A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations

<table>The zero and nonzero terms in the equation of each orthogonal column, where <svg height="14.925" id="M47" style="vertical-align:-2.05875pt" version="1.1" viewbox="0 0 21.2125 14.925" width="21.2125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.925)">
<g transform="translate(72,-60.06)">
<text transform="matrix(1,0,0,-1,-71.95,62.16)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑞</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.37,66.65)">
<tspan style="font-size: 7.85px; " x="0" y="0">(</tspan>
<tspan style="font-size: 7.85px; " x="2.6127181" y="0">𝑚</tspan>
<tspan style="font-size: 7.85px; " x="8.1833782" y="0">)</tspan>
</text>
</g>
</g>
</svg> is the <svg height="6.4124999" id="M48" style="vertical-align:-0.09pt" version="1.1" viewbox="0 0 10.075 6.4124999" width="10.075" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
<g transform="translate(72,-66.87)">
<text transform="matrix(1,0,0,-1,-71.95,67.01)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑚</tspan>
</text>
</g>
</g>
</svg>th column of orthogonal matrix and <svg height="12.5" id="M49" style="vertical-align:-0.1125pt" version="1.1" viewbox="0 0 21.275 12.5" width="21.275" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.5)">
<g transform="translate(72,-62)">
<text transform="matrix(1,0,0,-1,-71.95,62.16)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑎</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.32,66.65)">
<tspan style="font-size: 7.85px; " x="0" y="0">(</tspan>
<tspan style="font-size: 7.85px; " x="2.6127181" y="0">𝑚</tspan>
<tspan style="font-size: 7.85px; " x="8.1833782" y="0">)</tspan>
</text>
</g>
</g>
</svg> is <svg height="6.4124999" id="M50" style="vertical-align:-0.09pt" version="1.1" viewbox="0 0 10.075 6.4124999" width="10.075" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
<g transform="translate(72,-66.87)">
<text transform="matrix(1,0,0,-1,-71.95,67.01)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑚</tspan>
</text>
</g>
</g>
</svg>th column of matrix <svg height="9.4750004" id="M51" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 10.175 9.4750004" width="10.175" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.475)">
<g transform="translate(72,-64.42)">
<text transform="matrix(1,0,0,-1,-71.95,64.47)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝐴</tspan>
</text>
</g>
</g>
</svg>. Also, the zero terms are crossed out.</table>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations 