Numerical Methods in Computational Finance. Daniel J. Duffy

Чтение книги онлайн.

Читать онлайн книгу Numerical Methods in Computational Finance - Daniel J. Duffy страница 54

Numerical Methods in Computational Finance - Daniel J. Duffy

Скачать книгу

equation in one dimension:

      where:

StartLayout 1st Row psi equals Schr modifying above o with double dot dinger wave function period 2nd Row upper V left-parenthesis x right-parenthesis equals one hyphen dimensional potential period 3rd Row i equals StartRoot negative 1 EndRoot period EndLayout

      We rewrite Equation (5.23) in the equivalent form:

      while the implicit Euler BTCS scheme is given by:

i StartFraction psi Superscript n plus 1 Baseline minus psi Superscript n Baseline Over k EndFraction equals upper H psi Superscript n plus 1

      or

      or

left-parenthesis 1 plus italic i upper H k right-parenthesis psi Superscript n plus 1 Baseline equals psi Superscript n Baseline period

      (5.27)integral Subscript negative infinity Superscript infinity Baseline StartAbsoluteValue psi left-parenthesis x right-parenthesis EndAbsoluteValue squared italic d x equals 1 period

      A remedy for this is to use the Cayley form (this is essentially the Crank–Nicolson scheme):

i StartFraction psi Superscript n plus 1 Baseline minus psi Superscript n Baseline Over k EndFraction equals StartFraction upper H Over 2 EndFraction left-parenthesis psi Superscript n plus 1 Baseline plus psi Superscript n Baseline right-parenthesis

      or

      This scheme is unitary; you can check this by a bit of arithmetic using complex arithmetic.

      The solution of (5.24) is:

      (5.29)psi left-parenthesis x comma t right-parenthesis equals e Superscript negative italic i upper H t Baseline psi left-parenthesis x comma 0 right-parenthesis

      Matrix theory is too important to be ignored or given short shrift in any book on numerical analysis and its applications. For this reason, we gave a reasonably detailed exposition of matrix theory as a companion to the other chapters in this book (and it could possibly be a companion to other books).

      Конец ознакомительного фрагмента.

      Текст предоставлен ООО «ЛитРес».

      Прочитайте эту книгу целиком, купив полную легальную версию на ЛитРес.

      Безопасно оплатить книгу можно банковской картой Visa, MasterCard, Maestro, со счета мобильного телефона, с платежного терминала, в салоне МТС или Связной, через PayPal, WebMoney, Яндекс.Деньги,

Скачать книгу