐t ✉♥❞ [x]i,+ := {x ∈ [x] , xi = (x0 )i + ti } ✭✶✳✶✻✮ [x]i,− := {x ∈ [x] , xi = (x0 )i − si } ✭✶✳✶✼✮ ✶✳✸✳ ❙❚❊■●❯◆●❊◆ ✷✾ ❜❡③❡✐❝❤♥❡t✳ ❋❛❧❧s ❢ür ❛❧❧❡ i = 1, .

X0 )n = fi x1 , . . , xn − fi x1 , . . , xn−1 , (x0 )n + fi x1 , . . , xn−1 , (x0 )n −fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n + fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n − + · · · + fi x1 , (x0 )2 , . . , (x0 )n − fi (x0 )1 , . . , (x0 )n xj − (x0 )j . ❉✉r❝❤ P❡r♠✉t❛t✐♦♥ ❡r❣❡❜❡♥ s✐❝❤ ❡♥ts♣r❡❝❤❡♥❞ ❢ür ❡✐♥❡ ❋✉♥❦t✐♦♥ f : Rn → Rn ✐♥ ❥❡❞❡r ❑♦♠♣♦♥❡♥t❡ fi ❣❡♥❛✉ n! ▼ö❣❧✐❝❤❦❡✐t❡♥ ③✉r ❇✐❧❞✉♥❣ ❡✐♥❡r ❍❛♥s❡♥✲❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥✱ ✐♥s❣❡s❛♠t ❛❧s♦ n · n! ▼ö❣❧✐❝❤❦❡✐t❡♥✳ ✷✻ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ✭❋♦rts❡t③✉♥❣✮✿ ❲✐r ❜❡tr❛❝❤t❡♥ ❞✐❡ ❋✉♥❦t✐♦♥ f ❛✉s ♦❜✐❣❡♠ ❇❡✐s♣✐❡❧ ✉♥❞ ❜❡r❡❝❤♥❡♥ ❡✐♥❡ ❍❛♥s❡♥✲ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 = (0, 0) ♠✐t ❍✐❧❢❡ ❛♥❞❡r❡r ❡✐♥❣❡❢ü❣t❡r ❙✉♠✲ ♠❛♥❞❡♥ ✭s✐❡❤❡ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✽✮✱ ✉♥❞ ③✇❛r δf (x, x0 )ij = fi (x1 , .

