{"text_format":"3","workbook":"xlsx","n_sheets":6,"sheets":["Regularize 2D Formula Example","10-Point Example","21-Point Example","Projections","Backsolving","Second-Derivative Equation"],"not_rendered":{"charts":3,"images":1,"macros":1},"read":"whole","date_system":"1900"}

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["This worksheet shows how to use the VBA function as an Excel array function and gives five examples of its use."]
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[{"$nulls":7},3.59480000000011,1.270617353206894]
[{"$nulls":7},3.59520000000011,1.270142132360596]
[{"$nulls":7},3.59560000000011,1.2696669115142787]
[{"$nulls":7},3.59600000000011,1.269191690667944]
[{"$nulls":7},3.59640000000011,1.2687164698215951]
[{"$nulls":7},3.59680000000011,1.2682412489752348]
[{"$nulls":7},3.59720000000011,1.2677660281288663]
[{"$nulls":7},3.59760000000011,1.2672908072824924]
[{"$nulls":7},3.59800000000011,1.266815586436116]
[{"$nulls":7},3.59840000000011,1.2663403655897385]
[{"$nulls":7},3.59880000000011,1.265865144743361]
[{"$nulls":7},3.59920000000011,1.265389923896984]
[{"$nulls":7},3.59960000000011,1.2649147030506072]
[{"$nulls":7},3.60000000000011,1.2644394822042304]

{"sheet":"10-Point Example","index":1,"n_rows":89,"n_cols":28,"rows_kept":89,"merged":[[2,1,2,2],[10,6,13,9],[11,12,16,13],[16,1,19,3],[29,12,35,13],[47,12,52,13],[52,1,56,9],[63,19,72,22],[79,19,88,22]]}
["This worksheet shows the details of an example calculation, with the answer in Cells AB80:AB89.  This is the example used in the Math for Mere Mortals article."]
[]
[null,"Original Dataset",null,null,"Output Grid",null,"Interpolation:","Original Data"]
[null,"x","y",null,"x",null,"x","Low Index","High Index","Low Value","High Value","Low Weight","High Weight",null,null,1,2,3,4,5,6,7,8,9,10]
[null,0,1,null,-0.5,1,{"$f":"B5","v":0},{"$f":"IF(AND(G5>=$H$17,G5<=$H$19),IF(G5<$H$19,MATCH(G5,$E$5:$E$17,1),$H$18-1),0)","v":2},{"$f":"IF(H5=0,0,H5+1)","v":3},{"$f":"INDEX($E$5:$E$17,H5)","v":0},0.5,{"$f":"(K5-G5)/($K5-$J5)","v":1},{"$f":"(G5-J5)/($K5-$J5)","v":0},null,null,{"$f":"IF($H5=P$4,$L5,0)+IF($I5=P$4,$M5,0)","v":0},1,0,0,0,0,0,0,0,0,null,null,{"$f":"C5","v":1}]
[null,0.55,1.1,null,0,2,{"$f":"B6","v":0.55},{"$f":"IF(AND(G6>=$H$17,G6<=$H$19),IF(G6<$H$19,MATCH(G6,$E$5:$E$17,1),$H$18-1),0)","v":3},{"$f":"IF(H6=0,0,H6+1)","v":4},0.5,1,{"$f":"(K6-G6)/($K6-$J6)","v":0.8999999999999999},{"$f":"(G6-J6)/($K6-$J6)","v":0.10000000000000009},null,null,0,0,0.8999999999999999,0.10000000000000009,0,0,0,0,0,0,null,null,{"$f":"C6","v":1.1}]
[null,1.1,1.5,null,0.5,3,{"$f":"B7","v":1.1},{"$f":"IF(AND(G7>=$H$17,G7<=$H$19),IF(G7<$H$19,MATCH(G7,$E$5:$E$17,1),$H$18-1),0)","v":4},{"$f":"IF(H7=0,0,H7+1)","v":5},1,1.5,{"$f":"(K7-G7)/($K7-$J7)","v":0.7999999999999998},{"$f":"(G7-J7)/($K7-$J7)","v":0.20000000000000018},null,null,0,0,0,0.7999999999999998,0.20000000000000018,0,0,0,0,0,null,null,{"$f":"C7","v":1.5}]
[null,2.6,2.5,null,1,4,{"$f":"B8","v":2.6},{"$f":"IF(AND(G8>=$H$17,G8<=$H$19),IF(G8<$H$19,MATCH(G8,$E$5:$E$17,1),$H$18-1),0)","v":7},{"$f":"IF(H8=0,0,H8+1)","v":8},2.5,3,{"$f":"(K8-G8)/($K8-$J8)","v":0.7999999999999998},{"$f":"(G8-J8)/($K8-$J8)","v":0.20000000000000018},null,null,0,0,0,0,0,0,0.7999999999999998,0.20000000000000018,0,0,null,null,{"$f":"C8","v":2.5}]
[null,2.99,1.9,null,1.5,5,{"$f":"B9","v":2.99},{"$f":"IF(AND(G9>=$H$17,G9<=$H$19),IF(G9<$H$19,MATCH(G9,$E$5:$E$17,1),$H$18-1),0)","v":7},{"$f":"IF(H9=0,0,H9+1)","v":8},2.5,3,{"$f":"(K9-G9)/($K9-$J9)","v":0.019999999999999574},{"$f":"(G9-J9)/($K9-$J9)","v":0.9800000000000004},null,null,0,0,0,0,0,0,0.019999999999999574,0.9800000000000004,0,0,null,null,{"$f":"C9","v":1.9}]
[{"$nulls":4},2,{"$nulls":10},{"$f":"2*Q$24*Q$23","v":4},{"$f":"-2*Q$23*Q$25","v":-8},{"$f":"2*Q$25*Q$24","v":4},0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},2.5,null,"Above is the interpolation that maps the five input points onto the ten grid points that will be the output curve.  Each output point is represented by a column in the matrix to the right.",{"$nulls":8},0,{"$f":"2*R$24*R$23","v":4},{"$f":"-2*R$23*R$25","v":-8},{"$f":"2*R$25*R$24","v":4},0,0,0,0,0,0,null,null,0]
[{"$nulls":4},3,{"$nulls":7},"To the right is the precursor to Matrix A showing the fidelity equations in the first five rows and unscaled derivatives in the following rows.",null,null,0,0,{"$f":"2*S$24*S$23","v":4},{"$f":"-2*S$23*S$25","v":-8},{"$f":"2*S$25*S$24","v":4},0,0,0,0,0,null,null,0]
[null,"Stiffness:",0.1,null,3.3,{"$nulls":10},0,0,0,{"$f":"2*T$24*T$23","v":4},{"$f":"-2*T$23*T$25","v":-8},{"$f":"2*T$25*T$24","v":4},0,0,0,0,null,null,0]
[null,"Equation Ratio:",{"$f":"SQRT(COUNT(P5:P9)/COUNT(P10:P17))","v":0.7905694150420949},null,3.6,{"$nulls":10},0,0,0,0,{"$f":"2*U$24*U$23","v":4},{"$f":"-2*U$23*U$25","v":-8},{"$f":"2*U$25*U$24","v":4},0,0,0,null,null,0]
[null,"Constant:",{"$f":"C13*C14","v":0.0790569415042095},{"$nulls":12},0,0,0,0,0,{"$f":"2*V$24*V$23","v":4},{"$f":"-2*V$23*V$25","v":-8},{"$f":"2*V$25*V$24","v":4},0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,{"$f":"2*W$24*W$23","v":5.000000000000001},{"$f":"-2*W$23*W$25","v":-13.333333333333341},{"$f":"2*W$25*W$24","v":8.33333333333334},0,null,null,0]
[null,"Stiffness represents the weighing of the goodness of fit against the smoothness.  1 means equal weighting, 2 gives double weight to the smoothness, etc.",{"$nulls":4},"Min:",{"$f":"MIN(E5:E17)","v":-0.5},{"$nulls":7},0,0,0,0,0,0,0,{"$f":"2*X$24*X$23","v":11.111111111111116},{"$f":"-2*X$23*X$25","v":-22.222222222222214},{"$f":"2*X$25*X$24","v":11.1111111111111},null,null,0]
[{"$nulls":6},"LastIndex:",{"$f":"COUNT(E5:E17)","v":10}]
