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HW 02

Question One

Computing $\epsilon_{\text{mac}}$ for single precision numbers

  (load "../lizfcm.asd")
  (ql:quickload :lizfcm)

  (let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
                                                     1.0
                                                     1.0)))
    (lizfcm.utils:table (:headers '("a" "h" "err")
                         :domain-order (a h err)
                         :domain-values domain-values)))
a h err
1.0 0.5 0.5
1.0 0.25 0.25
1.0 0.125 0.125
1.0 0.0625 0.0625
1.0 0.03125 0.03125
1.0 0.015625 0.015625
1.0 0.0078125 0.0078125
1.0 0.00390625 0.00390625
1.0 0.001953125 0.001953125
1.0 0.0009765625 0.0009765625
1.0 0.00048828125 0.00048828125
1.0 0.00024414063 0.00024414063
1.0 0.00012207031 0.00012207031
1.0 6.1035156e-05 6.1035156e-05
1.0 3.0517578e-05 3.0517578e-05
1.0 1.5258789e-05 1.5258789e-05
1.0 7.6293945e-06 7.6293945e-06
1.0 3.8146973e-06 3.8146973e-06
1.0 1.9073486e-06 1.9073486e-06
1.0 9.536743e-07 9.536743e-07
1.0 4.7683716e-07 4.7683716e-07
1.0 2.3841858e-07 2.3841858e-07
1.0 1.1920929e-07 1.1920929e-07
1.0 5.9604645e-08 0.0

(with many rows truncated)

a h err
1.0 0.5 0.5
1.0 0.25 0.25
1.0 0.125 0.125
1.0 0.0625 0.0625
1.0 1.9073486e-06 1.9073486e-06
1.0 9.536743e-07 9.536743e-07
1.0 4.7683716e-07 4.7683716e-07
1.0 2.3841858e-07 2.3841858e-07
1.0 1.1920929e-07 1.1920929e-07
1.0 5.9604645e-08 0.0

$\epsilon_{\text{mac}}$ ≈ 5.9604 ⋅ 10-8

Question Two

Computing $\epsilon_{\text{mac}}$ for double precision numbers:

  (let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
                                                     1.0d0
                                                     1.0d0)))
    (lizfcm.utils:table (:headers '("a" "h" "err")
                         :domain-order (a h err)
                         :domain-values domain-values)))
a h err
1.0d0 0.5d0 0.5d0
1.0d0 0.25d0 0.25d0
1.0d0 0.125d0 0.125d0
1.0d0 0.0625d0 0.0625d0
1.0d0 0.03125d0 0.03125d0
1.0d0 0.015625d0 0.015625d0
1.0d0 0.0078125d0 0.0078125d0
1.0d0 0.00390625d0 0.00390625d0
1.0d0 0.001953125d0 0.001953125d0
1.0d0 9.765625d-4 9.765625d-4
1.0d0 4.8828125d-4 4.8828125d-4
1.0d0 2.44140625d-4 2.44140625d-4
1.0d0 1.220703125d-4 1.220703125d-4
1.0d0 6.103515625d-5 6.103515625d-5
1.0d0 3.0517578125d-5 3.0517578125d-5
1.0d0 1.52587890625d-5 1.52587890625d-5
1.0d0 7.62939453125d-6 7.62939453125d-6
1.0d0 3.814697265625d-6 3.814697265625d-6
1.0d0 1.9073486328125d-6 1.9073486328125d-6
1.0d0 9.5367431640625d-7 9.5367431640625d-7
1.0d0 4.76837158203125d-7 4.76837158203125d-7
1.0d0 2.384185791015625d-7 2.384185791015625d-7
1.0d0 1.1920928955078125d-7 1.1920928955078125d-7
1.0d0 5.960464477539063d-8 5.960464477539063d-8
1.0d0 2.9802322387695313d-8 2.9802322387695313d-8
1.0d0 1.4901161193847656d-8 1.4901161193847656d-8
1.0d0 7.450580596923828d-9 7.450580596923828d-9
1.0d0 3.725290298461914d-9 3.725290298461914d-9
1.0d0 1.862645149230957d-9 1.862645149230957d-9
1.0d0 9.313225746154785d-10 9.313225746154785d-10
1.0d0 4.656612873077393d-10 4.656612873077393d-10
1.0d0 2.3283064365386963d-10 2.3283064365386963d-10
1.0d0 1.1641532182693481d-10 1.1641532182693481d-10
1.0d0 5.820766091346741d-11 5.820766091346741d-11
1.0d0 2.9103830456733704d-11 2.9103830456733704d-11
1.0d0 1.4551915228366852d-11 1.4551915228366852d-11
1.0d0 7.275957614183426d-12 7.275957614183426d-12
1.0d0 3.637978807091713d-12 3.637978807091713d-12
1.0d0 1.8189894035458565d-12 1.8189894035458565d-12
1.0d0 9.094947017729282d-13 9.094947017729282d-13
1.0d0 4.547473508864641d-13 4.547473508864641d-13
1.0d0 2.2737367544323206d-13 2.2737367544323206d-13
1.0d0 1.1368683772161603d-13 1.1368683772161603d-13
1.0d0 5.684341886080802d-14 5.684341886080802d-14
1.0d0 2.842170943040401d-14 2.842170943040401d-14
1.0d0 1.4210854715202004d-14 1.4210854715202004d-14
1.0d0 7.105427357601002d-15 7.105427357601002d-15
1.0d0 3.552713678800501d-15 3.552713678800501d-15
1.0d0 1.7763568394002505d-15 1.7763568394002505d-15
1.0d0 8.881784197001252d-16 8.881784197001252d-16
1.0d0 4.440892098500626d-16 4.440892098500626d-16
1.0d0 2.220446049250313d-16 2.220446049250313d-16
1.0d0 1.1102230246251565d-16 0.0d0