X0 ) | x ∈ [x] } , ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ n−t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ xn−1 , . . , x0 ✳ ■st δ2 f ([x] ; x1 , x0 ) = δ2 f , δ2 f ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x]✱ s♦ ❣✐❧t ♠✐t ❞❡r ❊✐♥s❝❤❧✐❡ÿ✉♥❣s❡✐❣❡♥s❝❤❛❢t ✶✳✷✳✸ s♦✇✐❡ ❞❡r ❙✉❜❞✐str✐❜✉t✐✈✐tät ❞❡r ■♥t❡r✈❛❧❧r❡❝❤♥✉♥❣ ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✷ f (x) = f (x0 ) + δf (x1 , x0 ) + δ2 f (x; x1 , x0 ) · (x − x1 ) · (x − x0 ) ∈ f (x0 ) + δf (x1 , x0 ) + δ2 f ([x] ; x1 , x0 ) · ([x] − x1 ) · ([x] − x0 ) ⊆ f (x0 ) + δf (x1 , x0 ) · ([x] − x0 ) + δ2 f ([x] ; x1 , x0 ) · ([x] − x1 ) · ([x] − x0 ) ❢ür ❛❧❧❡ x ∈ [x]✳ ❋❡r♥❡r s❝❤❧✐❡ÿ❡♥ ❞✐❡ ❜❡✐❞❡♥ P❛r❛❜❡❧♥ g (x) := f (x0 ) + δf (x1 , x0 ) · (x − x0 ) + δ2 f · (x − x1 ) · (x − x0 ) ✉♥❞ h (x) := f (x0 ) + δf (x1 , x0 ) · (x − x0 ) + δ2 f · (x − x1 ) · (x − x0 ) ❢ür ❥❡❞❡s x ∈ [x] ❞❡♥ ❋✉♥❦t✐♦♥s✇❡rt f (x) ❡✐♥ ✭s✐❡❤❡ ❆❜❜✐❧❞✉♥❣ ✶✳✷✮✳ ✷✵ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ f(x) h f g _x x0 _ x x x1 ❆❜❜✐❧❞✉♥❣ ✶✳✷✿ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ♠✐t ❍✐❧❢❡ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐✲ ❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x1 ✉♥❞ x0 δ2 f (x; x1 , x0 ) ❤ä♥❣t ✈♦♥ ❞❡r ❲❛❤❧ ❞❡r ❜❡✐❞❡♥ P✉♥❦t❡ x1 ✉♥❞ x0 ❛❜✳ ■♠ ❋♦❧❣❡♥❞❡♥ ❜❡tr❛❝❤t❡♥ ✇✐r ❞❡♥ ❋❛❧❧ x1 = x0 ✳ ❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✸ ❊s s❡✐ f : D ⊆ R → R st❡t✐❣✱ [x] ⊆ D ✉♥❞ x0 ∈ [x] ❢❡st✳ ❛✮ ❊s ❡①✐st✐❡r❡ f (x0 )✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δ2 f : D → R ♠✐t f (x) = f (x0 ) + f (x0 ) · (x − x0 ) + δ2 f (x; x0 , x0 ) · (x − x0 )2 , x ∈ D, ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 ✳ ❜✮ ❉✐❡ ✑●r❡♥③✲■♥t❡r✈❛❧❧st❡✐❣✉♥❣✑ δflim ([x0 ]) ❡①✐st✐❡r❡✳ ❊✐♥ δ2 f ([x] ; x0 , x0 ) ∈ IR ♠✐t f (x) ∈ f (x0 ) + δflim ([x0 ]) · (x − x0 ) + δ2 f ([x] ; x0 , x0 ) · (x − x0 )2 , x ∈ [x] , ✭✶✳✾✮ ❤❡✐ÿt ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❩✉r ❆❜❦ür③✉♥❣ s❡✐ δ2 f (x; x0 ) := δ2 f (x; x0 , x0 ) s♦✇✐❡ δ2 f ([x] ; x0 ) := δ2 f ([x] ; x0 , x0 ) ❣❡s❡t③t✳ ❊s s❡✐ f : D ⊆ R → R st❡t✐❣✱ [x] ⊆ D ✉♥❞ f ③✇❡✐♠❛❧ st❡t✐❣ ❞✐✛❡r❡♥③✐❡r❜❛r ✐♥ ❡✐♥❡r ❯♠❣❡❜✉♥❣ ✈♦♥ x0 ∈ [x]✳ ❉❛♥♥ ✐st ❞✐❡ ❋✉♥❦t✐♦♥ δ2 f : D ⊆ R → R ♠✐t  f (x) − f (x0 ) − f (x0 ) (x − x0 )   ❢ür x = x0  (x − x0 )2 δ2 f (x; x0 ) = ✭✶✳✶✵✮    1 ❢ür x = x0 2 f (x0 ) ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✹ ✶✳✸✳ ❙❚❊■●❯◆●❊◆ ✷✶ ♥❛❝❤ ❞❡♠ ❙❛t③ ✈♦♥ ❚❛②❧♦r st❡t✐❣ ✐♥ x0 ✉♥❞ ❢♦❧❣❧✐❝❤ ❛✉❝❤ st❡t✐❣ ❛✉❢ ❣❛♥③ D✳ ❉❛♠✐t ❡①✐st✐❡rt ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ δ2 f ([x] ; x0 ) ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❊s ❣❡❧t❡ ✭✶✳✾✮ ♠✐t δflim ([x0 ]) = δf x0 , δf x0 ✉♥❞ δ2 f ([x] ; x0 ) = δ2 f , δ2 f ✳ ❉❛♥♥ ✐st ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✺ f (x) ∈ f (x0 ) + δflim ([x0 ]) · ([x] − x0 ) + δ2 f ([x] ; x0 ) · ([x] − x0 )2 ✭✶✳✶✶✮ ❡✐♥❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ❛✉❢ [x]✳ ❋❡r♥❡r ❢♦❧❣t ❛✉s ✭✶✳✾✮✱ ❞❛ss ❞✐❡ P❛r❛❜❡❧stü❝❦❡ gl (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x ≤ x ≤ x0 , hl (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x ≤ x ≤ x0 , gr (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x0 ≤ x ≤ x hr (x) := f (x0 ) + δf x0 · (x − x0 ) + δ2 f · (x − x0 )2 ❢ür x0 ≤ x ≤ x ✉♥❞ ❢ür ❥❡❞❡s x ∈ [x] ❞❡♥ ❋✉♥❦t✐♦♥s✇❡rt f (x) ❡✐♥s❝❤❧✐❡ÿ❡♥ ✭s✐❡❤❡ ❆❜❜✐❧❞✉♥❣ ✶✳✸✮✳ f(x) hr hl f gr gl _x x0 _ x x ❆❜❜✐❧❞✉♥❣ ✶✳✸✿ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ❞❡s ❲❡rt❡❜❡r❡✐❝❤s ✈♦♥ f ♠✐t ❍✐❧❢❡ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐✲ ❣✉♥❣ ③✇❡✐t❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✷✷ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ✶✳✸✳✸ ❙t❡✐❣✉♥❣❡♥ ✈♦♥ ❋✉♥❦t✐♦♥❡♥ ♠❡❤r❡r❡r ❱❛r✐❛❜❧❡♥ ❊s s❡✐ f : D ⊆ Rn → Rm st❡t✐❣ ✉♥❞ x0 ∈ D ❜❡❧✐❡❜✐❣✱ ❛❜❡r ❢❡st✳ ❊✐♥❡ ❋✉♥❦t✐♦♥ δf : D → Rm×n ♠✐t ❉❡✜♥✐t✐♦♥ ✶✳✸✳✷✻ f (x) = f (x0 ) + δf (x; x0 ) · (x − x0 ) , ❤❡✐ÿt ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ ✭✶✳✶✷✮ x ∈ D, x0 ✳ ❊✐♥❡ ■♥t❡r✈❛❧❧♠❛tr✐① δf ([x] ; x0 ) ∈ IRm×n ♠✐t δf ([x] ; x0 ) ⊇ {δf (x; x0 ) | x ∈ [x] } ♥❡♥♥❡♥ ✇✐r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ❉✐❡ ❆✉ss❛❣❡♥ ❛✉s ❇❡♠❡r❦✉♥❣ ✶✳✸✳✶✷ ❣❡❧t❡♥ s♦♠✐t ❡♥ts♣r❡❝❤❡♥❞ ❢ür ❞✐❡ ❙t❡✐❣✉♥❣❡♥ ✈♦♥ ❋✉♥❦t✐♦♥❡♥ ♠❡❤r❡r❡r ❱❛r✐❛❜❧❡♥✳ ❇❡r❡❝❤♥✉♥❣ ✈♦♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞✲ ♥✉♥❣ ❋ür ❞✐❡ ❇❡r❡❝❤♥✉♥❣ ✈♦♥ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞♥✉♥❣ ❣✐❜t ❡s ✈❡rs❝❤✐❡❞❡♥❡ ▼ö❣❧✐❝❤❦❡✐t❡♥✿ ❛✮ ❊s s❡✐ D ⊆ Rn ♦✛❡♥✱ f = (fi ) : D → Rm st❡t✐❣ ❞✐✛❡r❡♥③✐❡r❜❛r ✉♥❞ [x] ⊆ D✳ ❆✉❢ ●r✉♥❞ ❞❡s ▼✐tt❡❧✇❡rts❛t③❡s ✶✳✶✳✶ ❣✐❧t ❢ür ❛❧❧❡ x ∈ [x]   f (x) − f (x0 ) =  ∂f1 ∂x1 (x0 ∂fm ∂x1 (x0 + t1 (x − x0 )) ✳✳ ✳ ··· + tm (x − x0 )) · · · ∂f1 ∂xn (x0  + t1 (x − x0 ))  ✳✳  (x − x0 ) ✳ ∂fm ∂xn (x0 + tm (x − x0 )) ♠✐t ti ∈ [0, 1]✳ ❲❡❣❡♥ ❞❡r ❙t❡t✐❣❦❡✐t ✈♦♥ f ❡①✐st✐❡rt ❡✐♥❡ ■♥t❡r✈❛❧❧♠❛tr✐① δf ([x] ; x0 ) ∈ IRm×n ♠✐t δf ([x] ; x0 ) ⊇ f (x) | x ∈ [x] .

Download PDF sample

ALGOL 60 Implementation: The Translation and Use of ALGOL 60 Programs on a Computer by Brian Randell


by Donald
4.0

Rated 4.54 of 5 – based on 15 votes