[{"$nulls":6},"Max:",{"$f":"MAX(E5:E17)","v":3.6}]
[]
[{"$nulls":15},{"$f":"INDEX($E$5:$E$17,P4,1)","v":-0.5},0,0.5,1,1.5,2,2.5,3,3.3,3.6]
[]
[{"$nulls":14},"Derivatives:",null,{"$f":"1/(Q21-P21)","v":2},{"$f":"1/(R21-Q21)","v":2},2,2,2,2,2,3.3333333333333353,3.3333333333333304]
[{"$nulls":16},{"$f":"1/(R21-P21)","v":1},{"$f":"1/(S21-Q21)","v":1},1,1,1,1,1.2500000000000002,1.6666666666666665,-0.30303030303030304]
[{"$nulls":16},{"$f":"1/(R21-Q21)","v":2},{"$f":"1/(S21-R21)","v":2},2,2,2,2,3.3333333333333353,3.3333333333333304,-0.2777777777777778]
[]
[]
[]
[]
[{"$nulls":12},"This is the matrix A.  The first five rows are the fidelity equations based on the five input points.  The bottom eight rows are scaled second derivatives.",null,null,{"$f":"P5","v":0},1,0,0,0,0,0,0,0,0]
[{"$nulls":15},{"$f":"P6","v":0},0,0.8999999999999999,0.10000000000000009,0,0,0,0,0,0]
[{"$nulls":15},{"$f":"P7","v":0},0,0,0.7999999999999998,0.20000000000000018,0,0,0,0,0]
[{"$nulls":15},{"$f":"P8","v":0},0,0,0,0,0,0.7999999999999998,0.20000000000000018,0,0]
[{"$nulls":15},{"$f":"P9","v":0},0,0,0,0,0,0.019999999999999574,0.9800000000000004,0,0]
[{"$nulls":15},{"$f":"P10*$C$15","v":0.316227766016838},-0.632455532033676,0.316227766016838,0,0,0,0,0,0,0]
[{"$nulls":15},{"$f":"P11*$C$15","v":0},0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,0]
[{"$nulls":15},{"$f":"P12*$C$15","v":0},0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0]
[{"$nulls":15},{"$f":"P13*$C$15","v":0},0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0]
[{"$nulls":15},{"$f":"P14*$C$15","v":0},0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0]
[{"$nulls":15},{"$f":"P15*$C$15","v":0},0,0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0]
[{"$nulls":15},{"$f":"P16*$C$15","v":0},0,0,0,0,0,0.39528470752104755,-1.0540925533894605,0.658807845868413,0]
[{"$nulls":15},{"$f":"P17*$C$15","v":0},0,0,0,0,0,0,0.8784104611578837,-1.756820922315766,0.8784104611578825]
[]
[]
[]
[]
[]
[{"$nulls":12},"This is the matrix A projected onto itself.  To the right is the vector b after being projected onto A.  In 2D this is always pentadiagonal.",null,null,{"$f":"MMULT(TRANSPOSE(P30:Y42),P30:Y42)","v":0.10000000000000003},-0.20000000000000007,0.10000000000000003,0,0,0,0,0,0,0,null,null,{"$f":"MMULT(TRANSPOSE(P30:Y42),AB5:AB17)","v":0}]
[{"$nulls":15},-0.20000000000000007,1.5000000000000002,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,0,null,null,1]
[{"$nulls":15},0.10000000000000003,-0.40000000000000013,1.4100000000000001,-0.31000000000000005,0.10000000000000003,0,0,0,0,0,null,null,0.99]
[{"$nulls":15},0,0.10000000000000003,-0.31000000000000005,1.25,-0.24000000000000002,0.10000000000000003,0,0,0,0,null,null,1.3099999999999998]
[{"$nulls":15},0,0,0.10000000000000003,-0.24000000000000002,0.6400000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,null,null,0.30000000000000027]
[null,"As long as the algorithm can maintain numerical stability, all three of these datasets represent the same curves characteristic of the original five datapoints and the stiffness constant.  The only differences between them comes from the quantization error of 7 points vs. 10,000 points.  The trick to keeping the curves consistent is the scaling factor in Cell C15, which adjusts the magnitudes of the residuals from the derivative equations so that they do not dominate the calculations when there are many of them.",{"$nulls":13},0,0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0.10000000000000003,-0.40000000000000013,1.29665,-0.43706666666666744,0.260416666666667,0,null,null,2.037999999999999]
[{"$nulls":15},0,0,0,0,0,0.10000000000000003,-0.43706666666666744,2.98311604938272,-2.237654320987656,0.7716049382716048,null,null,2.362000000000001]
[{"$nulls":15},0,0,0,0,0,0,0.260416666666667,-2.237654320987656,3.5204475308641974,-1.5432098765432085,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0.7716049382716048,-1.5432098765432085,0.7716049382716037,null,null,0]
[]
[]
[]
[]
[]
[]
[{"$nulls":19},"We have to excuse Excel for having a matrix inverse function.  That is the only way to solve a linear system of equations using native Excel formulas; there is no MSOLVE(A,b) function in Excel.",{"$nulls":6},"This is the Excel solution.  -->",{"$f":"MMULT(MINVERSE(P48:Y57),AB48:AB57)","v":0.8634409505624512}]
[{"$nulls":27},0.9788708474855378]
[{"$nulls":27},1.094300744408625]
[{"$nulls":27},1.4210221664763332]
[{"$nulls":27},1.9275706592947177]
[{"$nulls":27},2.3768539615343527]
[{"$nulls":27},2.4871160817857914]
[{"$nulls":27},1.9766010286395814]
[{"$nulls":27},1.670291996751856]
[{"$nulls":27},1.3639829648641308]
[]
[]
[]
[]
[]
[]
[{"$nulls":19},"The VBA solution does this much more efficiently.  It creates the A'A matrix directly from the input points and solves the resulting system using a pentadiagonal algorithm.  Of course it arrives at the same solution as the Excel method.",{"$nulls":6},"This is the VBA solution.  -->",{"$f":"Regularizedata2D(B5:B9,C5:C9,C13,E5:E14)","v":0.8634409505624503}]
[{"$nulls":27},0.9788708474855379]
[{"$nulls":27},1.0943007444086255]
[{"$nulls":27},1.4210221664763336]
[{"$nulls":27},1.927570659294717]
[{"$nulls":27},2.376853961534352]
[{"$nulls":27},2.487116081785791]
[{"$nulls":27},1.9766010286395836]
[{"$nulls":27},1.6702919967518617]
[{"$nulls":27},1.3639829648641388]

{"sheet":"21-Point Example","index":2,"n_rows":145,"n_cols":39,"rows_kept":145,"merged":[[2,1,2,2],[10,6,13,9],[11,12,16,13],[16,1,19,3],[41,12,47,13],[70,12,75,13],[97,19,106,22],[124,19,133,22]]}
["This worksheet shows the details of another example calculation with 21 output points, with the answer in Cells AM125:AM145."]
[]
[null,"Original Dataset",null,null,"Output Grid",null,"Interpolation:","Original Data"]
[null,"x","y",null,"x",null,"x","Low Index","High Index","Low Value","High Value","Low Weight","High Weight",null,null,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]
[null,0,1,null,-0.4,1,{"$f":"B5","v":0},{"$f":"IF(AND(G5>=$H$17,G5<=$H$19),IF(G5<$H$19,MATCH(G5,$E$5:$E$25,1),$H$18-1),0)","v":3},{"$f":"IF(H5=0,0,H5+1)","v":4},{"$f":"INDEX($E$5:$E$25,H5)","v":0},0.2,{"$f":"(K5-G5)/($K5-$J5)","v":1},{"$f":"(G5-J5)/($K5-$J5)","v":0},null,null,{"$f":"IF($H5=P$4,$L5,0)+IF($I5=P$4,$M5,0)","v":0},0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"C5","v":1}]
[null,0.55,1.1,null,-0.2,2,{"$f":"B6","v":0.55},{"$f":"IF(AND(G6>=$H$17,G6<=$H$19),IF(G6<$H$19,MATCH(G6,$E$5:$E$25,1),$H$18-1),0)","v":5},{"$f":"IF(H6=0,0,H6+1)","v":6},0.4,0.6,{"$f":"(K6-G6)/($K6-$J6)","v":0.24999999999999972},{"$f":"(G6-J6)/($K6-$J6)","v":0.7500000000000002},null,null,0,0,0,0,0.24999999999999972,0.7500000000000002,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"C6","v":1.1}]
[null,1.1,1.5,null,0,3,{"$f":"B7","v":1.1},{"$f":"IF(AND(G7>=$H$17,G7<=$H$19),IF(G7<$H$19,MATCH(G7,$E$5:$E$25,1),$H$18-1),0)","v":8},{"$f":"IF(H7=0,0,H7+1)","v":9},1,1.2,{"$f":"(K7-G7)/($K7-$J7)","v":0.49999999999999944},{"$f":"(G7-J7)/($K7-$J7)","v":0.5000000000000006},null,null,0,0,0,0,0,0,0,0.49999999999999944,0.5000000000000006,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"C7","v":1.5}]
[null,2.6,2.5,null,0.2,4,{"$f":"B8","v":2.6},{"$f":"IF(AND(G8>=$H$17,G8<=$H$19),IF(G8<$H$19,MATCH(G8,$E$5:$E$25,1),$H$18-1),0)","v":16},{"$f":"IF(H8=0,0,H8+1)","v":17},2.6,2.8,{"$f":"(K8-G8)/($K8-$J8)","v":1},{"$f":"(G8-J8)/($K8-$J8)","v":0},null,null,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,null,null,{"$f":"C8","v":2.5}]
[null,2.99,1.9,null,0.4,5,{"$f":"B9","v":2.99},{"$f":"IF(AND(G9>=$H$17,G9<=$H$19),IF(G9<$H$19,MATCH(G9,$E$5:$E$25,1),$H$18-1),0)","v":17},{"$f":"IF(H9=0,0,H9+1)","v":18},2.8,3,{"$f":"(K9-G9)/($K9-$J9)","v":0.04999999999999889},{"$f":"(G9-J9)/($K9-$J9)","v":0.9500000000000011},null,null,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.04999999999999889,0.9500000000000011,0,0,0,null,null,{"$f":"C9","v":1.9}]
[{"$nulls":4},0.6,{"$nulls":10},{"$f":"2*Q$36*Q$35","v":25},{"$f":"-2*Q$35*Q$37","v":-50},{"$f":"2*Q$37*Q$36","v":25},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},0.8,null,"Above is the interpolation that maps the five input points onto the ten grid points that will be the output curve.  Each output point is represented by a column in the matrix to the right.",{"$nulls":8},0,{"$f":"2*R$36*R$35","v":25},{"$f":"-2*R$35*R$37","v":-50},{"$f":"2*R$37*R$36","v":25},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},1,{"$nulls":7},"To the right is the precursor to Matrix A showing the fidelity equations in the first five rows and unscaled derivatives in the following rows.",null,null,0,0,{"$f":"2*S$36*S$35","v":25},{"$f":"-2*S$35*S$37","v":-50},{"$f":"2*S$37*S$36","v":25},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[null,"Stiffness:",0.1,null,1.2,{"$nulls":10},0,0,0,{"$f":"2*T$36*T$35","v":25},{"$f":"-2*T$35*T$37","v":-50.00000000000001},{"$f":"2*T$37*T$36","v":25.000000000000004},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[null,"Equation Ratio:",{"$f":"SQRT(COUNT(P5:P9)/COUNT(P10:P28))","v":0.512989176042577},null,1.4,{"$nulls":10},0,0,0,0,{"$f":"2*U$36*U$35","v":25.000000000000004},{"$f":"-2*U$35*U$37","v":-49.99999999999999},{"$f":"2*U$37*U$36","v":24.999999999999993},0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[null,"Constant:",{"$f":"C13*C14","v":0.0512989176042577},null,1.6,{"$nulls":10},0,0,0,0,0,{"$f":"2*V$36*V$35","v":24.999999999999993},{"$f":"-2*V$35*V$37","v":-49.99999999999999},{"$f":"2*V$37*V$36","v":25.000000000000004},0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},1.8,{"$nulls":10},0,0,0,0,0,0,{"$f":"2*W$36*W$35","v":25.000000000000007},{"$f":"-2*W$35*W$37","v":-50.000000000000014},{"$f":"2*W$37*W$36","v":25.000000000000007},0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[null,"Stiffness represents the weighing of the goodness of fit against the smoothness.  1 means equal weighting, 2 gives double weight to the smoothness, etc.",null,null,2,null,"Min:",{"$f":"MIN(E5:E25)","v":-0.4},{"$nulls":7},0,0,0,0,0,0,0,{"$f":"2*X$36*X$35","v":25.000000000000007},{"$f":"-2*X$35*X$37","v":-50.000000000000014},{"$f":"2*X$37*X$36","v":25.000000000000007},0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},2.2,null,"LastIndex:",{"$f":"COUNT(E5:E25)","v":21},{"$nulls":7},0,0,0,0,0,0,0,0,{"$f":"2*Y$36*Y$35","v":24.999999999999996},{"$f":"-2*Y$35*Y$37","v":-49.999999999999964},{"$f":"2*Y$37*Y$36","v":24.999999999999968},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},2.4,null,"Max:",{"$f":"MAX(E5:E25)","v":3.6},{"$nulls":7},0,0,0,0,0,0,0,0,0,{"$f":"2*Z$36*Z$35","v":24.999999999999968},{"$f":"-2*Z$35*Z$37","v":-49.999999999999964},{"$f":"2*Z$37*Z$36","v":24.999999999999996},0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},2.6,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,{"$f":"2*AA$36*AA$35","v":25.000000000000007},{"$f":"-2*AA$35*AA$37","v":-50.000000000000014},{"$f":"2*AA$37*AA$36","v":25.000000000000007},0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},2.8,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AB$36*AB$35","v":24.999999999999996},{"$f":"-2*AB$35*AB$37","v":-49.999999999999964},{"$f":"2*AB$37*AB$36","v":24.999999999999968},0,0,0,0,0,0,0,null,null,0]
[{"$nulls":4},3,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AC$36*AC$35","v":24.999999999999982},{"$f":"-2*AC$35*AC$37","v":-50.00000000000003},{"$f":"2*AC$37*AC$36","v":25.00000000000004},0,0,0,0,0,0,null,null,0]
[{"$nulls":4},3.2,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AD$36*AD$35","v":25.00000000000004},{"$f":"-2*AD$35*AD$37","v":-50.00000000000003},{"$f":"2*AD$37*AD$36","v":24.999999999999982},0,0,0,0,0,null,null,0]
[{"$nulls":4},3.4,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AE$36*AE$35","v":24.999999999999982},{"$f":"-2*AE$35*AE$37","v":-50.00000000000003},{"$f":"2*AE$37*AE$36","v":25.00000000000004},0,0,0,0,null,null,0]
