數(shù)值線性代數(shù)上機作業(yè)1

大連理工大學數(shù)學科學學院大連理工大學數(shù)學科學學院數(shù)值線性代數(shù)上機作業(yè)小組成員:尹樂平(200811114)李春惠(200811018) 班級:0803經(jīng)典格拉程序姆施密特%clgs.m文件名正交化方法function[Q,R]=clgs(A)forj=1:nv=A(:,j);fori=1:j-1R(i,j)=Q(:,i)'*A(:,j);endR(j,j)=norm(v);Q(:,j)=v/R(j,j);QRendQR穩(wěn)定的。2,修正格拉程序姆施密特%mgs.m文件名正交化方法function[Q,R]=mgs(A)forj=1:nv=A(:,j);fori=1:j-1R(i,j)=Q(:,i)'*v;endR(j,j)=norm(v);Q(:,j)=v/R(j,j);QRendvA(:,j)，m-(j-1)的正交投影轉(zhuǎn)化為j-1m-1的正交投影算子，提高精度。3,對比魔方

魔方矩陣未改進算法結果：內(nèi)置QR內(nèi)置QR分A=解351626192433272123253192222720828331710153053412141643629131811Q=-0.1702-0.2070-0.1702-0.20700.4998-0.20620.28820.5740-0.45000.21060.64870.2096-0.0011-0.4460-0.4537-0.2062-0.40600.47330.37630.5034-0.33290.2620-0.06950.6287-0.20960.5220-0.41910.64240.1373-0.4501-0.3329-0.681116.469330.045939.096938.032138.671054.219634.879723.166925.260923.2963032.4907-8.9182-11.2895-7.9245007.6283-3.91147.43390003.41976.839300000.00000.05320.55020.14200.53240.0710R=56.347100000魔方矩陣改進算法結果（加重點部分為不相同結果：A=magic(6)[Q,R]=mgs(A)A=351626192433272123253192222720828331710153053412141643629131811Q=-0.1702-0.20700.4998-0.2062-0.15130.5740-0.45000.21060.6487-0.6431-0.0011-0.4460-0.4537-0.2062-0.13240.47330.37630.5034-0.3329-0.4729-0.06950.6287-0.20960.5220-0.01890.64240.1373-0.4501-0.3329-0.5675R=56.347116.469330.045939.096938.032138.6710054.219634.879723.166925.260923.29630032.4907-8.9182-11.2895-7.92450007.6283-3.91147.433900003.41976.8393000000.0000魔方矩陣用matlab內(nèi)置QR分解：A=magic(6)[Q,R]=qr(A)A=351626192433272123253192222720828331710153053412141643629131811Q=-0.62110.1702-0.2070-0.49980.2062-0.5000-0.0532-0.5740-0.4500-0.2106-0.64870.0000-0.55020.0011-0.44600.45370.20620.5000-0.1420-0.47330.3763-0.50340.33290.5000-0.53240.06950.62870.2096-0.52200.0000-0.0710-0.64240.13730.45010.3329-0.5000R=-56.3471-16.4693-30.0459-39.0969-38.0321-38.67100-54.2196-34.8797-23.1669-25.2609-23.29630032.4907-8.9182-11.2895-7.9245000-7.62833.9114-7.43390000-3.4197-6.8393000000.0000希爾伯特矩陣未改進算法結果：A=hilb(6)[Q,R]=clgs(A)A=1.00000.50000.33330.25000.20000.16670.50000.33330.25000.20000.16670.14290.33330.25000.20000.16670.14290.12500.25000.20000.16670.14290.12500.11110.20000.16670.14290.12500.11110.10000.16670.14290.12500.11110.10000.0909大連理工大學數(shù)學科學學院大連理工大學數(shù)學科學學院Q=0.8189-0.53970.1893-0.04820.0090-0.00110.40940.3320-0.70240.4489-0.16170.03320.27300.4219-0.1529-0.57230.5854-0.23210.20470.40670.2015-0.3866-0.46870.61890.16380.37350.39630.0915-0.4285-0.69610.13650.33990.49980.55740.47730.2784希爾伯特矩陣改進算法結果（加重點部分為不相同結果：A=hilb(6)[Q,R]=mgs(A)A=1.00000.50000.33330.25000.20000.16670.50000.33330.25000.20000.16670.14290.33330.25000.20000.16670.14290.12500.25000.20000.16670.14290.12500.11110.20000.16670.14290.12500.11110.10000.16670.14290.12500.11110.10000.0909R=1.22120.70190.50450.39700.32840.280600.13850.15110.14440.13400.1237000.00960.01520.01810.01950000.00050.00100.001400000.00000.0000000000.0000Q=0.8189-0.53970.1893-0.04820.0090-0.00110.40940.3320-0.70240.4489-0.16170.03310.27300.4219-0.1529-0.57230.5854-0.23200.20470.40670.2015-0.3866-0.46870.61880.16380.37350.39630.0915-0.4285-0.69610.13650.33990.49980.55740.47730.2785R=1.22120.70190.50450.39700.32840.280600.13850.15110.14440.13400.1237000.00960.01520.01810.01950000.00050.00100.001400000.00000.0000000000.0000希爾伯特矩陣用matlab內(nèi)置QR分解：A=hilb(6)[Q,R]=qr(A)A=1.00000.50000.33330.25000.20000.16670.50000.33330.25000.20000.16670.14290.33330.25000.20000.16670.14290.12500.25000.20000.16670.14290.12500.11110.20000.16670.14290.12500.11110.10000.16670.14290.12500.11110.10000.0909Q=-0.81890.5397-0.1893-0.04820.0090-0.0011-0.4094-0.33200.70240.4489-0.16170.0331-0.2730-0.42190.1529-0.57230.5854-0.2320-0.2047-0.4067-0.2015-0.3866-0.46870.6188-0.1638-0.3735-0.39630.0915-0.4285-0.6961-0.1365-0.3399-0.49980.55740.47730.2785R=-1.2212-0.7019-0.5045-0.3970-0.3284-0.28060-0.1385-0.1511-0.1444-0.1340-0.123700-0.0096-0.0152-0.0181-0.01950000.00050.00100.001400000.00000.0000000000由數(shù)字實驗可以看出系統(tǒng)內(nèi)置QR分解和標準格拉姆施密特算法以及修正6Q的正交性造成。4,通過實驗測試經(jīng)典和修正格拉姆-施密特算發(fā)之間的穩(wěn)定性的區(qū)別

經(jīng)典算法：%clgs.m文件名function[QC,RC]=clgs(A)forj=1:nv=A(:,j);fori=1:j-1RC(i,j)=QC(:,i)'*A(:,j);v=v-RC(i,j)*QC(:,i);endRC(j,j)=norm(v);QCRCend改進算法：%mgs.m文件名function[QM,RM]=clgs(A)forj=1:nv=A(:,j);fori=1:j-1RM(i,j)=QM(:,i)'*A(:,j);v=v-RM(i,j)*QM(:,i);endRM(j,j)=norm(v);QM(:,j)=v/RM(j,j);QMRMend構造A畫圖：formatlong[V,X]=qr(randn(80));S=diag(2.^(-1:-1:-80));A=U*S*V;[QC,RC]=clgs(A);[QM,RM]=m