(with many rows truncated)

a h err
1.0d0 0.5d0 0.5d0
1.0d0 0.25d0 0.25d0
1.0d0 0.125d0 0.125d0
1.0d0 0.0625d0 0.0625d0
1.0d0 0.03125d0 0.03125d0
1.0d0 0.015625d0 0.015625d0
1.0d0 0.0078125d0 0.0078125d0
1.0d0 0.00390625d0 0.00390625d0
1.0d0 0.001953125d0 0.001953125d0
1.0d0 7.105427357601002d-15 7.105427357601002d-15
1.0d0 3.552713678800501d-15 3.552713678800501d-15
1.0d0 1.7763568394002505d-15 1.7763568394002505d-15
1.0d0 8.881784197001252d-16 8.881784197001252d-16
1.0d0 4.440892098500626d-16 4.440892098500626d-16
1.0d0 2.220446049250313d-16 2.220446049250313d-16
1.0d0 1.1102230246251565d-16 0.0d0

Thus, $\epsilon_{\text{mac}}$ ≈ 1.1102 ⋅ 10-16

Question Three - |v|_2 

  (let ((vs '((1 1) (2 3) (4 5) (-1 2)))
        (2-norm (lizfcm.vector:p-norm 2)))
    (lizfcm.utils:table (:headers '("x" "y" "2norm")
                         :domain-order (x y)
                         :domain-values vs)
      (funcall 2-norm (list x y))))
x y 2norm
1 1 1.4142135
2 3 3.6055512
4 5 6.4031243
-1 2 2.236068

Question Four - |v|_1 

  (let ((vs '((1 1) (2 3) (4 5) (-1 2)))
        (1-norm (lizfcm.vector:p-norm 1)))
    (lizfcm.utils:table (:headers '("x" "y" "1norm")
                         :domain-order (x y)
                         :domain-values vs)
      (funcall 1-norm (list x y))))
x y 1norm
1 1 2
2 3 5
4 5 9
-1 2 3

Question Five - |v|∞ 

  (let ((vs '((1 1) (2 3) (4 5) (-1 2))))
    (lizfcm.utils:table (:headers '("x" "y" "max-norm")
                         :domain-order (x y)
                         :domain-values vs)
      (lizfcm.vector:max-norm (list x y))))
x y infty-norm
1 1 1
2 3 3
4 5 5
-1 2 2

Question Six - ||v - u|| via |v|2 

  (let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
         (vs2 '((7 9) (2 2) (8 -1) (4 4)))
         (2-norm (lizfcm.vector:p-norm 2)))
    (lizfcm.utils:table (:headers '("v1" "v2" "2-norm-d")
                         :domain-order (v1 v2)
                         :domain-values (mapcar (lambda (v1 v2)
                                                  (list v1 v2))
                                                vs
                                                vs2))
      (lizfcm.vector:distance v1 v2 2-norm)))
v1 v2 2-norm
(1 1) (7 9) 10.0
(2 3) (2 2) 1.0
(4 5) (8 -1) 7.2111025
(-1 2) (4 4) 5.3851647

Question Seven - ||v - u|| via |v|1 

  (let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
         (vs2 '((7 9) (2 2) (8 -1) (4 4)))
         (1-norm (lizfcm.vector:p-norm 1)))
    (lizfcm.utils:table (:headers '("v1" "v2" "1-norm-d")
                         :domain-order (v1 v2)
                         :domain-values (mapcar (lambda (v1 v2)
                                                  (list v1 v2))
                                                vs
                                                vs2))
      (lizfcm.vector:distance v1 v2 1-norm)))
v1 v2 1-norm-d
(1 1) (7 9) 14
(2 3) (2 2) 1
(4 5) (8 -1) 10
(-1 2) (4 4) 7

Question Eight - ||v - u|| via |v|∞ 

  (let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
         (vs2 '((7 9) (2 2) (8 -1) (4 4))))
    (lizfcm.utils:table (:headers '("v1" "v2" "max-norm-d")
                         :domain-order (v1 v2)
                         :domain-values (mapcar (lambda (v1 v2)
                                                  (list v1 v2))
                                                vs
                                                vs2))
      (lizfcm.vector:distance v1 v2 'lizfcm.vector:max-norm)))
v1 v2 max-norm-d
(1 1) (7 9) -6
(2 3) (2 2) 1
(4 5) (8 -1) 6
(-1 2) (4 4) -2