[{"$nulls":4},3.6,{"$nulls":10},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AF$36*AF$35","v":25.00000000000004},{"$f":"-2*AF$35*AF$37","v":-50.00000000000003},{"$f":"2*AF$37*AF$36","v":24.999999999999982},0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AG$36*AG$35","v":24.999999999999957},{"$f":"-2*AG$35*AG$37","v":-49.999999999999915},{"$f":"2*AG$37*AG$36","v":24.999999999999957},0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AH$36*AH$35","v":24.999999999999982},{"$f":"-2*AH$35*AH$37","v":-50.00000000000003},{"$f":"2*AH$37*AH$36","v":25.00000000000004},0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,{"$f":"2*AI$36*AI$35","v":25.00000000000004},{"$f":"-2*AI$35*AI$37","v":-50.00000000000003},{"$f":"2*AI$37*AI$36","v":24.999999999999982},null,null,0]
[]
[]
[]
[]
[{"$nulls":15},{"$f":"INDEX($E$5:$E$25,P4,1)","v":-0.4},{"$f":"INDEX($E$5:$E$25,Q4,1)","v":-0.2},0,0.2,0.4,0.6,0.8,1,1.2,1.4,1.6,1.8,2,2.2,2.4,2.6,2.8,3,3.2,3.4,3.6]
[]
[{"$nulls":14},"Derivatives:",null,{"$f":"1/(Q33-P33)","v":5},5,5,5,5.000000000000001,4.999999999999998,5.000000000000001,5.000000000000001,5.000000000000001,4.999999999999996,5.000000000000001,5.000000000000001,4.999999999999996,5.000000000000007,4.999999999999996,5.000000000000007,4.999999999999996,4.999999999999996,5.000000000000007,4.999999999999996]
[{"$nulls":16},{"$f":"1/(R33-P33)","v":2.5},2.5,2.5,2.5,2.5,2.5,2.5000000000000004,2.5000000000000004,2.499999999999999,2.499999999999999,2.5000000000000004,2.499999999999999,2.5000000000000004,2.5000000000000004,2.5000000000000004,2.5000000000000004,2.499999999999998,2.5000000000000004,2.5000000000000004,-0.29411764705882354]
[{"$nulls":16},{"$f":"1/(R33-Q33)","v":5},5,5,5.000000000000001,4.999999999999998,5.000000000000001,5.000000000000001,5.000000000000001,4.999999999999996,5.000000000000001,5.000000000000001,4.999999999999996,5.000000000000007,4.999999999999996,5.000000000000007,4.999999999999996,4.999999999999996,5.000000000000007,4.999999999999996,-0.2777777777777778]
[]
[]
[]
[]
[{"$nulls":12},"This is the matrix A.  The first five rows are the fidelity equations based on the five input points.  The bottom eight rows are scaled second derivatives.",null,null,{"$f":"P5","v":0},{"$f":"Q5","v":0},1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"AM5","v":1}]
[{"$nulls":15},{"$f":"P6","v":0},0,0,0,0.24999999999999972,0.7500000000000002,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"AM6","v":1.1}]
[{"$nulls":15},{"$f":"P7","v":0},0,0,0,0,0,0,0.49999999999999944,0.5000000000000006,0,0,0,0,0,0,0,0,0,0,0,0,null,null,1.5]
[{"$nulls":15},{"$f":"P8","v":0},0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,null,null,2.5]
[{"$nulls":15},{"$f":"P9","v":0},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.04999999999999889,0.9500000000000011,0,0,0,null,null,1.9]
[{"$nulls":15},{"$f":"P10*$C$15","v":1.2824729401064425},-2.564945880212885,1.2824729401064425,0,0,0,0,0,0,0,{"$f":"Z10*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,1.2824729401064425,-2.564945880212885,1.2824729401064425,0,0,0,0,0,0,{"$f":"Z11*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,1.2824729401064425,-2.564945880212885,1.2824729401064425,0,0,0,0,0,{"$f":"Z12*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,1.2824729401064425,-2.5649458802128855,1.2824729401064427,0,0,0,0,{"$f":"Z13*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,1.2824729401064427,-2.5649458802128846,1.282472940106442,0,0,0,{"$f":"Z14*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,1.282472940106442,-2.5649458802128846,1.2824729401064427,0,0,{"$f":"Z15*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,1.2824729401064427,-2.5649458802128855,1.2824729401064427,0,{"$f":"Z16*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,{"$f":"W17*$C$15","v":1.2824729401064427},-2.5649458802128855,1.2824729401064427,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P18*$C$15","v":0},0,0,0,0,0,0,{"$f":"W18*$C$15","v":0},1.2824729401064423,-2.5649458802128833,1.2824729401064408,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P19*$C$15","v":0},0,0,0,0,0,0,{"$f":"W19*$C$15","v":0},0,1.2824729401064408,-2.5649458802128833,1.2824729401064423,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P20*$C$15","v":0},0,0,0,0,0,0,{"$f":"W20*$C$15","v":0},0,0,1.2824729401064427,-2.5649458802128855,1.2824729401064427,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P21*$C$15","v":0},0,0,0,0,0,0,{"$f":"W21*$C$15","v":0},0,0,0,1.2824729401064423,-2.5649458802128833,1.2824729401064408,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P22*$C$15","v":0},0,0,0,0,0,0,{"$f":"W22*$C$15","v":0},0,0,0,0,1.2824729401064416,-2.5649458802128864,1.2824729401064445,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P23*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB23*$C$15","v":0},1.2824729401064445,-2.5649458802128864,1.2824729401064416,0,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P24*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB24*$C$15","v":0},0,1.2824729401064416,-2.5649458802128864,1.2824729401064445,0,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P25*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB25*$C$15","v":0},0,0,1.2824729401064445,-2.5649458802128864,1.2824729401064416,0,0,0,null,null,0]
[{"$nulls":15},{"$f":"P26*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB26*$C$15","v":0},0,0,0,1.2824729401064403,-2.5649458802128806,1.2824729401064403,0,0,null,null,0]
[{"$nulls":15},{"$f":"P27*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB27*$C$15","v":0},0,0,0,0,1.2824729401064416,-2.5649458802128864,1.2824729401064445,0,null,null,0]
[{"$nulls":15},{"$f":"P28*$C$15","v":0},0,0,0,0,0,0,0,0,0,0,0,{"$f":"AB28*$C$15","v":0},0,0,0,0,0,1.2824729401064445,-2.5649458802128864,1.2824729401064416,null,null,0]
[]
[]
[]
[]
[]
[{"$nulls":12},"This is the matrix A projected onto itself.  To the right is the vector b after being projected onto A.  In 2D this is always pentadiagonal.",null,null,{"$f":"MMULT(TRANSPOSE(P42:AJ65),P42:AJ65)","v":1.6447368421052628},-3.2894736842105257,1.6447368421052628,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,{"$f":"MMULT(TRANSPOSE(P42:AJ65),AM42:AM65)","v":0}]
[{"$nulls":15},-3.2894736842105257,8.223684210526315,-6.578947368421051,1.6447368421052628,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},1.6447368421052628,-6.578947368421051,10.868421052631579,-6.578947368421051,1.6447368421052628,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,1]
[{"$nulls":15},0,1.6447368421052628,-6.578947368421051,9.868421052631579,-6.578947368421052,1.6447368421052633,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,1.6447368421052628,-6.578947368421052,9.93092105263158,-6.391447368421053,1.6447368421052626,0,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0.2749999999999997]
[{"$nulls":15},0,0,0,1.6447368421052633,-6.391447368421053,10.430921052631575,-6.578947368421049,1.6447368421052626,0,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0.8250000000000003]
[{"$nulls":15},0,0,0,0,1.6447368421052626,-6.578947368421049,9.868421052631575,-6.578947368421053,1.6447368421052635,0,0,0,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,1.6447368421052626,-6.578947368421053,10.11842105263158,-6.328947368421054,1.6447368421052635,0,0,0,0,0,0,0,0,0,0,0,null,null,0.7499999999999991]
[{"$nulls":15},0,0,0,0,0,0,1.6447368421052635,-6.328947368421054,10.11842105263158,-6.5789473684210495,1.6447368421052604,0,0,0,0,0,0,0,0,0,0,null,null,0.7500000000000009]
[{"$nulls":15},0,0,0,0,0,0,0,1.6447368421052635,-6.5789473684210495,9.868421052631565,-6.578947368421038,1.6447368421052604,0,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,1.6447368421052604,-6.578947368421038,9.868421052631565,-6.5789473684210495,1.6447368421052635,0,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,1.6447368421052604,-6.5789473684210495,9.868421052631579,-6.5789473684210495,1.6447368421052604,0,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,1.6447368421052635,-6.5789473684210495,9.868421052631566,-6.578947368421044,1.6447368421052644,0,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,1.6447368421052604,-6.578947368421044,9.868421052631584,-6.5789473684210655,1.6447368421052644,0,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-6.5789473684210655,9.868421052631586,-6.57894736842105,1.6447368421052644,0,0,0,0,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-6.57894736842105,10.868421052631586,-6.5789473684210655,1.6447368421052644,0,0,0,null,null,2.5]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-6.5789473684210655,9.870921052631584,-6.531447368421041,1.6447368421052573,0,0,null,null,0.09499999999999789]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-6.531447368421041,10.770921052631552,-6.57894736842104,1.6447368421052644,0,null,null,1.805000000000002]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052573,-6.57894736842104,9.868421052631582,-6.5789473684210655,1.6447368421052644,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-6.5789473684210655,8.223684210526326,-3.289473684210525,null,null,0]
[{"$nulls":15},0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1.6447368421052644,-3.289473684210525,1.6447368421052606,null,null,0]
[]
[]
[]
[]
[]
[]
[{"$nulls":19},"We have to excuse Excel for having a matrix inverse function.  That is the only way to solve a linear system of equations using native Excel formulas; there is no MSOLVE(A,b) function in Excel.",{"$nulls":17},"This is the Excel solution.  -->",{"$f":"MMULT(MINVERSE(P71:AJ91),AM71:AM91)","v":0.8970231968744855}]
[{"$nulls":38},0.9366275616375082]
[{"$nulls":38},0.9762319264005311]
[{"$nulls":38},1.0158362911635535]
[{"$nulls":38},1.0698916446750508]
[{"$nulls":38},1.1528489756834972]
[{"$nulls":38},1.274278607211795]
[{"$nulls":38},1.4291088651061377]
[{"$nulls":38},1.6122680752127152]
[{"$nulls":38},1.8123952684492521]
[{"$nulls":38},2.01184018080501]
[{"$nulls":38},2.192952548269247]
[{"$nulls":38},2.3380821068312203]
[{"$nulls":38},2.42957859248019]
[{"$nulls":38},2.4497917412054115]
[{"$nulls":38},2.3810712889961416]
[{"$nulls":38},2.2057669718416415]
[{"$nulls":38},1.9785371820215172]
[{"$nulls":38},1.7513073922013978]
[{"$nulls":38},1.5240776023812794]
[{"$nulls":38},1.2968478125611624]
[]
[]
[]
[]
[]
[]
[{"$nulls":19},"The VBA solution does this much more efficiently.  It creates the A'A matrix directly from the input points and solves the resulting system using a pentadiagonal algorithm.  Of course it arrives at the same solution as the Excel method.",{"$nulls":17},"This is the VBA solution.  -->",{"$f":"Regularizedata2D(B5:B9,C5:C9,C13,E5:E25)","v":0.8970231968744852}]
[{"$nulls":38},0.9366275616375085]
[{"$nulls":38},0.9762319264005318]
[{"$nulls":38},1.0158362911635546]
[{"$nulls":38},1.0698916446750524]
[{"$nulls":38},1.1528489756835003]
[{"$nulls":38},1.274278607211802]
[{"$nulls":38},1.4291088651061488]
[{"$nulls":38},1.6122680752127339]
[{"$nulls":38},1.8123952684492808]
[{"$nulls":38},2.0118401808050455]
[{"$nulls":38},2.1929525482692847]
[{"$nulls":38},2.3380821068312563]
[{"$nulls":38},2.429578592480218]
[{"$nulls":38},2.449791741205429]
[{"$nulls":38},2.3810712889961487]
[{"$nulls":38},2.2057669718416406]
[{"$nulls":38},1.9785371820215096]
[{"$nulls":38},1.7513073922013822]
[{"$nulls":38},1.5240776023812563]
[{"$nulls":38},1.29684781256113]

{"sheet":"Projections","index":3,"n_rows":168,"n_cols":46,"rows_kept":168,"merged":[[8,16,9,25],[19,36,21,39],[20,3,21,12],[20,16,21,25],[47,3,48,12],[47,16,48,25],[78,3,81,12]]}
["This worksheet shows how the projections ATA and ATb are calculated directly."]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10,{"$nulls":3},"Nonzero Columns:"]
[{"$nulls":3},0,1,0,0,0,0,0,0,0,0,{"$nulls":18},1]
[{"$nulls":3},0,0,0.8999999999999999,0.10000000000000009,0,0,0,0,0,0,{"$nulls":3},1,2,3,4,5,6,7,8,9,10,{"$nulls":5},1.1]
[{"$nulls":3},0,0,0,0.7999999999999998,0.20000000000000018,0,0,0,0,0,{"$nulls":3},{"$f":"(COUNTIF(D4:D8,\"<>0\")>0)","v":false},{"$f":"(COUNTIF(E4:E8,\"<>0\")>0)","v":true},true,true,true,false,true,true,false,false,{"$nulls":5},1.5]
[{"$nulls":3},0,0,0,0,0,0,0.7999999999999998,0.20000000000000018,0,0,{"$nulls":18},2.5]
[{"$nulls":3},0,0,0,0,0,0,0.019999999999999574,0.9800000000000004,0,0,{"$nulls":18},1.9]
[{"$nulls":3},0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,0,0,{"$nulls":3},"<-- Stiffness = 0.1.  All of these values can be reproduced on the worksheet \"10-Point Example\" with a stiffness constant of 0.1.",{"$nulls":14},0]
[{"$nulls":3},0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0,0,0,0,0.39528470752104755,-1.0540925533894605,0.658807845868413,0,{"$nulls":18},0]
[{"$nulls":3},0,0,0,0,0,0,0,0.8784104611578837,-1.756820922315766,0.8784104611578825,{"$nulls":18},0]
[]
[]
[]
[{"$nulls":36},"This shows that a single second derivative changes the values of nine cells after it is projected onto itself."]
[{"$nulls":3},"If we just look at the five fidelity equations, we get the matrix A (partial) below.",{"$nulls":12},"If we just look at the derivative equations, we get the matrix A (partial) below."]
[]
[]
[]
[{"$nulls":3},0,1,0,0,0,0,0,0,0,0,{"$nulls":3},0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,0,0,{"$nulls":10},0.1,-0.2,0.3,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0.8999999999999999,0.10000000000000009,0,0,0,0,0,0,{"$nulls":3},0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0.7999999999999998,0.20000000000000018,0,0,0,0,0,{"$nulls":3},0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0.7999999999999998,0.20000000000000018,0,0,{"$nulls":3},0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0.019999999999999574,0.9800000000000004,0,0,{"$nulls":3},0,0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":16},0,0,0,0,0,0.316227766016838,-0.632455532033676,0.316227766016838,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":16},0,0,0,0,0,0,0.39528470752104755,-1.0540925533894605,0.658807845868413,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":16},0,0,0,0,0,0,0,0.8784104611578837,-1.756820922315766,0.8784104611578825,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[]
[]
[{"$nulls":3},{"$f":"MMULT(TRANSPOSE(D25:M29),D25:M29)","v":0},0,0,0,0,0,0,0,0,0,{"$nulls":3},{"$f":"MMULT(TRANSPOSE(Q25:Z32),Q25:Z32)","v":0.10000000000000003},-0.20000000000000007,0.10000000000000003,0,0,0,0,0,0,0,{"$nulls":10},{"$f":"MMULT(TRANSPOSE(AK25:AT32),AK25:AT32)","v":0.010000000000000002},-0.020000000000000004,0.03,0,0,0,0,0,0,0]
[{"$nulls":3},0,1,0,0,0,0,0,0,0,0,{"$nulls":3},-0.20000000000000007,0.5000000000000002,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,0,{"$nulls":10},-0.020000000000000004,0.04000000000000001,-0.06,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0.8099999999999998,0.09000000000000007,0,0,0,0,0,0,{"$nulls":3},0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,{"$nulls":10},0.03,-0.06,0.09,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0.09000000000000007,0.6499999999999997,0.16000000000000011,0,0,0,0,0,{"$nulls":3},0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0.16000000000000011,0.04000000000000007,0,0,0,0,0,{"$nulls":3},0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0,0,0,0,{"$nulls":3},0,0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0.6403999999999996,0.1795999999999997,0,0,{"$nulls":3},0,0,0,0,0.10000000000000003,-0.40000000000000013,0.6562500000000003,-0.6166666666666671,0.260416666666667,0,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0.1795999999999997,1.0004000000000008,0,0,{"$nulls":3},0,0,0,0,0,0.10000000000000003,-0.6166666666666671,1.9827160493827187,-2.237654320987656,0.7716049382716048,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0,0,0,0,{"$nulls":3},0,0,0,0,0,0,0.260416666666667,-2.237654320987656,3.5204475308641974,-1.5432098765432085,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0,0,0,0,0,0,{"$nulls":3},0,0,0,0,0,0,0,0.7716049382716048,-1.5432098765432085,0.7716049382716037,{"$nulls":10},0,0,0,0,0,0,0,0,0,0]
[]
[]
[]
[{"$nulls":3},"We can take the two partial A matrices and add them together.  The VBA code does this.",{"$nulls":12},"The A'A matrix from the other worksheet has these values, which should match the sum to the left."]
[]
[{"$nulls":34},"(Check)"]
[{"$nulls":3},"Sum",{"$nulls":12},"Check",{"$nulls":14},"RHS",null,null,"Solution"]
[{"$nulls":3},{"$f":"D35:M44+Q35:Z44","v":0.10000000000000003},-0.20000000000000007,0.10000000000000003,0,0,0,0,0,0,0,{"$nulls":3},0.10000000000000003,-0.20000000000000007,0.10000000000000003,0,0,0,0,0,0,0,{"$nulls":5},0,null,null,{"$f":"MMULT(MINVERSE(D52:M61),AF52:AF61)","v":0.8634409505624512}]
[{"$nulls":3},-0.20000000000000007,1.5000000000000002,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,0,{"$nulls":3},-0.20000000000000007,1.5000000000000002,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,0,{"$nulls":5},{"$f":"SUMPRODUCT(E$25:E$29,$AF$4:$AF$8)","v":1},null,null,0.9788708474855378]
[{"$nulls":3},0.10000000000000003,-0.40000000000000013,1.4100000000000001,-0.31000000000000005,0.10000000000000003,0,0,0,0,0,{"$nulls":3},0.10000000000000003,-0.40000000000000013,1.4100000000000001,-0.31000000000000005,0.10000000000000003,0,0,0,0,0,{"$nulls":5},{"$f":"SUMPRODUCT(F$25:F$29,$AF$4:$AF$8)","v":0.99},null,null,1.0943007444086252]
[{"$nulls":3},0,0.10000000000000003,-0.31000000000000005,1.25,-0.24000000000000002,0.10000000000000003,0,0,0,0,{"$nulls":3},0,0.10000000000000003,-0.31000000000000005,1.25,-0.24000000000000002,0.10000000000000003,0,0,0,0,{"$nulls":5},{"$f":"SUMPRODUCT(G$25:G$29,$AF$4:$AF$8)","v":1.3099999999999998},null,null,1.4210221664763332]
[{"$nulls":3},0,0,0.10000000000000003,-0.24000000000000002,0.6400000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,{"$nulls":3},0,0,0.10000000000000003,-0.24000000000000002,0.6400000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,{"$nulls":5},{"$f":"SUMPRODUCT(H$25:H$29,$AF$4:$AF$8)","v":0.30000000000000027},null,null,1.9275706592947177]
[{"$nulls":3},0,0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,{"$nulls":3},0,0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,{"$nulls":5},0,null,null,2.376853961534353]
[{"$nulls":3},0,0,0,0,0.10000000000000003,-0.40000000000000013,1.29665,-0.43706666666666744,0.260416666666667,0,{"$nulls":3},0,0,0,0,0.10000000000000003,-0.40000000000000013,1.29665,-0.43706666666666744,0.260416666666667,0,{"$nulls":5},{"$f":"SUMPRODUCT(J$25:J$29,$AF$4:$AF$8)","v":2.037999999999999},null,null,2.4871160817857914]
[{"$nulls":3},0,0,0,0,0,0.10000000000000003,-0.43706666666666744,2.9831160493827196,-2.237654320987656,0.7716049382716048,{"$nulls":3},0,0,0,0,0,0.10000000000000003,-0.43706666666666744,2.98311604938272,-2.237654320987656,0.7716049382716048,{"$nulls":5},{"$f":"SUMPRODUCT(K$25:K$29,$AF$4:$AF$8)","v":2.362000000000001},null,null,1.9766010286395812]
[{"$nulls":3},0,0,0,0,0,0,0.260416666666667,-2.237654320987656,3.5204475308641974,-1.5432098765432085,{"$nulls":3},0,0,0,0,0,0,0.260416666666667,-2.237654320987656,3.5204475308641974,-1.5432098765432085,{"$nulls":5},0,null,null,1.6702919967518546]
[{"$nulls":3},0,0,0,0,0,0,0,0.7716049382716048,-1.5432098765432085,0.7716049382716037,{"$nulls":3},0,0,0,0,0,0,0,0.7716049382716048,-1.5432098765432085,0.7716049382716037,{"$nulls":5},0,null,null,1.3639829648641282]
[]
[]
[]
[]
[{"$nulls":3},0,0,0,0,0,null,"<-- This is just the above A (partial) matrix in compressed form.  This was used to check the VBA code."]
[{"$nulls":3},0,0,1,0,0]
[{"$nulls":3},0,0,0.81,0.0900000000000001,0]
[{"$nulls":3},0,0.09,0.65,0.16,0]
[{"$nulls":3},0,0.16,0.0400000000000001,0,0]
[{"$nulls":3},0,0,0,0,0]
[{"$nulls":3},0,0,0.6404,0.1796,0]
[{"$nulls":3},0,0.1796,1.0004,0,0]
[{"$nulls":3},0,0,0,0,0]
[{"$nulls":3},0,0,0,0,0]
[]
[]
[]
[{"$nulls":3},"Below is a series of calculations showing how A'b (the vector on the right side of the equation) is constructed from the input values and the fidelity equations."]
[]
[]
[]
[]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,1,null,0,1,0,0,0,0,0,0,0,0,{"$nulls":4},1]
[null,2,null,0,0,0.8999999999999999,0.10000000000000009,0,0,0,0,0,0,{"$nulls":4},1.1]
[null,3,null,0,0,0,0.7999999999999998,0.20000000000000018,0,0,0,0,0,{"$nulls":4},1.5]
[null,4,null,0,0,0,0,0,0,0.7999999999999998,0.20000000000000018,0,0,{"$nulls":4},2.5]
[null,5,null,0,0,0,0,0,0,0.019999999999999574,0.9800000000000004,0,0,{"$nulls":4},1.9]
{"$rows_empty":9}
[{"$nulls":3},"Start with a blank RHS and begin adding elements."]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,1,null,null,1,{"$nulls":12},0]
[{"$nulls":17},{"$f":"E102*R86","v":1}]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[]
[]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,2,{"$nulls":3},0.9,0.1,{"$nulls":10},{"$f":"R102","v":0}]
[{"$nulls":17},{"$f":"R103","v":1}]
[{"$nulls":17},{"$f":"R104+F116*R87","v":0.9900000000000001}]
[{"$nulls":17},{"$f":"R104+G116*R87","v":0.11000000000000001}]
[{"$nulls":17},{"$f":"R106","v":0}]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[]
[]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,3,{"$nulls":4},0.8,0.2,{"$nulls":9},{"$f":"R116","v":0}]
[{"$nulls":17},{"$f":"R117","v":1}]
[{"$nulls":17},0.9900000000000001]
[{"$nulls":17},{"$f":"R119+G130*R88","v":1.3100000000000003}]
[{"$nulls":17},{"$f":"R120+H130*R88","v":0.30000000000000004}]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[{"$nulls":17},0]
[]
[]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,4,{"$nulls":7},0.8,0.2,{"$nulls":6},{"$f":"R130","v":0}]
[{"$nulls":17},{"$f":"R131","v":1}]
[{"$nulls":17},0.9900000000000001]
[{"$nulls":17},1.3100000000000003]
[{"$nulls":17},0.30000000000000004]
[{"$nulls":17},0]
[{"$nulls":17},{"$f":"R136+J144*R89","v":2}]
[{"$nulls":17},{"$f":"R137+K144*R89","v":0.5}]
[{"$nulls":17},0]
[{"$nulls":17},0]
[]
[]
[]
[]
[{"$nulls":3},1,2,3,4,5,6,7,8,9,10]
[null,5,{"$nulls":7},0.02,0.98,{"$nulls":6},{"$f":"R144","v":0},"<-- This is the right hand side, the vector b."]
[{"$nulls":17},{"$f":"R145","v":1}]
[{"$nulls":17},0.9900000000000001]
[{"$nulls":17},1.3100000000000003]
[{"$nulls":17},0.30000000000000004]
[{"$nulls":17},0]
[{"$nulls":17},{"$f":"R150+J159*R90","v":2.038}]
[{"$nulls":17},{"$f":"R151+K159*R90","v":2.362}]
[{"$nulls":17},0]
[{"$nulls":17},0]

{"sheet":"Backsolving","index":4,"n_rows":139,"n_cols":29,"rows_kept":139,"merged":[[3,3,4,12],[3,24,4,28],[137,12,137,13],[138,12,138,13]]}
["This worksheet demonstrates most of the process of compressing and solving a pentadiagonal matrix as is done in the VBA code."]
[]
[]
[{"$nulls":3},"Uncompressed Matrix",{"$nulls":12},"RHS",{"$nulls":3},"Solution",{"$nulls":3},"Compressed Matrix"]
[]
[{"$nulls":3},0.10000000000000003,-0.20000000000000007,0.10000000000000003,0,0,0,0,0,0,0,{"$nulls":3},0,{"$nulls":3},{"$f":"MMULT(MINVERSE(D6:M15),Q6:Q15)","v":0.8634409505624512},{"$nulls":3},{"$f":"B6","v":0},{"$f":"C6","v":0},{"$f":"D6","v":0.10000000000000003},{"$f":"E6","v":-0.20000000000000007},{"$f":"F6","v":0.10000000000000003}]
[{"$nulls":3},-0.20000000000000007,1.5000000000000002,-0.40000000000000013,0.10000000000000003,0,0,0,0,0,0,{"$nulls":3},1,{"$nulls":3},0.9788708474855378,{"$nulls":3},{"$f":"C7","v":0},{"$f":"D7","v":-0.20000000000000007},{"$f":"E7","v":1.5000000000000002},{"$f":"F7","v":-0.40000000000000013},{"$f":"G7","v":0.10000000000000003}]
[{"$nulls":3},0.10000000000000003,-0.40000000000000013,1.4100000000000001,-0.31000000000000005,0.10000000000000003,0,0,0,0,0,{"$nulls":3},0.99,{"$nulls":3},1.0943007444086252,{"$nulls":3},{"$f":"D8","v":0.10000000000000003},{"$f":"E8","v":-0.40000000000000013},{"$f":"F8","v":1.4100000000000001},{"$f":"G8","v":-0.31000000000000005},{"$f":"H8","v":0.10000000000000003}]
[{"$nulls":3},0,0.10000000000000003,-0.31000000000000005,1.25,-0.24000000000000002,0.10000000000000003,0,0,0,0,{"$nulls":3},1.3099999999999998,{"$nulls":3},1.4210221664763332,{"$nulls":3},{"$f":"E9","v":0.10000000000000003},{"$f":"F9","v":-0.31000000000000005},{"$f":"G9","v":1.25},{"$f":"H9","v":-0.24000000000000002},{"$f":"I9","v":0.10000000000000003}]
[{"$nulls":3},0,0,0.10000000000000003,-0.24000000000000002,0.6400000000000003,-0.40000000000000013,0.10000000000000003,0,0,0,{"$nulls":3},0.30000000000000027,{"$nulls":3},1.9275706592947177,{"$nulls":3},{"$f":"F10","v":0.10000000000000003},{"$f":"G10","v":-0.24000000000000002},{"$f":"H10","v":0.6400000000000003},{"$f":"I10","v":-0.40000000000000013},{"$f":"J10","v":0.10000000000000003}]
[{"$nulls":3},0,0,0,0.10000000000000003,-0.40000000000000013,0.6000000000000003,-0.40000000000000013,0.10000000000000003,0,0,{"$nulls":3},0,{"$nulls":3},2.376853961534353,{"$nulls":3},{"$f":"G11","v":0.10000000000000003},{"$f":"H11","v":-0.40000000000000013},{"$f":"I11","v":0.6000000000000003},{"$f":"J11","v":-0.40000000000000013},{"$f":"K11","v":0.10000000000000003}]
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[{"$nulls":3},0,0,0,0,0,0.3456250937091149,-0.3355566783533487,0.10000000000000003,0,0,{"$nulls":3},0.18459206496891425,0.184592064968914,null,null,2.3768539615343527,{"$nulls":3},{"$f":"G128","v":0},{"$f":"H128","v":0},{"$f":"I128","v":0.3456250937091149},{"$f":"J128","v":-0.3355566783533487},{"$f":"K128","v":0.10000000000000003}]
[{"$nulls":3},0,0,0,0,0,0,0.9539767624449127,-0.33997976977724187,0.260416666666667,0,{"$nulls":3},2.135618459022471,2.13561845902247,null,null,2.4871160817857914,{"$nulls":3},{"$f":"H129","v":0},{"$f":"I129","v":0},{"$f":"J129","v":0.9539767624449127},{"$f":"K129","v":-0.33997976977724187},{"$f":"L129","v":0.260416666666667}]
[{"$nulls":3},0,0,0,0,0,0,0,2.8330204289665244,-2.1448466113401135,0.7716049382716048,{"$nulls":3},3.0696869562779576,3.06968695627796,null,null,1.9766010286395814,{"$nulls":3},{"$f":"I130","v":0},{"$f":"J130","v":0},{"$f":"K130","v":2.8330204289665244},{"$f":"L130","v":-2.1448466113401135},{"$f":"M130","v":0.7716049382716048}]
[{"$nulls":3},0,0,0,0,0,0,0,0,1.82552069675722,-0.9590368080317055,{"$nulls":3},1.741042740865539,1.74104274086554,null,null,1.6702919967518557,{"$nulls":3},{"$f":"J131","v":0},{"$f":"K131","v":0},{"$f":"L131","v":1.82552069675722},{"$f":"M131","v":-0.9590368080317055},{"$f":"N131","v":0}]
["Scale1:",{"$f":"-L119/L118","v":0.5253497315781186},null,0,0,0,0,0,0,0,0,{"$f":"L119+$B$132*L118","v":0},{"$f":"M119+$B$132*M118","v":0.05761993591590897},{"$nulls":3},{"$f":"Q119+$B$132*Q118","v":0.07859261102586268},{"v":0.0785926110258631,"d":"7.86E-02"},null,null,1.36398296486413,{"$nulls":3},{"$f":"K132","v":0},{"$f":"L132","v":0},{"$f":"M132","v":0.05761993591590897},{"$f":"N132","v":0},{"$f":"O132","v":0}]
[]
[]
[]
[{"$nulls":20},{"$f":"Q132/M132","v":1.36398296486413},null,"<-- Just checking."]
[]
[{"$nulls":12},{"$f":"(Q131-U132*M131)/L131","v":{"v":1.670291996751856,"d":"1.67029"}},null,null,"<-- Just checking."]
[{"$nulls":12},{"$f":"Q132/M132","v":1.36398296486413},null,null,"<-- Just checking."]

{"sheet":"Second-Derivative Equation","index":5,"n_rows":22,"n_cols":5,"rows_kept":22,"merged":[[21,2,21,3]]}
["This worksheet shows how the derivative coefficients are calculated.  This topic is explained in the Math for Mere Mortals article \"A Numerical Second Derivative from Three Points.\""]
[]
[]
[]
[null,"x","y"]
[null,0.32,{"$f":"B6^2","v":0.1024}]
[null,0.36,{"$f":"B7^2","v":0.1296}]
[null,0.4,{"$f":"B8^2","v":0.16000000000000003}]
[]
[]
[null,"x3-x2:",{"$f":"B8-B7","v":0.040000000000000036}]
[null,"x3-x1:",{"$f":"B8-B6","v":0.08000000000000002}]
[null,"x2-x1:",{"$f":"B7-B6","v":0.03999999999999998}]
[]
[]
[{"$nulls":4},{"$f":"2/(C12*C13)","v":625.0000000000002}]
[{"$nulls":4},{"$f":"-2/(C11*C13)","v":-1249.9999999999995}]
[{"$nulls":4},{"$f":"2/(C12*C11)","v":624.9999999999993}]
[]
[]
[]
[null,null,"Second Derivative:",null,{"$f":"2*(C6/(C12*C13)-C7/(C11*C13)+C8/(C12*C11))","v":2.000000000000014}]